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Properties

Label 1-967-967.574-r0-0-0
Degree $1$
Conductor $967$
Sign $0.858 - 0.512i$
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  + (−0.775 + 0.631i)2-s + (0.962 − 0.269i)3-s + (0.203 − 0.979i)4-s + (−0.917 − 0.398i)5-s + (−0.576 + 0.816i)6-s + (0.682 − 0.730i)7-s + (0.460 + 0.887i)8-s + (0.854 − 0.519i)9-s + (0.962 − 0.269i)10-s + (0.854 − 0.519i)11-s + (−0.0682 − 0.997i)12-s + (0.854 + 0.519i)13-s + (−0.0682 + 0.997i)14-s + (−0.990 − 0.136i)15-s + (−0.917 − 0.398i)16-s + (−0.990 + 0.136i)17-s + ⋯
L(s)  = 1  + (−0.775 + 0.631i)2-s + (0.962 − 0.269i)3-s + (0.203 − 0.979i)4-s + (−0.917 − 0.398i)5-s + (−0.576 + 0.816i)6-s + (0.682 − 0.730i)7-s + (0.460 + 0.887i)8-s + (0.854 − 0.519i)9-s + (0.962 − 0.269i)10-s + (0.854 − 0.519i)11-s + (−0.0682 − 0.997i)12-s + (0.854 + 0.519i)13-s + (−0.0682 + 0.997i)14-s + (−0.990 − 0.136i)15-s + (−0.917 − 0.398i)16-s + (−0.990 + 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.858 - 0.512i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (574, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.858 - 0.512i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.473766652 - 0.4066369496i\)
\(L(\frac12)\) \(\approx\) \(1.473766652 - 0.4066369496i\)
\(L(1)\) \(\approx\) \(1.084281261 - 0.07274901863i\)
\(L(1)\) \(\approx\) \(1.084281261 - 0.07274901863i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.775 + 0.631i)T \)
3 \( 1 + (0.962 - 0.269i)T \)
5 \( 1 + (-0.917 - 0.398i)T \)
7 \( 1 + (0.682 - 0.730i)T \)
11 \( 1 + (0.854 - 0.519i)T \)
13 \( 1 + (0.854 + 0.519i)T \)
17 \( 1 + (-0.990 + 0.136i)T \)
19 \( 1 + (0.962 - 0.269i)T \)
23 \( 1 + (0.203 - 0.979i)T \)
29 \( 1 + (0.854 + 0.519i)T \)
31 \( 1 + (0.460 + 0.887i)T \)
37 \( 1 + (0.854 + 0.519i)T \)
41 \( 1 + (-0.576 + 0.816i)T \)
43 \( 1 + (-0.990 - 0.136i)T \)
47 \( 1 + (-0.990 - 0.136i)T \)
53 \( 1 + (0.203 - 0.979i)T \)
59 \( 1 + (0.203 + 0.979i)T \)
61 \( 1 + (-0.576 + 0.816i)T \)
67 \( 1 + (0.460 - 0.887i)T \)
71 \( 1 + (-0.775 - 0.631i)T \)
73 \( 1 + (0.203 + 0.979i)T \)
79 \( 1 + (0.203 + 0.979i)T \)
83 \( 1 + (-0.576 + 0.816i)T \)
89 \( 1 + (0.460 - 0.887i)T \)
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

−21.70528295576419282627337890123, −20.68017519055697168092949411944, −20.194450515399096246279147069407, −19.5528508980554013525318781172, −18.770427581886893227321204516346, −18.148298149815996799862639793533, −17.38023573173860181240200679313, −16.03001346810208308834634711972, −15.560569885122425911472535057373, −14.88714920838543109350840814988, −13.84362723105308051132900469419, −12.94937384328846999713822883786, −11.81825546611719738016946817136, −11.45922402283813453904420323113, −10.48221397069924222742562313931, −9.50653748849652780541176905708, −8.83971628383278649494401101096, −8.06315021739266174162477035849, −7.54370123418561526826964543603, −6.47685115527124255090530878699, −4.750465778329119400137316263319, −3.89736384338224342922020858998, −3.160003932078011662378032162388, −2.22806822607004115265341290360, −1.242002976921074286485233665225, 0.961775238358241315255186368072, 1.52672272772436691006011700538, 3.088020175099915698890093524789, 4.190456226847892691629894560323, 4.85332789035353082876648649107, 6.60486574647994992118404434475, 6.91064886464798856482269329274, 8.07615164604742247912815846200, 8.489601687614927204859585821645, 9.07098951572314863416651526540, 10.201709618795751758547631400067, 11.2163737961629784112230878467, 11.7945672417878905837865304921, 13.241311818647315336302068992021, 13.9298536747806755458611302652, 14.621389333301245227714074710129, 15.38319981816666066833168682851, 16.23546956441066482605341585548, 16.75653827379994876791257430197, 17.97638070102988451653176910378, 18.430897722935854284685767254278, 19.55960492011896279015593361108, 19.83441322270365125991422779852, 20.47762148085676112829587941444, 21.39928672442137524310476257353

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