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Properties

Label 1-79-79.18-r0-0-0
Degree $1$
Conductor $79$
Sign $0.404 + 0.914i$
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  + (−0.354 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.404 + 0.914i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.404 + 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8825092257 + 0.5743154543i\)
\(L(\frac12)\) \(\approx\) \(0.8825092257 + 0.5743154543i\)
\(L(1)\) \(\approx\) \(0.9969157971 + 0.4748933399i\)
\(L(1)\) \(\approx\) \(0.9969157971 + 0.4748933399i\)

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

−30.84482824408036474276959846067, −29.78515802619556594875497919112, −29.34476903388655032380439084415, −27.505500067922392954478311069623, −26.62136593079335436790221273430, −26.07589626630201634788032572434, −24.53246169255018862581270553356, −23.37852655594423011440469586090, −21.75997774583398856754900287395, −21.18913558387276585937975271661, −19.74433033753024191621040889187, −19.105763770027033308821173633257, −18.09132599791252677909162510199, −17.054410811713837561311977936326, −14.84589884475634245278082470249, −14.06264340199736664466962973085, −13.02667534828304421750170396101, −11.457077042734238032970215689050, −10.580949317193561250623091998228, −9.110696614198989771609296550336, −8.03653370179187525338536577109, −6.8121249878789989953803934053, −4.28381858289459688994744020315, −2.94429597105843792808958976361, −1.740094964088105303903951054026, 1.95280110995363658275752394134, 4.50200033444467481723880395957, 5.16220309966862697904267318810, 7.27251236141769240788990873697, 8.43428219798663900946804761583, 9.14960237639424294042112153192, 10.35795667792409209210820927409, 12.49331919601952477879568025953, 13.75665063417813676064016283315, 14.96752033860511401070889074995, 15.55461312344631652933985228440, 16.97735251855233792238782064184, 17.85182062425873516419579449586, 19.36471915386636520906519540238, 20.36731778177274815817965536947, 21.42898027444215658441664051307, 22.855622398605110112039289144655, 24.38184510756829906484883552138, 24.90035449848640417703474123639, 25.73108637144433168504888332162, 27.32079585867227904306628821343, 27.50759194577995216457048366127, 28.823337423625366558544232656131, 30.779792338055195009190281293332, 31.57994726255304891733945484343

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