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Properties

Label 1-967-967.138-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.476 + 0.878i$
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  + (0.929 + 0.368i)2-s + (0.924 + 0.380i)3-s + (0.728 + 0.684i)4-s + (−0.906 + 0.422i)5-s + (0.719 + 0.694i)6-s + (0.602 − 0.797i)7-s + (0.425 + 0.905i)8-s + (0.710 + 0.703i)9-s + (−0.998 + 0.0585i)10-s + (−0.844 + 0.536i)11-s + (0.413 + 0.910i)12-s + (−0.0422 + 0.999i)13-s + (0.854 − 0.519i)14-s + (−0.998 + 0.0455i)15-s + (0.0617 + 0.998i)16-s + (−0.967 + 0.250i)17-s + ⋯
L(s)  = 1  + (0.929 + 0.368i)2-s + (0.924 + 0.380i)3-s + (0.728 + 0.684i)4-s + (−0.906 + 0.422i)5-s + (0.719 + 0.694i)6-s + (0.602 − 0.797i)7-s + (0.425 + 0.905i)8-s + (0.710 + 0.703i)9-s + (−0.998 + 0.0585i)10-s + (−0.844 + 0.536i)11-s + (0.413 + 0.910i)12-s + (−0.0422 + 0.999i)13-s + (0.854 − 0.519i)14-s + (−0.998 + 0.0455i)15-s + (0.0617 + 0.998i)16-s + (−0.967 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.476 + 0.878i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.476 + 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.564071614 + 2.628281433i\)
\(L(\frac12)\) \(\approx\) \(1.564071614 + 2.628281433i\)
\(L(1)\) \(\approx\) \(1.762353718 + 1.136941652i\)
\(L(1)\) \(\approx\) \(1.762353718 + 1.136941652i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.929 + 0.368i)T \)
3 \( 1 + (0.924 + 0.380i)T \)
5 \( 1 + (-0.906 + 0.422i)T \)
7 \( 1 + (0.602 - 0.797i)T \)
11 \( 1 + (-0.844 + 0.536i)T \)
13 \( 1 + (-0.0422 + 0.999i)T \)
17 \( 1 + (-0.967 + 0.250i)T \)
19 \( 1 + (-0.971 - 0.238i)T \)
23 \( 1 + (0.987 + 0.155i)T \)
29 \( 1 + (0.993 + 0.116i)T \)
31 \( 1 + (-0.927 - 0.374i)T \)
37 \( 1 + (0.754 + 0.655i)T \)
41 \( 1 + (0.203 + 0.979i)T \)
43 \( 1 + (-0.941 + 0.337i)T \)
47 \( 1 + (-0.658 - 0.752i)T \)
53 \( 1 + (0.898 + 0.439i)T \)
59 \( 1 + (0.216 - 0.976i)T \)
61 \( 1 + (0.746 - 0.665i)T \)
67 \( 1 + (0.787 + 0.615i)T \)
71 \( 1 + (-0.145 + 0.989i)T \)
73 \( 1 + (0.898 - 0.439i)T \)
79 \( 1 + (-0.618 + 0.785i)T \)
83 \( 1 + (0.516 - 0.856i)T \)
89 \( 1 + (0.139 - 0.990i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−21.16361103347928617267800853900, −20.89257086000358661890261778832, −19.92924809530858938689407656152, −19.43036412426259883418920541809, −18.6282413916554955349095348830, −17.87882888071924561491625241608, −16.34304294859541331985335855180, −15.48048885727970141505681297287, −15.15534174889607857518199599030, −14.405028668537575057816582875329, −13.27360822805900746197243052428, −12.82522008626091840737071713896, −12.139201751445263954360991211056, −11.16269317833956275008263410332, −10.485296142854138054709235665483, −9.06191713407473154753659016886, −8.385201145321614348219596427959, −7.6469592819044489007472340535, −6.65954334590695258915357322872, −5.460916052870420149496300304173, −4.71695908231319170913316181232, −3.73835148911375792597218909653, −2.82437937405419666386574064795, −2.162209225378881242044622906262, −0.82405766172420529360157798547, 1.87163963568221695310945101010, 2.72309367006428074107476996830, 3.736992546563818351900749489008, 4.48314184806245921767537150811, 4.85659572180958991255448139485, 6.719372836103315645742002694140, 7.11694122021556848014430487371, 8.07342135650547030066544395691, 8.579499441847178137797575248148, 10.0374777435224996244102372314, 11.00215037015023213901478389975, 11.41323381355860146965007982117, 12.80321531397446092705369221311, 13.30525507466298416928425992586, 14.26239765434697256465149764795, 14.90226527462097083884903527745, 15.33587077949298566003903564591, 16.22348645716461276710614569583, 16.94938727692901578807917584498, 18.10039750953505356612528264126, 19.14461082900015013853608183520, 20.00425686798077672095486320201, 20.37220147153394871050267580, 21.44594633868727455448075930561, 21.723545443617157052648593828361

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