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Properties

Label 1-183-183.35-r0-0-0
Degree $1$
Conductor $183$
Sign $0.850 - 0.526i$
Analytic cond. $0.849848$
Root an. cond. $0.849848$
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  + (−0.406 − 0.913i)2-s + (−0.669 + 0.743i)4-s + (−0.978 − 0.207i)5-s + (0.994 − 0.104i)7-s + (0.951 + 0.309i)8-s + (0.207 + 0.978i)10-s + i·11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.104 − 0.994i)16-s + (0.743 + 0.669i)17-s + (0.104 − 0.994i)19-s + (0.809 − 0.587i)20-s + (0.913 − 0.406i)22-s + (0.951 − 0.309i)23-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.669 + 0.743i)4-s + (−0.978 − 0.207i)5-s + (0.994 − 0.104i)7-s + (0.951 + 0.309i)8-s + (0.207 + 0.978i)10-s + i·11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.104 − 0.994i)16-s + (0.743 + 0.669i)17-s + (0.104 − 0.994i)19-s + (0.809 − 0.587i)20-s + (0.913 − 0.406i)22-s + (0.951 − 0.309i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $0.850 - 0.526i$
Analytic conductor: \(0.849848\)
Root analytic conductor: \(0.849848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{183} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 183,\ (0:\ ),\ 0.850 - 0.526i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7876177431 - 0.2240359007i\)
\(L(\frac12)\) \(\approx\) \(0.7876177431 - 0.2240359007i\)
\(L(1)\) \(\approx\) \(0.7686205279 - 0.2329881240i\)
\(L(1)\) \(\approx\) \(0.7686205279 - 0.2329881240i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.406 - 0.913i)T \)
5 \( 1 + (-0.978 - 0.207i)T \)
7 \( 1 + (0.994 - 0.104i)T \)
11 \( 1 + iT \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.743 + 0.669i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (0.951 - 0.309i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (0.406 - 0.913i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.743 - 0.669i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (0.406 + 0.913i)T \)
67 \( 1 + (-0.207 + 0.978i)T \)
71 \( 1 + (-0.207 - 0.978i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.743 + 0.669i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (0.587 - 0.809i)T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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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

−27.15991512710439214651874251744, −26.76437672148418640537910530278, −25.18566102732033145874192107299, −24.64435589030134435816932941269, −23.55864577672483960260102713999, −22.99583484580640665598648868799, −21.75919205282271854738439421220, −20.448233923302659086063460317114, −19.30019274739310312528877323235, −18.56970660708212880364902209292, −17.59557226382651973840801103318, −16.50349330714047335999602467138, −15.667343895387494173357647812646, −14.67178803956187365470761919748, −14.02280851117526003842220000846, −12.42450759681506438133398535789, −11.22044145962560527670533194563, −10.27647733347692627071666289061, −8.73700191051586561603525115671, −7.97250587633482331399681282776, −7.20314469163512867237615540359, −5.67069919211999854447371552888, −4.7476639163210782441690367203, −3.25271294145493615525582824008, −0.9809464827903113111565129124, 1.23764487753203287736662079039, 2.65067031725008218003282543846, 4.261771996226873887557936098279, 4.7758536203544226624191994932, 7.156235043640934223351715526, 8.00215497891517613970964288710, 9.015720181381417505546059220878, 10.22072067726243285991098190053, 11.39331441995330443260895344339, 11.95226432369072628703617940024, 12.98930987556873696476697999203, 14.368826088133862759463062406131, 15.33341160250402400081441460034, 16.83273721066252227388696879158, 17.47859511197179019519646853850, 18.7131389928483249247828728996, 19.46466287245816718888863951397, 20.42962404651412809785740581638, 21.12834191167291629680731631616, 22.24231815884229097515171805129, 23.34689888565233650938882227062, 24.07480640843930368886104506557, 25.49664003797355416057947077724, 26.62936527763249184491430390356, 27.27301367050538788123821073091

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