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Properties

Label 1-967-967.190-r0-0-0
Degree $1$
Conductor $967$
Sign $0.972 + 0.232i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (−0.877 + 0.480i)2-s + (−0.917 + 0.398i)3-s + (0.538 − 0.842i)4-s + (0.995 + 0.0909i)5-s + (0.613 − 0.789i)6-s + (0.983 − 0.181i)7-s + (−0.0682 + 0.997i)8-s + (0.682 − 0.730i)9-s + (−0.917 + 0.398i)10-s + (0.682 − 0.730i)11-s + (−0.158 + 0.987i)12-s + (0.291 − 0.956i)13-s + (−0.775 + 0.631i)14-s + (−0.949 + 0.313i)15-s + (−0.419 − 0.907i)16-s + (0.203 + 0.979i)17-s + ⋯
L(s)  = 1  + (−0.877 + 0.480i)2-s + (−0.917 + 0.398i)3-s + (0.538 − 0.842i)4-s + (0.995 + 0.0909i)5-s + (0.613 − 0.789i)6-s + (0.983 − 0.181i)7-s + (−0.0682 + 0.997i)8-s + (0.682 − 0.730i)9-s + (−0.917 + 0.398i)10-s + (0.682 − 0.730i)11-s + (−0.158 + 0.987i)12-s + (0.291 − 0.956i)13-s + (−0.775 + 0.631i)14-s + (−0.949 + 0.313i)15-s + (−0.419 − 0.907i)16-s + (0.203 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.972 + 0.232i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.972 + 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.130831690 + 0.1335255184i\)
\(L(\frac12)\) \(\approx\) \(1.130831690 + 0.1335255184i\)
\(L(1)\) \(\approx\) \(0.8201115405 + 0.1396291619i\)
\(L(1)\) \(\approx\) \(0.8201115405 + 0.1396291619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.877 + 0.480i)T \)
3 \( 1 + (-0.917 + 0.398i)T \)
5 \( 1 + (0.995 + 0.0909i)T \)
7 \( 1 + (0.983 - 0.181i)T \)
11 \( 1 + (0.682 - 0.730i)T \)
13 \( 1 + (0.291 - 0.956i)T \)
17 \( 1 + (0.203 + 0.979i)T \)
19 \( 1 + (0.113 - 0.993i)T \)
23 \( 1 + (0.460 + 0.887i)T \)
29 \( 1 + (0.682 + 0.730i)T \)
31 \( 1 + (0.898 - 0.439i)T \)
37 \( 1 + (0.291 - 0.956i)T \)
41 \( 1 + (-0.990 - 0.136i)T \)
43 \( 1 + (0.746 + 0.665i)T \)
47 \( 1 + (-0.949 + 0.313i)T \)
53 \( 1 + (-0.998 - 0.0455i)T \)
59 \( 1 + (-0.998 + 0.0455i)T \)
61 \( 1 + (0.377 + 0.926i)T \)
67 \( 1 + (-0.0682 - 0.997i)T \)
71 \( 1 + (0.854 - 0.519i)T \)
73 \( 1 + (-0.998 + 0.0455i)T \)
79 \( 1 + (-0.998 + 0.0455i)T \)
83 \( 1 + (0.613 - 0.789i)T \)
89 \( 1 + (-0.829 + 0.557i)T \)
97 \( 1 + T \)
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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

−21.554576280154244055654548221111, −20.90186970827321534808962409676, −20.32766050485062876958291614661, −18.91576621125421145691372202110, −18.546793330854111807434565245544, −17.77124835507199707423800984783, −17.16584283357172014248967923673, −16.69349046398548037778244887222, −15.73069217133862097108476627748, −14.40456851879065080840703256833, −13.674233975877058412929427369742, −12.5607195175606738600618733108, −11.88496641415878903569532758601, −11.37836882776734291426438651165, −10.32394239711495317560316406856, −9.767616461477040845906724215608, −8.79478561690585950263950933921, −7.89741444463502712742490561559, −6.80754815812335175008044373571, −6.35194427823945489541513945620, −5.053988802458065607338896499641, −4.31236044616310437924417975780, −2.59716814899506528638826617730, −1.67140522068796457360717261825, −1.173954724330493401357001513, 0.95709823866298578801387401486, 1.566738737581715041882002989718, 3.09356604800025044770488021122, 4.62857971698238634379309932564, 5.46434829009519978688741527689, 6.06283669547733534288469082255, 6.83908849717599531202224743507, 7.94558693950660740897007315229, 8.874390592161762784530219024731, 9.62356506875749064627215006071, 10.61601564379718628130273907320, 10.927541130083579240984129440173, 11.78194762973382334210057872010, 13.046304159719510687889701056201, 14.05031814237842904699925146542, 14.857448868517517194468941305, 15.571961157225366741622545659257, 16.532207019699366419846318867470, 17.34342204356695806062215090406, 17.560338809081698278918233996584, 18.21717689492281131535751342603, 19.24796950819491381684844368207, 20.19214868730933240085304583358, 21.18778640258201015961697621256, 21.577289803002862293638498342869

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