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Properties

Label 1-967-967.25-r0-0-0
Degree $1$
Conductor $967$
Sign $0.558 - 0.829i$
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  + (0.152 − 0.988i)2-s + (−0.998 + 0.0585i)3-s + (−0.953 − 0.300i)4-s + (0.999 + 0.0130i)5-s + (−0.0941 + 0.995i)6-s + (0.643 + 0.765i)7-s + (−0.442 + 0.896i)8-s + (0.993 − 0.116i)9-s + (0.165 − 0.986i)10-s + (0.527 − 0.849i)11-s + (0.969 + 0.244i)12-s + (0.471 − 0.881i)13-s + (0.854 − 0.519i)14-s + (−0.998 + 0.0455i)15-s + (0.818 + 0.574i)16-s + (0.981 + 0.193i)17-s + ⋯
L(s)  = 1  + (0.152 − 0.988i)2-s + (−0.998 + 0.0585i)3-s + (−0.953 − 0.300i)4-s + (0.999 + 0.0130i)5-s + (−0.0941 + 0.995i)6-s + (0.643 + 0.765i)7-s + (−0.442 + 0.896i)8-s + (0.993 − 0.116i)9-s + (0.165 − 0.986i)10-s + (0.527 − 0.849i)11-s + (0.969 + 0.244i)12-s + (0.471 − 0.881i)13-s + (0.854 − 0.519i)14-s + (−0.998 + 0.0455i)15-s + (0.818 + 0.574i)16-s + (0.981 + 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.558 - 0.829i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.558 - 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.314743875 - 0.6998449279i\)
\(L(\frac12)\) \(\approx\) \(1.314743875 - 0.6998449279i\)
\(L(1)\) \(\approx\) \(0.9903181864 - 0.4373549444i\)
\(L(1)\) \(\approx\) \(0.9903181864 - 0.4373549444i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.152 - 0.988i)T \)
3 \( 1 + (-0.998 + 0.0585i)T \)
5 \( 1 + (0.999 + 0.0130i)T \)
7 \( 1 + (0.643 + 0.765i)T \)
11 \( 1 + (0.527 - 0.849i)T \)
13 \( 1 + (0.471 - 0.881i)T \)
17 \( 1 + (0.981 + 0.193i)T \)
19 \( 1 + (-0.0162 + 0.999i)T \)
23 \( 1 + (-0.822 - 0.568i)T \)
29 \( 1 + (-0.945 + 0.325i)T \)
31 \( 1 + (0.672 + 0.739i)T \)
37 \( 1 + (-0.807 + 0.589i)T \)
41 \( 1 + (0.203 + 0.979i)T \)
43 \( 1 + (0.994 + 0.103i)T \)
47 \( 1 + (0.880 - 0.474i)T \)
53 \( 1 + (0.898 + 0.439i)T \)
59 \( 1 + (-0.999 + 0.00650i)T \)
61 \( 1 + (0.746 - 0.665i)T \)
67 \( 1 + (-0.977 - 0.212i)T \)
71 \( 1 + (-0.932 - 0.362i)T \)
73 \( 1 + (0.898 - 0.439i)T \)
79 \( 1 + (0.228 + 0.973i)T \)
83 \( 1 + (-0.837 + 0.547i)T \)
89 \( 1 + (0.304 + 0.952i)T \)
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

−22.131059618893524674685246449177, −21.232799518155019148355043921699, −20.7133769435159041672417007468, −19.18721800931315439413937656117, −18.31679152911439827750022678460, −17.51382274093773946070938618512, −17.28615246331170875146517168289, −16.55191655702654222087121953390, −15.70738595507165612969200272396, −14.67614258642922356833075750404, −13.90280534193256454985925107917, −13.37016229539340991391669349458, −12.35033996342748969142219680187, −11.52945534053739200782608956552, −10.42940002330281793662715955874, −9.687041957181909786512646573875, −8.94191213038657333018068354745, −7.46348812790758672622280066043, −7.10958194916634002931313676186, −6.098090242371874941512870523958, −5.48324288111680379797377231595, −4.52859290015855396347954604131, −3.93366172472257609244758253172, −1.943919315093381648174205529203, −0.939903511539417665720343390298, 1.07254143480124736036848200220, 1.691682443891480219687426338407, 2.940080379379590130193520437419, 3.97878257228049062881362102197, 5.1782371407559765923241660337, 5.71604652142603991745137671042, 6.22616193552589164721226904509, 7.9752673503141738193559735651, 8.81420701475038212328517591741, 9.79500025385279181459906472888, 10.49979753705865988668458016710, 11.081693294675218500029196218696, 12.163827129115064262779937986603, 12.42331647274666468690558189161, 13.55114299861916503425887330401, 14.26309314855510605690012277875, 15.10532976763642938571031708801, 16.377830252442382041022537251999, 17.11504771257668556788412535399, 17.90364143608455925380626334876, 18.4589920771949678843218479591, 18.9882159583438132689481731, 20.36830434534462292995365896504, 21.116487026172746537764221843739, 21.51724247951227617226975777820

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