The text below is a report automatically generated by Pari scripts. For each algebraic number field studied, it lists the diagnostic values. A short explanation of the diagnostic values: A lemma in the paper (please see the web page) states that if the value next to "Simple zero:" is less than 1, then zetak(s) has a simple zero within the distance of "Distance:" from rho. No other zetachi(s, chi) function has a zero within the distance of "Competition:" from rho. Theoretically it would be enough to have "Simple zero:" < 1 and "Distance:" < "Competition:", but due to the limited precision of the computations it is much better to have those criteria satisfied with a large safety margin. Defining polynomial: x^2 - 26 Im rho: 1.370583964578000000000000000 Func. eq. error: 3.000000000000000000000000000 E-28 (Should be close to 0.) Simple zero: 0.0000000006156083969573566383000000000 (Should be <1. Close to 0 if we are close to a zero of zeta_K(s).) Distance: 3.740062320487336835000000000 E-13 (The upper bound for the distance to a simple zero of zetak.) Competition: 0.004185676224708701314612234550 (Lower bound for the distance to the nearest zero of other Hecke zeta functions.) Contrad. check: 0.0009113087976802918688000000000 (Must be <1. The ratio between the lower and upper estimates for zetak zero distance.) |Zeta_K(rho)|: 3.237952426677979259164024777 E-13 (The absolute value of zeta_K(s) at rho.) Min |zetachi|: 1.255543300280701484613818486 (The smallest of zetachi values at rho. Should be non-zero.) Defining polynomial: x^2 + 65 Im rho: 1.053258936999224463261538680 Func. eq. error: 2.000000000000000000000000000 E-27 (Should be close to 0.) Simple zero: 1.343867798000000000000000000 E-23 (Should be <1. Close to 0 if we are close to a zero of zeta_K(s).) Distance: 5.645024846000000000000000000 E-27 (The upper bound for the distance to a simple zero of zetak.) Competition: 0.00007416807953754938274939876035 (Lower bound for the distance to the nearest zero of other Hecke zeta functions.) Contrad. check: 0.0006300870730000000000000000000 (Must be <1. The ratio between the lower and upper estimates for zetak zero distance.) |Zeta_K(rho)|: 8.538957618821757989864677754 E-27 (The absolute value of zeta_K(s) at rho.) Min |zetachi|: 0.04824189196203555221163115124 (The smallest of zetachi values at rho. Should be non-zero.) Defining polynomial: x^2 + 9982 Im rho: 0.2765970174871810881810812134 Func. eq. error: 1.000000000000000000000000000 E-26 (Should be close to 0.) Simple zero: 1.200131002000000000000000000 E-21 (Should be <1. Close to 0 if we are close to a zero of zeta_K(s).) Distance: 3.862449811000000000000000000 E-27 (The upper bound for the distance to a simple zero of zetak.) Competition: 0.000009706584611347068851341608490 (Lower bound for the distance to the nearest zero of other Hecke zeta functions.) Contrad. check: 0.000004827535250000000000000000000 (Must be <1. The ratio between the lower and upper estimates for zetak zero distance.) |Zeta_K(rho)|: 7.720169492075141479289840555 E-27 (The absolute value of zeta_K(s) at rho.) Min |zetachi|: 0.6511704204977774461499043511 (The smallest of zetachi values at rho. Should be non-zero.) Defining polynomial: x^3 - x^2 + 7*x + 8 Im rho: 1.350474195561608855571544170 Func. eq. error: 2.000000000000000000000000000 E-26 (Should be close to 0.) Simple zero: 1.034522229000000000000000000 E-20 (Should be <1. Close to 0 if we are close to a zero of zeta_K(s).) Distance: 1.438182659000000000000000000 E-25 (The upper bound for the distance to a simple zero of zetak.) Competition: 0.00004987980214547820274573741575 (Lower bound for the distance to the nearest zero of other Hecke zeta functions.) Contrad. check: 0.00002085285292000000000000000000 (Must be <1. The ratio between the lower and upper estimates for zetak zero distance.) |Zeta_K(rho)|: 1.615397005051238130557068432 E-25 (The absolute value of zeta_K(s) at rho.) Min |zetachi|: 0.8410140814137826909178257480 (The smallest of zetachi values at rho. Should be non-zero.) Defining polynomial: x^3 - x^2 - 97*x - 384 Im rho: 0.4306392812448931468310766248 Func. eq. error: 7.000000000000000000000000000 E-24 (Should be close to 0.) Simple zero: 2.760034804000000000000000000 E-18 (Should be <1. Close to 0 if we are close to a zero of zeta_K(s).) Distance: 7.744454184000000000000000000 E-25 (The upper bound for the distance to a simple zero of zetak.) Competition: 8.847246324658260626531366979 E-11 (Lower bound for the distance to the nearest zero of other Hecke zeta functions.) Contrad. check: 0.0000004208889416000000000000000000 (Must be <1. The ratio between the lower and upper estimates for zetak zero distance.) |Zeta_K(rho)|: 3.871798561905465709632528045 E-24 (The absolute value of zeta_K(s) at rho.) Min |zetachi|: 0.0001884119865728655394547874703 (The smallest of zetachi values at rho. Should be non-zero.)