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Properties

Label 1-967-967.57-r0-0-0
Degree $1$
Conductor $967$
Sign $0.754 - 0.656i$
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  + (0.998 − 0.0520i)2-s + (−0.883 − 0.468i)3-s + (0.994 − 0.103i)4-s + (0.692 − 0.721i)5-s + (−0.906 − 0.422i)6-s + (−0.807 + 0.589i)7-s + (0.987 − 0.155i)8-s + (0.560 + 0.828i)9-s + (0.653 − 0.756i)10-s + (0.996 + 0.0779i)11-s + (−0.927 − 0.374i)12-s + (−0.566 − 0.824i)13-s + (−0.775 + 0.631i)14-s + (−0.949 + 0.313i)15-s + (0.978 − 0.206i)16-s + (0.892 + 0.451i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0520i)2-s + (−0.883 − 0.468i)3-s + (0.994 − 0.103i)4-s + (0.692 − 0.721i)5-s + (−0.906 − 0.422i)6-s + (−0.807 + 0.589i)7-s + (0.987 − 0.155i)8-s + (0.560 + 0.828i)9-s + (0.653 − 0.756i)10-s + (0.996 + 0.0779i)11-s + (−0.927 − 0.374i)12-s + (−0.566 − 0.824i)13-s + (−0.775 + 0.631i)14-s + (−0.949 + 0.313i)15-s + (0.978 − 0.206i)16-s + (0.892 + 0.451i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.754 - 0.656i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.754 - 0.656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.361145609 - 0.8835268473i\)
\(L(\frac12)\) \(\approx\) \(2.361145609 - 0.8835268473i\)
\(L(1)\) \(\approx\) \(1.676421350 - 0.3929806155i\)
\(L(1)\) \(\approx\) \(1.676421350 - 0.3929806155i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.998 - 0.0520i)T \)
3 \( 1 + (-0.883 - 0.468i)T \)
5 \( 1 + (0.692 - 0.721i)T \)
7 \( 1 + (-0.807 + 0.589i)T \)
11 \( 1 + (0.996 + 0.0779i)T \)
13 \( 1 + (-0.566 - 0.824i)T \)
17 \( 1 + (0.892 + 0.451i)T \)
19 \( 1 + (-0.533 + 0.845i)T \)
23 \( 1 + (0.981 + 0.193i)T \)
29 \( 1 + (-0.145 + 0.989i)T \)
31 \( 1 + (0.216 - 0.976i)T \)
37 \( 1 + (0.152 + 0.988i)T \)
41 \( 1 + (-0.990 - 0.136i)T \)
43 \( 1 + (0.985 - 0.168i)T \)
47 \( 1 + (0.719 - 0.694i)T \)
53 \( 1 + (-0.998 - 0.0455i)T \)
59 \( 1 + (0.919 + 0.392i)T \)
61 \( 1 + (0.377 + 0.926i)T \)
67 \( 1 + (0.737 - 0.675i)T \)
71 \( 1 + (-0.544 + 0.838i)T \)
73 \( 1 + (-0.998 + 0.0455i)T \)
79 \( 1 + (0.177 - 0.984i)T \)
83 \( 1 + (-0.235 - 0.971i)T \)
89 \( 1 + (-0.953 - 0.300i)T \)
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

−22.037519358448251093395297286491, −21.33920504453892095239277376821, −20.66582799854446953985633149424, −19.42519304664792736444624540465, −18.94672261168014307363161483000, −17.419133894368137750446273760855, −17.06548574273043093893584335170, −16.32190960081779543869253093165, −15.4787107625082361378948026026, −14.51722987864150966597774541813, −14.04341745697527443353667210537, −13.03110517839423604148010235695, −12.26931761581991173068588226922, −11.3756012882392998104904749707, −10.77739626798991581269236661172, −9.870325512266805791948934738135, −9.26293177017405813232344284590, −7.24768990005156396701969879225, −6.75657473769038164703644715668, −6.19810681456367526686256593558, −5.22844876974617216383187616808, −4.30688534113128755380557041521, −3.50686775520533273232391981271, −2.54475095126225510275094005647, −1.160387804754952033377669108899, 1.075247000146461360765399738702, 1.94686340761169861677002965049, 3.09158144780581815599644274109, 4.25619393884663207586132398936, 5.31298788712077916299355792528, 5.77740356158214604888276616626, 6.469231485975728297086510813798, 7.36801276469546846901164284934, 8.56551656562158444761172092236, 9.846254829931171425806223423587, 10.37212556781505401771185674042, 11.625063852038466352891112438759, 12.30597784505754826015070039311, 12.75647169714584212671627485304, 13.36039079726807306091807797401, 14.44820229202430049433221282178, 15.238904257438844329839611578924, 16.28871767443360849064644704365, 16.89943066411637206413782858147, 17.3308190720405277754381027797, 18.71247741959554587671363522468, 19.297701979709865760414107535615, 20.24542495473479111237626200336, 21.085619868119562795300487754168, 22.05193171297355379406877209630

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