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Properties

Label 1-59-59.38-r1-0-0
Degree $1$
Conductor $59$
Sign $0.440 - 0.897i$
Analytic cond. $6.34043$
Root an. cond. $6.34043$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.440 - 0.897i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ 0.440 - 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.035876755 - 0.6453188005i\)
\(L(\frac12)\) \(\approx\) \(1.035876755 - 0.6453188005i\)
\(L(1)\) \(\approx\) \(0.8149500615 - 0.3896017542i\)
\(L(1)\) \(\approx\) \(0.8149500615 - 0.3896017542i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.468 - 0.883i)T \)
3 \( 1 + (-0.725 - 0.687i)T \)
5 \( 1 + (0.976 + 0.214i)T \)
7 \( 1 + (0.796 + 0.605i)T \)
11 \( 1 + (0.370 - 0.928i)T \)
13 \( 1 + (-0.0541 + 0.998i)T \)
17 \( 1 + (0.796 - 0.605i)T \)
19 \( 1 + (-0.947 - 0.319i)T \)
23 \( 1 + (0.856 + 0.515i)T \)
29 \( 1 + (0.468 - 0.883i)T \)
31 \( 1 + (0.947 - 0.319i)T \)
37 \( 1 + (0.994 - 0.108i)T \)
41 \( 1 + (-0.856 + 0.515i)T \)
43 \( 1 + (0.370 + 0.928i)T \)
47 \( 1 + (-0.976 + 0.214i)T \)
53 \( 1 + (0.267 - 0.963i)T \)
61 \( 1 + (-0.468 - 0.883i)T \)
67 \( 1 + (0.994 + 0.108i)T \)
71 \( 1 + (0.976 - 0.214i)T \)
73 \( 1 + (0.161 - 0.986i)T \)
79 \( 1 + (-0.725 + 0.687i)T \)
83 \( 1 + (-0.647 - 0.762i)T \)
89 \( 1 + (-0.468 + 0.883i)T \)
97 \( 1 + (0.161 + 0.986i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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