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Properties

Label 1-317-317.168-r0-0-0
Degree $1$
Conductor $317$
Sign $0.917 - 0.397i$
Analytic cond. $1.47214$
Root an. cond. $1.47214$
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  + (0.961 − 0.274i)2-s + (0.980 − 0.197i)3-s + (0.848 − 0.528i)4-s + (0.0992 + 0.995i)5-s + (0.888 − 0.459i)6-s + (0.216 − 0.976i)7-s + (0.671 − 0.741i)8-s + (0.921 − 0.387i)9-s + (0.368 + 0.929i)10-s + (0.0596 + 0.998i)11-s + (0.727 − 0.685i)12-s + (−0.804 + 0.594i)13-s + (−0.0596 − 0.998i)14-s + (0.293 + 0.955i)15-s + (0.441 − 0.897i)16-s + (−0.578 + 0.815i)17-s + ⋯
L(s)  = 1  + (0.961 − 0.274i)2-s + (0.980 − 0.197i)3-s + (0.848 − 0.528i)4-s + (0.0992 + 0.995i)5-s + (0.888 − 0.459i)6-s + (0.216 − 0.976i)7-s + (0.671 − 0.741i)8-s + (0.921 − 0.387i)9-s + (0.368 + 0.929i)10-s + (0.0596 + 0.998i)11-s + (0.727 − 0.685i)12-s + (−0.804 + 0.594i)13-s + (−0.0596 − 0.998i)14-s + (0.293 + 0.955i)15-s + (0.441 − 0.897i)16-s + (−0.578 + 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.917 - 0.397i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.035719872 - 0.6291134194i\)
\(L(\frac12)\) \(\approx\) \(3.035719872 - 0.6291134194i\)
\(L(1)\) \(\approx\) \(2.360348459 - 0.3747041518i\)
\(L(1)\) \(\approx\) \(2.360348459 - 0.3747041518i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.961 - 0.274i)T \)
3 \( 1 + (0.980 - 0.197i)T \)
5 \( 1 + (0.0992 + 0.995i)T \)
7 \( 1 + (0.216 - 0.976i)T \)
11 \( 1 + (0.0596 + 0.998i)T \)
13 \( 1 + (-0.804 + 0.594i)T \)
17 \( 1 + (-0.578 + 0.815i)T \)
19 \( 1 + (-0.888 - 0.459i)T \)
23 \( 1 + (-0.999 + 0.0397i)T \)
29 \( 1 + (-0.216 - 0.976i)T \)
31 \( 1 + (-0.827 - 0.561i)T \)
37 \( 1 + (0.754 - 0.656i)T \)
41 \( 1 + (-0.700 + 0.714i)T \)
43 \( 1 + (0.971 + 0.236i)T \)
47 \( 1 + (0.827 + 0.561i)T \)
53 \( 1 + (0.888 + 0.459i)T \)
59 \( 1 + (0.971 - 0.236i)T \)
61 \( 1 + (-0.610 - 0.792i)T \)
67 \( 1 + (-0.545 - 0.838i)T \)
71 \( 1 + (-0.293 + 0.955i)T \)
73 \( 1 + (-0.936 - 0.350i)T \)
79 \( 1 + (0.848 + 0.528i)T \)
83 \( 1 + (-0.255 + 0.966i)T \)
89 \( 1 + (-0.961 + 0.274i)T \)
97 \( 1 + (-0.754 + 0.656i)T \)
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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

−25.151284261783679806022069855776, −24.32392521063759697518986218480, −23.89260684260688040890994264805, −22.17158303752791785109998313278, −21.759900275702394112053689586760, −20.83880181524883795627912471898, −20.15106721166176556697244576554, −19.30788577022343747223303331334, −18.015221383765981264363115219453, −16.63445037280379853181773666897, −15.98142225156656819417811840737, −15.11408638622209337355753214210, −14.29913659944116303448448967315, −13.38082339351238233674180224317, −12.59084420824184082687616808794, −11.78585862979781287595894340492, −10.380652805365298423712749585620, −8.93837710412930515986471242873, −8.43218873870107333711827285927, −7.36147276516268418148355209051, −5.85033953569303974775732892610, −5.04605870969743945351137719959, −4.01754940515260799960063271828, −2.811471666971608101484653211185, −1.889280052072611153569447661933, 1.89724962110354198178294742401, 2.46313436015684318502159876828, 3.97514917818749788568061577750, 4.328233059578415829179108762099, 6.25611044414781843614625047515, 7.10999950264372893249621279505, 7.7401558582941137022327284144, 9.56884682660524850639800879880, 10.30511466180676045523138413999, 11.25839174738575405765962942311, 12.50591327930430481236916093509, 13.36791339568065859761578001747, 14.21389540012708165759353282989, 14.81834024847670067938292980607, 15.451015178825327315569919108474, 16.96019050218802967002264457651, 18.07566256089155268295208201113, 19.33895012875934983495541596506, 19.74578968069512091002612655520, 20.65462179368380469340468245190, 21.60080545620018472952851951840, 22.32021547048969194690167287990, 23.47619256776649680892139435320, 23.98416721403277262751088534159, 25.08121300645822245044094289066

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