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Properties

Label 1-967-967.116-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.918 + 0.394i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (0.152 + 0.988i)2-s + (0.998 + 0.0585i)3-s + (−0.953 + 0.300i)4-s + (−0.999 + 0.0130i)5-s + (0.0941 + 0.995i)6-s + (−0.643 + 0.765i)7-s + (−0.442 − 0.896i)8-s + (0.993 + 0.116i)9-s + (−0.165 − 0.986i)10-s + (0.527 + 0.849i)11-s + (−0.969 + 0.244i)12-s + (−0.471 − 0.881i)13-s + (−0.854 − 0.519i)14-s + (−0.998 − 0.0455i)15-s + (0.818 − 0.574i)16-s + (0.981 − 0.193i)17-s + ⋯
L(s)  = 1  + (0.152 + 0.988i)2-s + (0.998 + 0.0585i)3-s + (−0.953 + 0.300i)4-s + (−0.999 + 0.0130i)5-s + (0.0941 + 0.995i)6-s + (−0.643 + 0.765i)7-s + (−0.442 − 0.896i)8-s + (0.993 + 0.116i)9-s + (−0.165 − 0.986i)10-s + (0.527 + 0.849i)11-s + (−0.969 + 0.244i)12-s + (−0.471 − 0.881i)13-s + (−0.854 − 0.519i)14-s + (−0.998 − 0.0455i)15-s + (0.818 − 0.574i)16-s + (0.981 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.918 + 0.394i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.918 + 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4610216966 + 2.240768440i\)
\(L(\frac12)\) \(\approx\) \(0.4610216966 + 2.240768440i\)
\(L(1)\) \(\approx\) \(0.9566382480 + 0.8246414089i\)
\(L(1)\) \(\approx\) \(0.9566382480 + 0.8246414089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.152 + 0.988i)T \)
3 \( 1 + (0.998 + 0.0585i)T \)
5 \( 1 + (-0.999 + 0.0130i)T \)
7 \( 1 + (-0.643 + 0.765i)T \)
11 \( 1 + (0.527 + 0.849i)T \)
13 \( 1 + (-0.471 - 0.881i)T \)
17 \( 1 + (0.981 - 0.193i)T \)
19 \( 1 + (0.0162 + 0.999i)T \)
23 \( 1 + (0.822 - 0.568i)T \)
29 \( 1 + (0.945 + 0.325i)T \)
31 \( 1 + (0.672 - 0.739i)T \)
37 \( 1 + (0.807 + 0.589i)T \)
41 \( 1 + (-0.203 + 0.979i)T \)
43 \( 1 + (-0.994 + 0.103i)T \)
47 \( 1 + (-0.880 - 0.474i)T \)
53 \( 1 + (0.898 - 0.439i)T \)
59 \( 1 + (-0.999 - 0.00650i)T \)
61 \( 1 + (0.746 + 0.665i)T \)
67 \( 1 + (0.977 - 0.212i)T \)
71 \( 1 + (-0.932 + 0.362i)T \)
73 \( 1 + (0.898 + 0.439i)T \)
79 \( 1 + (-0.228 + 0.973i)T \)
83 \( 1 + (-0.837 - 0.547i)T \)
89 \( 1 + (-0.304 + 0.952i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−21.25763318843591685515150478323, −20.13768440273092553435521784733, −19.60290012083079525774112935362, −19.23824182002052406716965457010, −18.66352346222984947455320691339, −17.31534452256059104960592862301, −16.43753012469365343985374943067, −15.52976380980767650789378174422, −14.54891651123737603292868133884, −13.95667692734398664263890346802, −13.25290360731950027524979146473, −12.37701160134514027718823356324, −11.631270142204398264316318403386, −10.75342944573477186007425838108, −9.83799839361698678838081063791, −9.0891598958615934262850646942, −8.36593361340542156203951880167, −7.3924894025531551087023136875, −6.545873580519885132836316779725, −4.87956316547521187056713685541, −4.08478204552478291073170546651, −3.37373847194306949023516997783, −2.80515690502270430302706879351, −1.32819814797863896463416255263, −0.52127134648990806816953939272, 0.99220227564967570774724931449, 2.77944192643311011714178323011, 3.40398427660353865593016785352, 4.371636904402497450828879285183, 5.19870394104494415843847779343, 6.487882278881075188643183388362, 7.20392417166364985934891245177, 8.12896918444274561928848159062, 8.46603595403177053478863728982, 9.71031616581180556823612872245, 10.013174855146663674467540760792, 11.896713922439313439583144008215, 12.51796416238465609527189759213, 13.10520547684516028246210789715, 14.3423274696657377029735332185, 15.03196240606963417317456321635, 15.211884636857839366832082832379, 16.24170494758092287391239215216, 16.808371443417619476689232184959, 18.17623385121502917686225257344, 18.70821963001739585351864420127, 19.46943338203250285003757589160, 20.179917610086483435909017710573, 21.15654304788130611896872366447, 22.09440787494147370772667051626

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