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Properties

Label 1-79-79.35-r1-0-0
Degree $1$
Conductor $79$
Sign $0.828 + 0.559i$
Analytic cond. $8.48972$
Root an. cond. $8.48972$
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  + (0.799 − 0.600i)2-s + (−0.987 + 0.160i)3-s + (0.278 − 0.960i)4-s + (−0.845 + 0.534i)5-s + (−0.692 + 0.721i)6-s + (0.632 + 0.774i)7-s + (−0.354 − 0.935i)8-s + (0.948 − 0.316i)9-s + (−0.354 + 0.935i)10-s + (−0.0402 + 0.999i)11-s + (−0.120 + 0.992i)12-s + (0.692 + 0.721i)13-s + (0.970 + 0.239i)14-s + (0.748 − 0.663i)15-s + (−0.845 − 0.534i)16-s + (0.970 − 0.239i)17-s + ⋯
L(s)  = 1  + (0.799 − 0.600i)2-s + (−0.987 + 0.160i)3-s + (0.278 − 0.960i)4-s + (−0.845 + 0.534i)5-s + (−0.692 + 0.721i)6-s + (0.632 + 0.774i)7-s + (−0.354 − 0.935i)8-s + (0.948 − 0.316i)9-s + (−0.354 + 0.935i)10-s + (−0.0402 + 0.999i)11-s + (−0.120 + 0.992i)12-s + (0.692 + 0.721i)13-s + (0.970 + 0.239i)14-s + (0.748 − 0.663i)15-s + (−0.845 − 0.534i)16-s + (0.970 − 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.828 + 0.559i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ 0.828 + 0.559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.525588676 + 0.4670199416i\)
\(L(\frac12)\) \(\approx\) \(1.525588676 + 0.4670199416i\)
\(L(1)\) \(\approx\) \(1.179850630 + 0.01565607375i\)
\(L(1)\) \(\approx\) \(1.179850630 + 0.01565607375i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.799 - 0.600i)T \)
3 \( 1 + (-0.987 + 0.160i)T \)
5 \( 1 + (-0.845 + 0.534i)T \)
7 \( 1 + (0.632 + 0.774i)T \)
11 \( 1 + (-0.0402 + 0.999i)T \)
13 \( 1 + (0.692 + 0.721i)T \)
17 \( 1 + (0.970 - 0.239i)T \)
19 \( 1 + (0.428 + 0.903i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.200 + 0.979i)T \)
31 \( 1 + (-0.919 - 0.391i)T \)
37 \( 1 + (0.996 + 0.0804i)T \)
41 \( 1 + (-0.885 - 0.464i)T \)
43 \( 1 + (0.0402 + 0.999i)T \)
47 \( 1 + (0.996 - 0.0804i)T \)
53 \( 1 + (-0.987 - 0.160i)T \)
59 \( 1 + (-0.278 - 0.960i)T \)
61 \( 1 + (-0.568 + 0.822i)T \)
67 \( 1 + (0.120 - 0.992i)T \)
71 \( 1 + (0.354 + 0.935i)T \)
73 \( 1 + (0.692 - 0.721i)T \)
83 \( 1 + (0.278 - 0.960i)T \)
89 \( 1 + (-0.354 + 0.935i)T \)
97 \( 1 + (0.568 - 0.822i)T \)
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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

−30.504405751307315263508573695089, −30.1605469372098192389289125148, −28.689565529458620128856297356873, −27.4444185331351406779528145959, −26.64589229048219272796562261872, −24.93355928947061111529888649511, −23.829428925813330875960112884862, −23.57234146116530295709817334880, −22.41024381251541947356057548987, −21.21577675181119760057036888764, −20.14124630292668550806065744437, −18.4099779346687739926522741405, −17.09603208639407612389340101601, −16.379499573385855214929353437456, −15.41382547836922239167955728117, −13.85024498174565174324003297191, −12.77845438606901277687929190165, −11.64968838019086416913542780846, −10.805785100857055669080877376340, −8.30677355701410418914879860891, −7.41648683913503021829642236612, −5.936512087172196387495458165529, −4.820766071755920714732770076928, −3.662514288816754309914160988738, −0.74043855659084409066917564127, 1.61309134789354586248121809554, 3.62723283010648015902996360251, 4.84483065944185395418595527891, 6.03553960113402106804245178035, 7.45422824018945126818251652356, 9.665621640228371100385598725901, 10.957968577731114906702573772635, 11.801673322619054304892839597757, 12.45503955403316783157779010923, 14.32298487495648012105493615750, 15.29434953249227453817707233338, 16.26225851000495557536321935068, 18.19200762777176562789471515238, 18.74678353129606331149613480443, 20.339606309586518056133514811152, 21.42723212705899177020570262093, 22.35089632212880052279922105800, 23.33357238145802088994256529577, 23.87689213609513153230454150693, 25.4047797874469457851196877701, 27.32308195639324715886406196816, 27.87724206802077859881730318901, 28.828838853060335033070013556718, 30.05358190122667188806776107316, 30.92618456233347670872936054796

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