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Properties

Label 1-79-79.57-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.660 - 0.750i$
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  + (−0.354 − 0.935i)2-s + (−0.885 + 0.464i)3-s + (−0.748 + 0.663i)4-s + (0.120 + 0.992i)5-s + (0.748 + 0.663i)6-s + (−0.885 + 0.464i)7-s + (0.885 + 0.464i)8-s + (0.568 − 0.822i)9-s + (0.885 − 0.464i)10-s + (0.120 − 0.992i)11-s + (0.354 − 0.935i)12-s + (−0.748 + 0.663i)13-s + (0.748 + 0.663i)14-s + (−0.568 − 0.822i)15-s + (0.120 − 0.992i)16-s + (0.748 − 0.663i)17-s + ⋯
L(s)  = 1  + (−0.354 − 0.935i)2-s + (−0.885 + 0.464i)3-s + (−0.748 + 0.663i)4-s + (0.120 + 0.992i)5-s + (0.748 + 0.663i)6-s + (−0.885 + 0.464i)7-s + (0.885 + 0.464i)8-s + (0.568 − 0.822i)9-s + (0.885 − 0.464i)10-s + (0.120 − 0.992i)11-s + (0.354 − 0.935i)12-s + (−0.748 + 0.663i)13-s + (0.748 + 0.663i)14-s + (−0.568 − 0.822i)15-s + (0.120 − 0.992i)16-s + (0.748 − 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.660 - 0.750i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.660 - 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1389932670 - 0.3075866378i\)
\(L(\frac12)\) \(\approx\) \(0.1389932670 - 0.3075866378i\)
\(L(1)\) \(\approx\) \(0.4980249967 - 0.1084811588i\)
\(L(1)\) \(\approx\) \(0.4980249967 - 0.1084811588i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.354 - 0.935i)T \)
3 \( 1 + (-0.885 + 0.464i)T \)
5 \( 1 + (0.120 + 0.992i)T \)
7 \( 1 + (-0.885 + 0.464i)T \)
11 \( 1 + (0.120 - 0.992i)T \)
13 \( 1 + (-0.748 + 0.663i)T \)
17 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 + (-0.970 - 0.239i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.568 - 0.822i)T \)
31 \( 1 + (-0.354 - 0.935i)T \)
37 \( 1 + (0.970 + 0.239i)T \)
41 \( 1 + (-0.120 - 0.992i)T \)
43 \( 1 + (-0.120 - 0.992i)T \)
47 \( 1 + (0.970 - 0.239i)T \)
53 \( 1 + (-0.885 - 0.464i)T \)
59 \( 1 + (0.748 + 0.663i)T \)
61 \( 1 + (0.970 + 0.239i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (-0.885 - 0.464i)T \)
73 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (-0.748 + 0.663i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (-0.970 - 0.239i)T \)
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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

−31.61495434756636762335000151973, −29.94349529170666999254817634423, −28.83234790013491922141736583109, −28.05711805345502103784768789961, −27.14720656579651580024709599262, −25.49450613148545952509175323959, −24.96044824357733253198381871811, −23.58457724612405810134887738607, −23.14680512479338762527184206988, −21.888772524723215759477032784623, −20.01393395945200565998118636567, −19.04090596396680022035539057928, −17.58923548684618186278179330949, −16.95742801564062432340533976284, −16.170465970307625257087189342748, −14.76429113178861152915083667893, −12.93733626167002454864728863171, −12.65354636348018617026028834503, −10.47333374851145509335807944429, −9.48746006628103506140458860075, −7.87715550738072069202179210375, −6.7984322498652612730890350396, −5.579870454797090482301598725659, −4.478840285529506640393193816643, −1.24679241836164301455604394500, 0.23164422706190236757847202922, 2.630697876807432469328052707297, 3.90160613320901683087475888984, 5.68389809389448818440592212329, 7.07024858450092905270528899417, 9.15032480051172945783572943292, 10.055738815694986181135774956193, 11.14411697560900028372248061507, 11.9795788105470610561819334247, 13.3348949426141124685813991700, 14.876044183132099600590997893776, 16.41478942089533439769116045142, 17.29096672646635735887069070505, 18.83348222794778607394290512035, 19.01047703972510151988308943407, 20.95882281294814341464812328148, 21.940956127804868968022564436638, 22.404513576303640739538803269778, 23.551162332478216167661572357919, 25.50882502685470265347819895658, 26.64443080277576408123134678218, 27.276228259217734393638868495132, 28.55206988745933938990337736109, 29.38910840987557294383753207095, 29.94875770013581197002868534710

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