Properties

Label 1-967-967.158-r1-0-0
Degree $1$
Conductor $967$
Sign $0.314 - 0.949i$
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  + (−0.442 − 0.896i)2-s + (0.984 + 0.174i)3-s + (−0.608 + 0.793i)4-s + (−0.999 + 0.0390i)5-s + (−0.279 − 0.960i)6-s + (0.864 + 0.502i)7-s + (0.981 + 0.193i)8-s + (0.938 + 0.344i)9-s + (0.477 + 0.878i)10-s + (−0.995 + 0.0974i)11-s + (−0.737 + 0.675i)12-s + (0.995 + 0.0974i)13-s + (0.0682 − 0.997i)14-s + (−0.990 − 0.136i)15-s + (−0.260 − 0.965i)16-s + (0.833 − 0.552i)17-s + ⋯
L(s)  = 1  + (−0.442 − 0.896i)2-s + (0.984 + 0.174i)3-s + (−0.608 + 0.793i)4-s + (−0.999 + 0.0390i)5-s + (−0.279 − 0.960i)6-s + (0.864 + 0.502i)7-s + (0.981 + 0.193i)8-s + (0.938 + 0.344i)9-s + (0.477 + 0.878i)10-s + (−0.995 + 0.0974i)11-s + (−0.737 + 0.675i)12-s + (0.995 + 0.0974i)13-s + (0.0682 − 0.997i)14-s + (−0.990 − 0.136i)15-s + (−0.260 − 0.965i)16-s + (0.833 − 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.314 - 0.949i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.314 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.720021801 - 1.241705015i\)
\(L(\frac12)\) \(\approx\) \(1.720021801 - 1.241705015i\)
\(L(1)\) \(\approx\) \(1.067782279 - 0.3579587514i\)
\(L(1)\) \(\approx\) \(1.067782279 - 0.3579587514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.442 + 0.896i)T \)
3 \( 1 + (-0.984 - 0.174i)T \)
5 \( 1 + (0.999 - 0.0390i)T \)
7 \( 1 + (-0.864 - 0.502i)T \)
11 \( 1 + (0.995 - 0.0974i)T \)
13 \( 1 + (-0.995 - 0.0974i)T \)
17 \( 1 + (-0.833 + 0.552i)T \)
19 \( 1 + (0.0487 + 0.998i)T \)
23 \( 1 + (0.241 + 0.970i)T \)
29 \( 1 + (-0.544 - 0.838i)T \)
31 \( 1 + (0.799 + 0.600i)T \)
37 \( 1 + (0.316 - 0.948i)T \)
41 \( 1 + (-0.576 + 0.816i)T \)
43 \( 1 + (0.951 - 0.307i)T \)
47 \( 1 + (0.0876 + 0.996i)T \)
53 \( 1 + (-0.203 + 0.979i)T \)
59 \( 1 + (0.999 + 0.0195i)T \)
61 \( 1 + (0.576 - 0.816i)T \)
67 \( 1 + (-0.799 + 0.600i)T \)
71 \( 1 + (0.442 - 0.896i)T \)
73 \( 1 + (-0.203 - 0.979i)T \)
79 \( 1 + (-0.638 + 0.769i)T \)
83 \( 1 + (-0.165 + 0.986i)T \)
89 \( 1 + (-0.799 + 0.600i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−21.44003495556870318812234923650, −20.80210428753886270200974343843, −19.95947842348496165020853053598, −19.269969712762811004081294270702, −18.472083955564011799344818330135, −18.02489658550600143609790558037, −16.8350643436464861372863688062, −15.967877807869701098832758038699, −15.453363161740422053175638662135, −14.64736421416765404730650103393, −14.01699177715671362777652253867, −13.22870551224621176677537010650, −12.2558421380471480221035046499, −10.92043591758973509112243074886, −10.360186041170994633806440551840, −9.23631439706026710669330710163, −8.201442797530214732615392405, −7.93495230250526962194941004806, −7.42709954668507367332175541263, −6.175847858703284773106254591239, −5.08315751366946627014982215154, −4.06711156877389284849997535493, −3.40880435927094665448954237050, −1.734184665504875270741862483367, −0.90240117162035795539561720463, 0.58953076086137524709843453196, 1.777011598086052078678544197852, 2.7612238915888032005992638084, 3.42320505701740826275602369531, 4.460837302505205281135354495811, 5.11135817999458647552222854764, 7.078616488397511275255618588922, 7.865980609545809161575272538, 8.47493464258146185030101340783, 8.97172563566037901503579441833, 10.189268956545417791679758340, 10.88049880331173989815722999164, 11.65003712612662414312709139177, 12.514332268922244434174636327369, 13.30432526493791751855728738367, 14.18576458911062755186890791237, 15.04235319296200956279628815324, 15.85180645643371400205476221330, 16.52091317629938305199809893633, 17.97064597023444325720067175492, 18.55497109219687475754087913409, 18.913563295931265293105373509648, 20.11432360393348378568609160854, 20.39580663058053550975649711459, 21.17345914158546191168469355986

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