Properties

Label 1-79-79.62-r0-0-0
Degree $1$
Conductor $79$
Sign $0.896 - 0.442i$
Analytic cond. $0.366874$
Root an. cond. $0.366874$
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  + (0.885 + 0.464i)2-s + (0.120 − 0.992i)3-s + (0.568 + 0.822i)4-s + (−0.354 − 0.935i)5-s + (0.568 − 0.822i)6-s + (0.120 − 0.992i)7-s + (0.120 + 0.992i)8-s + (−0.970 − 0.239i)9-s + (0.120 − 0.992i)10-s + (−0.354 + 0.935i)11-s + (0.885 − 0.464i)12-s + (0.568 + 0.822i)13-s + (0.568 − 0.822i)14-s + (−0.970 + 0.239i)15-s + (−0.354 + 0.935i)16-s + (0.568 + 0.822i)17-s + ⋯
L(s)  = 1  + (0.885 + 0.464i)2-s + (0.120 − 0.992i)3-s + (0.568 + 0.822i)4-s + (−0.354 − 0.935i)5-s + (0.568 − 0.822i)6-s + (0.120 − 0.992i)7-s + (0.120 + 0.992i)8-s + (−0.970 − 0.239i)9-s + (0.120 − 0.992i)10-s + (−0.354 + 0.935i)11-s + (0.885 − 0.464i)12-s + (0.568 + 0.822i)13-s + (0.568 − 0.822i)14-s + (−0.970 + 0.239i)15-s + (−0.354 + 0.935i)16-s + (0.568 + 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.896 - 0.442i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.896 - 0.442i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.444398663 - 0.3372884727i\)
\(L(\frac12)\) \(\approx\) \(1.444398663 - 0.3372884727i\)
\(L(1)\) \(\approx\) \(1.516129310 - 0.1937475403i\)
\(L(1)\) \(\approx\) \(1.516129310 - 0.1937475403i\)

Euler product

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

−31.557948620789156469648676512190, −30.300438310636192518205359519975, −29.23512290393008067534111952500, −27.93588703472278199030458635688, −27.22834154251162789682972655934, −25.80336783484392374309166782199, −24.76254561864511906087696242877, −23.109301877516537801440928705570, −22.52703258802071864509857555113, −21.4308473258473635600634954379, −20.79224442027076361667761277597, −19.28693820218310258002995518314, −18.450819908174021513469299097272, −16.32711881948154317373118358704, −15.28272084882474959630071841818, −14.72886538401639194738722702363, −13.40990767043205680400057267411, −11.729297894168102012296507341849, −10.96882968680004616498087417560, −9.89949106246472966827637353262, −8.24419634717127714705239525990, −6.17482063839635000387675903527, −5.169118415177485236621129315605, −3.49718211521312403325656159375, −2.74313011299499568697965106567, 1.73486454520547861344368694780, 3.78190725048554460251150191097, 5.04535137775583192083814468222, 6.66882797796405377067145256426, 7.600266901038835589881114288, 8.7245492089306210619523868508, 11.0596664852716683513840574637, 12.37162842067562318863031654605, 13.068213422128560198578325241059, 14.05751416964848187365453029488, 15.35225671178249942907824716439, 16.82249566436177805509331027670, 17.39865161084635814614820760199, 19.24001375040785574164851423531, 20.34223330381299476662480417187, 21.10978913751476242283353994857, 23.040770010246387161091115050407, 23.559186966218491439982638726227, 24.27737721897493208806901452724, 25.48487658129374492339140061669, 26.29149039917001539589701019099, 28.12284707059795202667744218577, 29.192511660741367526025004224202, 30.355158838814652840865889258738, 30.974314135009070177081040194429

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