Properties

Degree 1
Conductor 79
Sign $0.404 - 0.914i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $0.404 - 0.914i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (22, \cdot )$
Sato-Tate  :  $\mu(13)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 79,\ (0:\ ),\ 0.404 - 0.914i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8825092257 - 0.5743154543i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8825092257 - 0.5743154543i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9969157971 - 0.4748933399i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9969157971 - 0.4748933399i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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