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Properties

Label 1-967-967.444-r0-0-0
Degree $1$
Conductor $967$
Sign $0.369 + 0.929i$
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  + (0.880 + 0.474i)2-s + (0.996 + 0.0779i)3-s + (0.549 + 0.835i)4-s + (0.190 + 0.981i)5-s + (0.840 + 0.541i)6-s + (−0.285 − 0.958i)7-s + (0.0876 + 0.996i)8-s + (0.987 + 0.155i)9-s + (−0.297 + 0.954i)10-s + (0.737 − 0.675i)11-s + (0.483 + 0.875i)12-s + (0.216 − 0.976i)13-s + (0.203 − 0.979i)14-s + (0.113 + 0.993i)15-s + (−0.395 + 0.918i)16-s + (−0.260 + 0.965i)17-s + ⋯
L(s)  = 1  + (0.880 + 0.474i)2-s + (0.996 + 0.0779i)3-s + (0.549 + 0.835i)4-s + (0.190 + 0.981i)5-s + (0.840 + 0.541i)6-s + (−0.285 − 0.958i)7-s + (0.0876 + 0.996i)8-s + (0.987 + 0.155i)9-s + (−0.297 + 0.954i)10-s + (0.737 − 0.675i)11-s + (0.483 + 0.875i)12-s + (0.216 − 0.976i)13-s + (0.203 − 0.979i)14-s + (0.113 + 0.993i)15-s + (−0.395 + 0.918i)16-s + (−0.260 + 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.369 + 0.929i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.369 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.250101483 + 2.205487861i\)
\(L(\frac12)\) \(\approx\) \(3.250101483 + 2.205487861i\)
\(L(1)\) \(\approx\) \(2.299525599 + 0.9976027251i\)
\(L(1)\) \(\approx\) \(2.299525599 + 0.9976027251i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.880 + 0.474i)T \)
3 \( 1 + (0.996 + 0.0779i)T \)
5 \( 1 + (0.190 + 0.981i)T \)
7 \( 1 + (-0.285 - 0.958i)T \)
11 \( 1 + (0.737 - 0.675i)T \)
13 \( 1 + (0.216 - 0.976i)T \)
17 \( 1 + (-0.260 + 0.965i)T \)
19 \( 1 + (0.152 - 0.988i)T \)
23 \( 1 + (0.279 + 0.960i)T \)
29 \( 1 + (-0.822 + 0.568i)T \)
31 \( 1 + (0.959 + 0.282i)T \)
37 \( 1 + (-0.618 + 0.785i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (0.0357 - 0.999i)T \)
47 \( 1 + (-0.533 - 0.845i)T \)
53 \( 1 + (-0.419 - 0.907i)T \)
59 \( 1 + (0.771 - 0.636i)T \)
61 \( 1 + (-0.247 - 0.968i)T \)
67 \( 1 + (-0.724 - 0.689i)T \)
71 \( 1 + (-0.0292 + 0.999i)T \)
73 \( 1 + (-0.419 + 0.907i)T \)
79 \( 1 + (-0.791 - 0.610i)T \)
83 \( 1 + (0.100 + 0.994i)T \)
89 \( 1 + (-0.235 + 0.971i)T \)
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

−21.2122797578085377425190208357, −20.95089041548979799174899387907, −20.259771304836705059298235766441, −19.35305714001227755550467888726, −18.899444882091828706180262865767, −17.90004877843997011131204985634, −16.42767531637461871292295601699, −15.97550003731865253074184188375, −14.999390511223405418490066873484, −14.35364583702231854953552250992, −13.62010581389576578833529984969, −12.78037417996859696160836300256, −12.20537145079074680118712588873, −11.56355008071651271051939596193, −10.07604653809752676196255516113, −9.285649085530892784948803123643, −8.976468757555050692411546515138, −7.66701683646211393748455444456, −6.59188275082743097925745800858, −5.77389965549338422979579931155, −4.538202636671465827239497704, −4.16027290384175283997246375104, −2.87919380072558678865046399031, −2.06889949636850767324424528201, −1.31122655390365316790887791591, 1.54144882914162792977580979562, 2.82877796415464829868838466198, 3.46517102228244325740808223095, 3.98152742141319366287200859237, 5.289318751722476965616025839107, 6.47060720223443045361187335429, 6.97524823202553870115015020571, 7.82422409714982272741943900132, 8.63772593521613576330945165513, 9.80624066881643581751144725854, 10.72257536871773915770482122867, 11.36658814419655534682179077021, 12.76183593386479363745239391189, 13.46415093650890978928022187453, 13.883558865770632749460198392428, 14.72701421576036927605112215542, 15.31508295619938714664734535952, 16.032477091149296842793610973879, 17.17894446523928430237295772672, 17.721829884562505562842708209747, 19.05728016783640169974695164889, 19.68232513130497944076466852522, 20.34268418588242382010034164750, 21.302486712538801976733319502688, 21.96457159045275027677399764659

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