Properties

Degree 1
Conductor 59
Sign $0.440 - 0.897i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.468 − 0.883i)2-s + (−0.725 − 0.687i)3-s + (−0.561 + 0.827i)4-s + (0.976 + 0.214i)5-s + (−0.267 + 0.963i)6-s + (0.796 + 0.605i)7-s + (0.994 + 0.108i)8-s + (0.0541 + 0.998i)9-s + (−0.267 − 0.963i)10-s + (0.370 − 0.928i)11-s + (0.976 − 0.214i)12-s + (−0.0541 + 0.998i)13-s + (0.161 − 0.986i)14-s + (−0.561 − 0.827i)15-s + (−0.370 − 0.928i)16-s + (0.796 − 0.605i)17-s + ⋯
L(s,χ)  = 1  + (−0.468 − 0.883i)2-s + (−0.725 − 0.687i)3-s + (−0.561 + 0.827i)4-s + (0.976 + 0.214i)5-s + (−0.267 + 0.963i)6-s + (0.796 + 0.605i)7-s + (0.994 + 0.108i)8-s + (0.0541 + 0.998i)9-s + (−0.267 − 0.963i)10-s + (0.370 − 0.928i)11-s + (0.976 − 0.214i)12-s + (−0.0541 + 0.998i)13-s + (0.161 − 0.986i)14-s + (−0.561 − 0.827i)15-s + (−0.370 − 0.928i)16-s + (0.796 − 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.440 - 0.897i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.440 - 0.897i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(59\)
\( \varepsilon \)  =  $0.440 - 0.897i$
motivic weight  =  \(0\)
character  :  $\chi_{59} (38, \cdot )$
Sato-Tate  :  $\mu(58)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 59,\ (1:\ ),\ 0.440 - 0.897i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.035876755 - 0.6453188005i$
$L(\frac12,\chi)$  $\approx$  $1.035876755 - 0.6453188005i$
$L(\chi,1)$  $\approx$  0.8149500615 - 0.3896017542i
$L(1,\chi)$  $\approx$  0.8149500615 - 0.3896017542i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−32.902808931167444421510725113182, −32.261529464902476249754759000278, −30.265652474417327583269178757225, −28.96988415117315021728699034516, −27.859909764071330012444836609809, −27.23044540490106481344626464693, −25.87953614680873273377302235765, −24.94766516647489143957100819556, −23.59238921643278206003264182706, −22.7238416448171816263274501631, −21.351929727255432744001077520747, −20.21371810801082768190459783460, −18.23185570451577405165627459377, −17.24977324738228291851268132378, −16.88681322478450765683853092754, −15.196902373712169434301375929935, −14.36298080138677002629428767330, −12.68049056306093827006589780433, −10.60213800463835965814735457747, −9.99705791865274959671491068284, −8.51505112733891380961797223861, −6.80938044269237815963336614541, −5.51243556935461679861625470927, −4.5015756217314782088879641878, −1.17845542193310647319223069255, 1.219399042906551735569398861005, 2.492207784236014755982806796713, 4.93066418432354089006232524253, 6.437076963500205652713186935726, 8.172011378337587797955037164027, 9.530663944224204880854873502, 11.07463430212767420757980980946, 11.76801393006723099227537412997, 13.195418562956429327492575715646, 14.20238817878740094142727180844, 16.66857088031049228624116350041, 17.46220782387067725720639322982, 18.5316610901363758469431726954, 19.21004004604939741939983623543, 21.27190978613041482388043557790, 21.58878584568699616537232494967, 22.990682487585118978060158555017, 24.48065904265683850439037315202, 25.483785748212500447657877838878, 26.98942331567229461939218766019, 28.0856361140150091005258074304, 29.021536927664676088419112057895, 29.81756220844021004524889782101, 30.66279039162717092956225736542, 31.98629159060747896760565103048

Graph of the $Z$-function along the critical line