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Properties

Label 1-967-967.667-r0-0-0
Degree $1$
Conductor $967$
Sign $0.465 - 0.885i$
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  + (0.682 + 0.730i)2-s + (−0.576 − 0.816i)3-s + (−0.0682 + 0.997i)4-s + (−0.990 − 0.136i)5-s + (0.203 − 0.979i)6-s + (0.962 − 0.269i)7-s + (−0.775 + 0.631i)8-s + (−0.334 + 0.942i)9-s + (−0.576 − 0.816i)10-s + (−0.334 + 0.942i)11-s + (0.854 − 0.519i)12-s + (−0.334 − 0.942i)13-s + (0.854 + 0.519i)14-s + (0.460 + 0.887i)15-s + (−0.990 − 0.136i)16-s + (0.460 − 0.887i)17-s + ⋯
L(s)  = 1  + (0.682 + 0.730i)2-s + (−0.576 − 0.816i)3-s + (−0.0682 + 0.997i)4-s + (−0.990 − 0.136i)5-s + (0.203 − 0.979i)6-s + (0.962 − 0.269i)7-s + (−0.775 + 0.631i)8-s + (−0.334 + 0.942i)9-s + (−0.576 − 0.816i)10-s + (−0.334 + 0.942i)11-s + (0.854 − 0.519i)12-s + (−0.334 − 0.942i)13-s + (0.854 + 0.519i)14-s + (0.460 + 0.887i)15-s + (−0.990 − 0.136i)16-s + (0.460 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.465 - 0.885i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.465 - 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8183209158 - 0.4944607238i\)
\(L(\frac12)\) \(\approx\) \(0.8183209158 - 0.4944607238i\)
\(L(1)\) \(\approx\) \(0.9626460192 + 0.06498937853i\)
\(L(1)\) \(\approx\) \(0.9626460192 + 0.06498937853i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.682 + 0.730i)T \)
3 \( 1 + (-0.576 - 0.816i)T \)
5 \( 1 + (-0.990 - 0.136i)T \)
7 \( 1 + (0.962 - 0.269i)T \)
11 \( 1 + (-0.334 + 0.942i)T \)
13 \( 1 + (-0.334 - 0.942i)T \)
17 \( 1 + (0.460 - 0.887i)T \)
19 \( 1 + (-0.576 - 0.816i)T \)
23 \( 1 + (-0.0682 + 0.997i)T \)
29 \( 1 + (-0.334 - 0.942i)T \)
31 \( 1 + (-0.775 + 0.631i)T \)
37 \( 1 + (-0.334 - 0.942i)T \)
41 \( 1 + (0.203 - 0.979i)T \)
43 \( 1 + (0.460 + 0.887i)T \)
47 \( 1 + (0.460 + 0.887i)T \)
53 \( 1 + (-0.0682 + 0.997i)T \)
59 \( 1 + (-0.0682 - 0.997i)T \)
61 \( 1 + (0.203 - 0.979i)T \)
67 \( 1 + (-0.775 - 0.631i)T \)
71 \( 1 + (0.682 - 0.730i)T \)
73 \( 1 + (-0.0682 - 0.997i)T \)
79 \( 1 + (-0.0682 - 0.997i)T \)
83 \( 1 + (0.203 - 0.979i)T \)
89 \( 1 + (-0.775 - 0.631i)T \)
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

−21.86957472288295096335573244041, −21.13468953127860402199094905923, −20.648087366187299662092875412672, −19.67224443032573832864507191097, −18.73789329333597943366170656179, −18.34634858268665409150628925599, −16.86167693505989351017833684781, −16.36246021956725677956744259030, −15.29599938332197090660238655431, −14.70570516677546622774846827345, −14.23824546363898079930483642634, −12.80237348693833772959343243763, −12.01786816601118414953418284833, −11.47511036187522975906776174928, −10.790201524274475370190614020394, −10.2286988315339481752903015099, −8.90541500680571687213156927361, −8.27934538130330126196116587914, −6.83976941132731985247819159722, −5.81869029695004307363739818143, −5.09082118967676688625371042046, −4.14162642709407382634947680578, −3.729689746869253367523438406399, −2.509378167945465492533160132254, −1.17536378221469303752986000881, 0.38649814198144316937316121836, 1.98664482618006378617266949687, 3.12045064610721580898330289366, 4.42725661232474764591321933463, 4.98049823190698280095124972305, 5.751289254204120883134617342237, 7.09050897434811876153086721821, 7.57283738688503081044205205821, 7.91716417631702951711637107522, 9.12423257488701796854518496927, 10.76202305354966008571583120375, 11.38646835000268280744405971724, 12.269497211147986867772697820181, 12.67874211260340933595542763109, 13.64816371784917702153492878498, 14.49010426748941003219424972925, 15.33336788952094801885011528, 15.91531297798810584421940448236, 16.98809613413228198441356356058, 17.618688128711791560754013850746, 18.085531730368262561027404580945, 19.243275997233810629989620150677, 20.16331805119655967407319843082, 20.813620563198260054059498822888, 21.898565132203114665001773568080

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