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
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
$p$ | $F_p(T)$ | |
---|---|---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.442 - 0.896i)T \) |
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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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