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Properties

Label 1-967-967.440-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.121 - 0.992i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (−0.857 − 0.514i)2-s + (−0.737 + 0.675i)3-s + (0.471 + 0.881i)4-s + (0.870 + 0.491i)5-s + (0.979 − 0.200i)6-s + (−0.991 + 0.129i)7-s + (0.0487 − 0.998i)8-s + (0.0876 − 0.996i)9-s + (−0.494 − 0.869i)10-s + (−0.724 − 0.689i)11-s + (−0.943 − 0.331i)12-s + (0.235 + 0.971i)13-s + (0.917 + 0.398i)14-s + (−0.974 + 0.225i)15-s + (−0.555 + 0.831i)16-s + (−0.145 − 0.989i)17-s + ⋯
L(s)  = 1  + (−0.857 − 0.514i)2-s + (−0.737 + 0.675i)3-s + (0.471 + 0.881i)4-s + (0.870 + 0.491i)5-s + (0.979 − 0.200i)6-s + (−0.991 + 0.129i)7-s + (0.0487 − 0.998i)8-s + (0.0876 − 0.996i)9-s + (−0.494 − 0.869i)10-s + (−0.724 − 0.689i)11-s + (−0.943 − 0.331i)12-s + (0.235 + 0.971i)13-s + (0.917 + 0.398i)14-s + (−0.974 + 0.225i)15-s + (−0.555 + 0.831i)16-s + (−0.145 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.121 - 0.992i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.121 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1413267918 - 0.1596788105i\)
\(L(\frac12)\) \(\approx\) \(0.1413267918 - 0.1596788105i\)
\(L(1)\) \(\approx\) \(0.5040424723 + 0.06715870903i\)
\(L(1)\) \(\approx\) \(0.5040424723 + 0.06715870903i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.857 - 0.514i)T \)
3 \( 1 + (-0.737 + 0.675i)T \)
5 \( 1 + (0.870 + 0.491i)T \)
7 \( 1 + (-0.991 + 0.129i)T \)
11 \( 1 + (-0.724 - 0.689i)T \)
13 \( 1 + (0.235 + 0.971i)T \)
17 \( 1 + (-0.145 - 0.989i)T \)
19 \( 1 + (-0.989 + 0.142i)T \)
23 \( 1 + (0.945 + 0.325i)T \)
29 \( 1 + (0.511 + 0.859i)T \)
31 \( 1 + (0.100 + 0.994i)T \)
37 \( 1 + (0.209 - 0.977i)T \)
41 \( 1 + (-0.854 - 0.519i)T \)
43 \( 1 + (0.566 + 0.824i)T \)
47 \( 1 + (-0.779 + 0.626i)T \)
53 \( 1 + (-0.648 + 0.761i)T \)
59 \( 1 + (0.254 + 0.967i)T \)
61 \( 1 + (-0.877 + 0.480i)T \)
67 \( 1 + (-0.811 - 0.584i)T \)
71 \( 1 + (0.874 + 0.485i)T \)
73 \( 1 + (-0.648 - 0.761i)T \)
79 \( 1 + (-0.886 + 0.462i)T \)
83 \( 1 + (-0.767 - 0.641i)T \)
89 \( 1 + (0.911 - 0.410i)T \)
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

−21.86943954170882644750003051266, −20.7718238614521859222066570368, −20.02832349378152721008495477039, −19.09768997698687700611588721963, −18.55958567438012636064340523849, −17.55804415005529908479992687039, −17.23469607709187524852552918474, −16.57243616663168438057751956581, −15.61844765350067070368287982014, −14.929912637606202901973107496852, −13.45031655952078035506011524535, −13.07155829947506380328715212173, −12.32904749647173923589772765062, −10.99696649989781772689918684192, −10.25373571462801029943420568754, −9.817344331821362173227471720603, −8.56321227948836572252302211793, −7.92713222856774201865215840695, −6.75803797835826853889961214290, −6.283413929552448413819621661278, −5.52562276042579210678974380565, −4.65126566054152663603328459536, −2.68407903395566429395689368470, −1.84212485183974631218442409908, −0.738341135985246074531006587606, 0.09347913348695071079651098879, 1.37136771355295466578067813947, 2.7544051917837499772271537479, 3.27666681452953189910098499605, 4.56769538676008103443177500799, 5.77258163591584710984140851278, 6.55179608736529719608294162619, 7.170342892084456460403565246775, 8.884419974464314463155976219951, 9.18980524840608467336025055808, 10.13311133286050188695703081933, 10.71983767384750998521914970028, 11.32782083911859421198952889718, 12.39512615599998302142204047485, 13.12785401540990996370591716183, 14.10178953820092785245070340286, 15.36000680923869660216045752645, 16.208549051532098347339308201580, 16.55602150315267597556117335863, 17.47838512641151473368477978075, 18.23483060624880925147424801868, 18.82172133424203631423914712608, 19.610099535806971332962386781270, 20.84503455445816880074669355621, 21.366292817771522676050587491495

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