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Properties

Label 1-317-317.181-r0-0-0
Degree $1$
Conductor $317$
Sign $-0.231 + 0.972i$
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  + (−0.545 + 0.838i)2-s + (0.804 + 0.594i)3-s + (−0.405 − 0.914i)4-s + (0.949 + 0.312i)5-s + (−0.936 + 0.350i)6-s + (0.848 + 0.528i)7-s + (0.987 + 0.158i)8-s + (0.293 + 0.955i)9-s + (−0.780 + 0.625i)10-s + (−0.905 + 0.423i)11-s + (0.216 − 0.976i)12-s + (0.888 − 0.459i)13-s + (−0.905 + 0.423i)14-s + (0.578 + 0.815i)15-s + (−0.671 + 0.741i)16-s + (0.754 − 0.656i)17-s + ⋯
L(s)  = 1  + (−0.545 + 0.838i)2-s + (0.804 + 0.594i)3-s + (−0.405 − 0.914i)4-s + (0.949 + 0.312i)5-s + (−0.936 + 0.350i)6-s + (0.848 + 0.528i)7-s + (0.987 + 0.158i)8-s + (0.293 + 0.955i)9-s + (−0.780 + 0.625i)10-s + (−0.905 + 0.423i)11-s + (0.216 − 0.976i)12-s + (0.888 − 0.459i)13-s + (−0.905 + 0.423i)14-s + (0.578 + 0.815i)15-s + (−0.671 + 0.741i)16-s + (0.754 − 0.656i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $-0.231 + 0.972i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ -0.231 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9540924036 + 1.207244410i\)
\(L(\frac12)\) \(\approx\) \(0.9540924036 + 1.207244410i\)
\(L(1)\) \(\approx\) \(1.017880834 + 0.7354732265i\)
\(L(1)\) \(\approx\) \(1.017880834 + 0.7354732265i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.545 + 0.838i)T \)
3 \( 1 + (0.804 + 0.594i)T \)
5 \( 1 + (0.949 + 0.312i)T \)
7 \( 1 + (0.848 + 0.528i)T \)
11 \( 1 + (-0.905 + 0.423i)T \)
13 \( 1 + (0.888 - 0.459i)T \)
17 \( 1 + (0.754 - 0.656i)T \)
19 \( 1 + (-0.936 - 0.350i)T \)
23 \( 1 + (-0.727 - 0.685i)T \)
29 \( 1 + (0.848 - 0.528i)T \)
31 \( 1 + (-0.331 - 0.943i)T \)
37 \( 1 + (0.511 + 0.859i)T \)
41 \( 1 + (-0.827 + 0.561i)T \)
43 \( 1 + (-0.177 + 0.984i)T \)
47 \( 1 + (-0.331 - 0.943i)T \)
53 \( 1 + (-0.936 - 0.350i)T \)
59 \( 1 + (-0.177 - 0.984i)T \)
61 \( 1 + (-0.0992 - 0.995i)T \)
67 \( 1 + (-0.999 + 0.0397i)T \)
71 \( 1 + (0.578 - 0.815i)T \)
73 \( 1 + (-0.869 + 0.494i)T \)
79 \( 1 + (-0.405 + 0.914i)T \)
83 \( 1 + (-0.980 + 0.197i)T \)
89 \( 1 + (-0.545 + 0.838i)T \)
97 \( 1 + (0.511 + 0.859i)T \)
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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

−25.39797542152695734240178227269, −23.93572041451044866386656596960, −23.43458815316506680656463480322, −21.55520313729662817692888593507, −21.18717573381416176981688289580, −20.540422161015964319738347088939, −19.5671396039714055143058230314, −18.58442267210353998094673843035, −17.969964206406058762295381285021, −17.176010121854425635389822494867, −16.076200687293200483939648294439, −14.40465578708033358383113683203, −13.76059124410770443418087718938, −13.02292675290458082908386406120, −12.12442922029086150946078082698, −10.764010412670715403632507320094, −10.1185082039668918601109802489, −8.79388378897513386907439788556, −8.34068261441096178216738820866, −7.308924345247828259525290837905, −5.85711259052303846448019733638, −4.32709137274419118109394995832, −3.18020376860406768392718756942, −1.8940781775049076635513846668, −1.306331046260974184224844880011, 1.72663899196709837990222293375, 2.733414157803784249226515065531, 4.5761927874568487234441095284, 5.37188338095173300161192275868, 6.444833156454235159285123645558, 7.93341991533348548375710377779, 8.36496430628336729038510966501, 9.5504286163930955811504386668, 10.21162499954510238215820360225, 11.09045836293994800681538696436, 13.060911697003390203816833258515, 13.86472727157701278722456276563, 14.72848158394560430887712152809, 15.31736773373085057064428096098, 16.26756831843577811463412256926, 17.34095948086651997920200486828, 18.3536201333771868243186995863, 18.66636727961062857383183783716, 20.20383665709984994714174516093, 20.91763830035842011374573808949, 21.73637250340791707848391422748, 22.83863502683595614397880790779, 23.90036597869597375670964327968, 25.013756884185917671646248640226, 25.432956733163324911281308671183

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