Dirichlet series
L(s) = 1 | + (0.145 − 0.989i)2-s + (0.719 − 0.694i)3-s + (−0.957 − 0.288i)4-s + (0.551 + 0.833i)5-s + (−0.581 − 0.813i)6-s + (−0.967 + 0.252i)7-s + (−0.424 + 0.905i)8-s + (0.0365 − 0.999i)9-s + (0.905 − 0.424i)10-s + (0.217 − 0.976i)11-s + (−0.889 + 0.457i)12-s + (−0.989 − 0.145i)13-s + (0.109 + 0.994i)14-s + (0.976 + 0.217i)15-s + (0.833 + 0.551i)16-s + (−0.946 + 0.322i)17-s + ⋯ |
L(s) = 1 | + (0.145 − 0.989i)2-s + (0.719 − 0.694i)3-s + (−0.957 − 0.288i)4-s + (0.551 + 0.833i)5-s + (−0.581 − 0.813i)6-s + (−0.967 + 0.252i)7-s + (−0.424 + 0.905i)8-s + (0.0365 − 0.999i)9-s + (0.905 − 0.424i)10-s + (0.217 − 0.976i)11-s + (−0.889 + 0.457i)12-s + (−0.989 − 0.145i)13-s + (0.109 + 0.994i)14-s + (0.976 + 0.217i)15-s + (0.833 + 0.551i)16-s + (−0.946 + 0.322i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(173\) |
Sign: | $-0.375 + 0.926i$ |
Analytic conductor: | \(18.5914\) |
Root analytic conductor: | \(18.5914\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{173} (104, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 173,\ (1:\ ),\ -0.375 + 0.926i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(-0.2390796228 - 0.3546719658i\) |
\(L(\frac12)\) | \(\approx\) | \(-0.2390796228 - 0.3546719658i\) |
\(L(1)\) | \(\approx\) | \(0.7164837256 - 0.5889299542i\) |
\(L(1)\) | \(\approx\) | \(0.7164837256 - 0.5889299542i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 173 | \( 1 \) |
good | 2 | \( 1 + (0.145 - 0.989i)T \) |
3 | \( 1 + (0.719 - 0.694i)T \) | |
5 | \( 1 + (0.551 + 0.833i)T \) | |
7 | \( 1 + (-0.967 + 0.252i)T \) | |
11 | \( 1 + (0.217 - 0.976i)T \) | |
13 | \( 1 + (-0.989 - 0.145i)T \) | |
17 | \( 1 + (-0.946 + 0.322i)T \) | |
19 | \( 1 + (-0.994 - 0.109i)T \) | |
23 | \( 1 + (-0.976 + 0.217i)T \) | |
29 | \( 1 + (-0.581 + 0.813i)T \) | |
31 | \( 1 + (0.694 - 0.719i)T \) | |
37 | \( 1 + (-0.109 + 0.994i)T \) | |
41 | \( 1 + (-0.252 - 0.967i)T \) | |
43 | \( 1 + (0.957 - 0.288i)T \) | |
47 | \( 1 + (-0.997 + 0.0729i)T \) | |
53 | \( 1 + (0.611 - 0.791i)T \) | |
59 | \( 1 + (-0.489 - 0.872i)T \) | |
61 | \( 1 + (0.946 + 0.322i)T \) | |
67 | \( 1 + (0.694 + 0.719i)T \) | |
71 | \( 1 + (-0.813 - 0.581i)T \) | |
73 | \( 1 + (-0.639 + 0.768i)T \) | |
79 | \( 1 + (0.0729 - 0.997i)T \) | |
83 | \( 1 + (0.391 + 0.920i)T \) | |
89 | \( 1 + (-0.744 + 0.667i)T \) | |
97 | \( 1 + (-0.920 - 0.391i)T \) | |
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Imaginary part of the first few zeros on the critical line
−27.75278980723116640846833854355, −26.56751281156011655048574607641, −25.9639922252673354265635008620, −25.04593206378706768968581300188, −24.48121290343645280231053709263, −23.04889459017707002775009252687, −22.17774592706671973557076166218, −21.32873998841015853560353845315, −20.11727844725122918125368345375, −19.354139928079002583034772850143, −17.72866177139291599935294807250, −16.86089395048067963814508239365, −16.073929199965709493326029446796, −15.1668297603816255172154573666, −14.15542559231544841741298911086, −13.21266617907290684376451227320, −12.41665550143515366583430663128, −10.076058036319099529695013058, −9.52842336564057662799885502346, −8.6388469707746644028841651985, −7.38257433707958345780291494582, −6.17003432920261229591671649561, −4.74540903686573983508051654767, −4.11630751158695460182202552637, −2.347855216831684376275995198041, 0.12200828245296004936295235926, 2.01530046831295775918184609695, 2.81213051420108572499374091860, 3.82408289580731300448006175287, 5.84916262721328418138110456444, 6.78820930412644789233966761493, 8.40567329241426223490012364925, 9.40594539396838759719869110181, 10.30137522035947324868159046252, 11.565872517800283263675424929710, 12.736084316031437415550449559272, 13.450511939939319559875326374146, 14.32581716322255190861221614961, 15.26648647925323932517785296152, 17.1783611396067978212024969607, 18.14850318609746879102890372677, 19.24069290716392183734521779498, 19.36167558606358954534333728016, 20.66980622054308332371075997615, 21.96191218790375605427100452594, 22.24695302605249006214572138217, 23.66444208471388720010142719302, 24.625508212832758037505554660696, 25.9592742807123321494021137935, 26.370710153306917526173679332112