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Properties

Label 1-79-79.34-r1-0-0
Degree $1$
Conductor $79$
Sign $0.760 + 0.649i$
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  + (−0.200 + 0.979i)2-s + (−0.428 + 0.903i)3-s + (−0.919 − 0.391i)4-s + (0.692 − 0.721i)5-s + (−0.799 − 0.600i)6-s + (0.996 − 0.0804i)7-s + (0.568 − 0.822i)8-s + (−0.632 − 0.774i)9-s + (0.568 + 0.822i)10-s + (0.278 − 0.960i)11-s + (0.748 − 0.663i)12-s + (0.799 − 0.600i)13-s + (−0.120 + 0.992i)14-s + (0.354 + 0.935i)15-s + (0.692 + 0.721i)16-s + (−0.120 − 0.992i)17-s + ⋯
L(s)  = 1  + (−0.200 + 0.979i)2-s + (−0.428 + 0.903i)3-s + (−0.919 − 0.391i)4-s + (0.692 − 0.721i)5-s + (−0.799 − 0.600i)6-s + (0.996 − 0.0804i)7-s + (0.568 − 0.822i)8-s + (−0.632 − 0.774i)9-s + (0.568 + 0.822i)10-s + (0.278 − 0.960i)11-s + (0.748 − 0.663i)12-s + (0.799 − 0.600i)13-s + (−0.120 + 0.992i)14-s + (0.354 + 0.935i)15-s + (0.692 + 0.721i)16-s + (−0.120 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.760 + 0.649i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ 0.760 + 0.649i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.342981938 + 0.4959136202i\)
\(L(\frac12)\) \(\approx\) \(1.342981938 + 0.4959136202i\)
\(L(1)\) \(\approx\) \(0.9345225689 + 0.4009393996i\)
\(L(1)\) \(\approx\) \(0.9345225689 + 0.4009393996i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.200 + 0.979i)T \)
3 \( 1 + (-0.428 + 0.903i)T \)
5 \( 1 + (0.692 - 0.721i)T \)
7 \( 1 + (0.996 - 0.0804i)T \)
11 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + (0.799 - 0.600i)T \)
17 \( 1 + (-0.120 - 0.992i)T \)
19 \( 1 + (-0.0402 + 0.999i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.987 - 0.160i)T \)
31 \( 1 + (0.948 - 0.316i)T \)
37 \( 1 + (0.845 + 0.534i)T \)
41 \( 1 + (0.970 + 0.239i)T \)
43 \( 1 + (-0.278 - 0.960i)T \)
47 \( 1 + (0.845 - 0.534i)T \)
53 \( 1 + (-0.428 - 0.903i)T \)
59 \( 1 + (0.919 - 0.391i)T \)
61 \( 1 + (-0.885 + 0.464i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (-0.568 + 0.822i)T \)
73 \( 1 + (0.799 + 0.600i)T \)
83 \( 1 + (-0.919 - 0.391i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (0.885 - 0.464i)T \)
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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

−30.35651388118727725761126265212, −30.025076639339389326925538993893, −28.431765388840263958117093667582, −28.19886247630438799261187975247, −26.489523136546334185983222013016, −25.554798787844721783886307538088, −24.12103413841027559091475272825, −23.000277972525784286148311717085, −22.00582921126371978348044209946, −20.98961344955570377353532235926, −19.69245235000990150909481217548, −18.49862417357144713339162903875, −17.85240984055586654525806518222, −17.09138780326974833208975812384, −14.65965646295544857927161566774, −13.70260697902687291052219868288, −12.58239757517847375083280219379, −11.356055348469096392374913701243, −10.62293638484590045836762584716, −9.00750041117053379848686100882, −7.61565833643022893237295253302, −6.145239095921927337880274622559, −4.50418512697428786725964243114, −2.375759432504012742078100326, −1.4576861452832930472662051051, 0.92705625072466339655328569418, 4.01531834902646804276169007457, 5.3207929183239134549971384652, 5.98740906726099376467316911630, 8.075041537880086552586156362996, 9.04773106906583958663763814934, 10.18755065674849831748725668335, 11.562330825356722780211338776875, 13.45740449070919542445978069259, 14.39388984705548576139046839204, 15.70177273901966536669570589504, 16.60247906006656333257012184549, 17.44605298369547669898858990505, 18.39860214718941598266175855571, 20.400536295242558609494339504777, 21.3097050868020636892398044824, 22.40627282671525517043425447236, 23.60590950727690577008369266024, 24.60415216163621417929241817628, 25.55341934711604727660409600192, 26.89925588101855999524690577153, 27.55953874291454693509763582400, 28.42877002849317817679902563659, 29.75557748010110324012814050584, 31.60443103251021103897568961884

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