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Properties

Label 1-317-317.152-r0-0-0
Degree $1$
Conductor $317$
Sign $0.498 + 0.866i$
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  + (0.980 − 0.197i)2-s + (0.727 + 0.685i)3-s + (0.921 − 0.387i)4-s + (−0.368 + 0.929i)5-s + (0.848 + 0.528i)6-s + (−0.869 + 0.494i)7-s + (0.827 − 0.561i)8-s + (0.0596 + 0.998i)9-s + (−0.177 + 0.984i)10-s + (0.754 − 0.656i)11-s + (0.936 + 0.350i)12-s + (−0.216 + 0.976i)13-s + (−0.754 + 0.656i)14-s + (−0.905 + 0.423i)15-s + (0.700 − 0.714i)16-s + (−0.441 − 0.897i)17-s + ⋯
L(s)  = 1  + (0.980 − 0.197i)2-s + (0.727 + 0.685i)3-s + (0.921 − 0.387i)4-s + (−0.368 + 0.929i)5-s + (0.848 + 0.528i)6-s + (−0.869 + 0.494i)7-s + (0.827 − 0.561i)8-s + (0.0596 + 0.998i)9-s + (−0.177 + 0.984i)10-s + (0.754 − 0.656i)11-s + (0.936 + 0.350i)12-s + (−0.216 + 0.976i)13-s + (−0.754 + 0.656i)14-s + (−0.905 + 0.423i)15-s + (0.700 − 0.714i)16-s + (−0.441 − 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.498 + 0.866i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.498 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.232849089 + 1.291135716i\)
\(L(\frac12)\) \(\approx\) \(2.232849089 + 1.291135716i\)
\(L(1)\) \(\approx\) \(1.960435458 + 0.5978038713i\)
\(L(1)\) \(\approx\) \(1.960435458 + 0.5978038713i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.980 - 0.197i)T \)
3 \( 1 + (0.727 + 0.685i)T \)
5 \( 1 + (-0.368 + 0.929i)T \)
7 \( 1 + (-0.869 + 0.494i)T \)
11 \( 1 + (0.754 - 0.656i)T \)
13 \( 1 + (-0.216 + 0.976i)T \)
17 \( 1 + (-0.441 - 0.897i)T \)
19 \( 1 + (-0.848 + 0.528i)T \)
23 \( 1 + (0.888 - 0.459i)T \)
29 \( 1 + (0.869 + 0.494i)T \)
31 \( 1 + (0.641 + 0.767i)T \)
37 \( 1 + (-0.671 - 0.741i)T \)
41 \( 1 + (0.992 - 0.119i)T \)
43 \( 1 + (-0.961 + 0.274i)T \)
47 \( 1 + (-0.641 - 0.767i)T \)
53 \( 1 + (0.848 - 0.528i)T \)
59 \( 1 + (-0.961 - 0.274i)T \)
61 \( 1 + (-0.0198 - 0.999i)T \)
67 \( 1 + (0.804 - 0.594i)T \)
71 \( 1 + (0.905 + 0.423i)T \)
73 \( 1 + (-0.405 - 0.914i)T \)
79 \( 1 + (0.921 + 0.387i)T \)
83 \( 1 + (-0.999 + 0.0397i)T \)
89 \( 1 + (-0.980 + 0.197i)T \)
97 \( 1 + (0.671 + 0.741i)T \)
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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

−24.930615940102986577033366280146, −24.158752754194765232255044345377, −23.28632878732043440182814102016, −22.681398955559407488915707996597, −21.36662964846344636538770938976, −20.45517326078630628418358481816, −19.648430070506163234615102604481, −19.43564720106579032885026980358, −17.41390982927228108552149613511, −16.95810178678232177412751871183, −15.46608070960318413053961480801, −15.15243066134745153685840463434, −13.82307047076794425086022458346, −12.9287204032918749357284167258, −12.68695539480684084044433721764, −11.616630952165412840529286854309, −10.11427312141161236479333083049, −8.85306759301043703069532477271, −7.89153155208249910663250901221, −6.93333802143465838897951883099, −6.09997418009023929231722325930, −4.572687487819759993917136988709, −3.75890418452722010306698842084, −2.638394303751516480188385293860, −1.229082063647815530602163786672, 2.19970147095430838639294088559, 3.090850036284350906404206732923, 3.81423179795607895793644036554, 4.90128378003720336541311377708, 6.40508888446259397018295188502, 6.9676958916984451502260552954, 8.55987734172945253388172749011, 9.61521858919615421769164459223, 10.62987934962029634495843991701, 11.46772057133141787250417656773, 12.481935713601131703617596175418, 13.78426832811951449450467844034, 14.31016198616812026128276506034, 15.14869162086137487402450172766, 15.9798925295176717067477158691, 16.64604465729151525922047441112, 18.66193452621407357215033638268, 19.36825861440561383605908542493, 19.83054476300028701570610130657, 21.30750514947379418012467776160, 21.63435346766898652908589037742, 22.61112955971980122911311294353, 23.12448255486091489154695086916, 24.61509433370730025330741993236, 25.177352142420515235337297178328

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