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Properties

Label 1-317-317.11-r0-0-0
Degree $1$
Conductor $317$
Sign $-0.631 - 0.775i$
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  + (0.848 − 0.528i)2-s + (0.921 − 0.387i)3-s + (0.441 − 0.897i)4-s + (−0.980 + 0.197i)5-s + (0.578 − 0.815i)6-s + (−0.905 − 0.423i)7-s + (−0.0992 − 0.995i)8-s + (0.700 − 0.714i)9-s + (−0.727 + 0.685i)10-s + (−0.992 + 0.119i)11-s + (0.0596 − 0.998i)12-s + (0.293 − 0.955i)13-s + (−0.992 + 0.119i)14-s + (−0.827 + 0.561i)15-s + (−0.610 − 0.792i)16-s + (−0.331 − 0.943i)17-s + ⋯
L(s)  = 1  + (0.848 − 0.528i)2-s + (0.921 − 0.387i)3-s + (0.441 − 0.897i)4-s + (−0.980 + 0.197i)5-s + (0.578 − 0.815i)6-s + (−0.905 − 0.423i)7-s + (−0.0992 − 0.995i)8-s + (0.700 − 0.714i)9-s + (−0.727 + 0.685i)10-s + (−0.992 + 0.119i)11-s + (0.0596 − 0.998i)12-s + (0.293 − 0.955i)13-s + (−0.992 + 0.119i)14-s + (−0.827 + 0.561i)15-s + (−0.610 − 0.792i)16-s + (−0.331 − 0.943i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $-0.631 - 0.775i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ -0.631 - 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8455036217 - 1.780096723i\)
\(L(\frac12)\) \(\approx\) \(0.8455036217 - 1.780096723i\)
\(L(1)\) \(\approx\) \(1.320968693 - 1.017568970i\)
\(L(1)\) \(\approx\) \(1.320968693 - 1.017568970i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.848 - 0.528i)T \)
3 \( 1 + (0.921 - 0.387i)T \)
5 \( 1 + (-0.980 + 0.197i)T \)
7 \( 1 + (-0.905 - 0.423i)T \)
11 \( 1 + (-0.992 + 0.119i)T \)
13 \( 1 + (0.293 - 0.955i)T \)
17 \( 1 + (-0.331 - 0.943i)T \)
19 \( 1 + (0.578 + 0.815i)T \)
23 \( 1 + (0.996 - 0.0794i)T \)
29 \( 1 + (-0.905 + 0.423i)T \)
31 \( 1 + (0.368 + 0.929i)T \)
37 \( 1 + (0.138 - 0.990i)T \)
41 \( 1 + (-0.0198 - 0.999i)T \)
43 \( 1 + (0.888 + 0.459i)T \)
47 \( 1 + (0.368 + 0.929i)T \)
53 \( 1 + (0.578 + 0.815i)T \)
59 \( 1 + (0.888 - 0.459i)T \)
61 \( 1 + (-0.255 + 0.966i)T \)
67 \( 1 + (-0.405 + 0.914i)T \)
71 \( 1 + (-0.827 - 0.561i)T \)
73 \( 1 + (0.754 + 0.656i)T \)
79 \( 1 + (0.441 + 0.897i)T \)
83 \( 1 + (-0.869 - 0.494i)T \)
89 \( 1 + (0.848 - 0.528i)T \)
97 \( 1 + (0.138 - 0.990i)T \)
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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

−25.595806928621759334282070764408, −24.407331350940534791536186996058, −23.877197310491855053508620321846, −22.84389510041305417737001756191, −21.95002998779045336252748523360, −21.12577722964817455498056645990, −20.29124080940073620713951001005, −19.351793586358353891143717239120, −18.61450426644183192602081180342, −16.83069900771885034069349502435, −16.069662707769972894712023076317, −15.370430542330473359186309214654, −14.903214241064942525798183065098, −13.346753081141520783250013883883, −13.15472519550041854735881647361, −11.86394182846555045853953394653, −10.849891433567229299519491744659, −9.323709547245151699371298633034, −8.47049245788451490429952602306, −7.57852807090291214131151992199, −6.60756288700347114734009417855, −5.184883605654436855756833204305, −4.16208085635744148476867215370, −3.33522904454004501418768942060, −2.39141210921311442819867923403, 0.86720237708926944021645781094, 2.714598946733087248577492644999, 3.233516657585905480715328092580, 4.2056740328825879611760709385, 5.589288682614696033765909842324, 7.056409368174581311392205561581, 7.55415377465830402358205863720, 8.999391834425926841051722693326, 10.160875143112051007394236643416, 10.978232242676415397615646215846, 12.35398617176958228934551861431, 12.86842366934717326090855336466, 13.732434506701669619487783614347, 14.692012593441162896253559053, 15.67952888937738386704227316577, 16.05310078830784808291530374793, 18.1553250563391532211140481950, 18.90517566405909199975143502486, 19.64990541020249878144094184037, 20.42738464009166468261752719924, 20.9128865623604378865764624556, 22.505417359629751225050649940180, 22.9775925837251675483655859590, 23.778960640214625490478565720018, 24.75292147607340796745970926096

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