Properties

Degree 1
Conductor 79
Sign $0.998 + 0.0565i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.948 + 0.316i)2-s + (0.996 + 0.0804i)3-s + (0.799 + 0.600i)4-s + (0.278 − 0.960i)5-s + (0.919 + 0.391i)6-s + (−0.428 − 0.903i)7-s + (0.568 + 0.822i)8-s + (0.987 + 0.160i)9-s + (0.568 − 0.822i)10-s + (0.692 − 0.721i)11-s + (0.748 + 0.663i)12-s + (−0.919 + 0.391i)13-s + (−0.120 − 0.992i)14-s + (0.354 − 0.935i)15-s + (0.278 + 0.960i)16-s + (−0.120 + 0.992i)17-s + ⋯
L(s,χ)  = 1  + (0.948 + 0.316i)2-s + (0.996 + 0.0804i)3-s + (0.799 + 0.600i)4-s + (0.278 − 0.960i)5-s + (0.919 + 0.391i)6-s + (−0.428 − 0.903i)7-s + (0.568 + 0.822i)8-s + (0.987 + 0.160i)9-s + (0.568 − 0.822i)10-s + (0.692 − 0.721i)11-s + (0.748 + 0.663i)12-s + (−0.919 + 0.391i)13-s + (−0.120 − 0.992i)14-s + (0.354 − 0.935i)15-s + (0.278 + 0.960i)16-s + (−0.120 + 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.998 + 0.0565i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.998 + 0.0565i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $0.998 + 0.0565i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (3, \cdot )$
Sato-Tate  :  $\mu(78)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 79,\ (1:\ ),\ 0.998 + 0.0565i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.981488930 + 0.1127215123i$
$L(\frac12,\chi)$  $\approx$  $3.981488930 + 0.1127215123i$
$L(\chi,1)$  $\approx$  2.495379437 + 0.1279703292i
$L(1,\chi)$  $\approx$  2.495379437 + 0.1279703292i

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

−30.82196548913587510345557011164, −30.12509302125876323314967027232, −29.166365992555334269212787140548, −27.7257704396535464251913037731, −26.31124593957576377151927776603, −25.13703508447381180533246587533, −24.759840014671910298998820080, −22.99226831733807102368124385709, −22.10566516193993192204228813250, −21.286861428655952380567090882368, −19.89806117855751889473094716929, −19.21886724264275295129066383370, −17.99012570716029076392075039066, −15.836337878031411002183841857545, −14.852967932515795301134989049442, −14.29120041465833185862743963087, −12.93976812435371181360903269670, −11.95012534850251695696008444317, −10.29689741281446110834422431235, −9.30331001768519665641411624227, −7.31258049336690335058498614136, −6.31465895881721791128850773416, −4.50510319126903745202170771668, −2.907638380526986143336438014857, −2.23933403024223979640157070862, 1.771627309976480566646381271320, 3.596389346409099070296424332605, 4.49098431525697461210130713234, 6.23830163421459634823575896394, 7.637236827933848018607246025, 8.828145656877882635852938153325, 10.27328236555705716873502166212, 12.14578640691844733650988825156, 13.225068922629716485554270564347, 13.95603366753778951259875768757, 15.05244354620252478059607102291, 16.44916481158917333255930249089, 17.0604896099547262415179252803, 19.475412010394310524613908039575, 20.02233673586323147737213883196, 21.22407302916777874114898438839, 21.96317881328653087327906968626, 23.68197591253712166020996072599, 24.326935386071937580781511072680, 25.37044523468246805451558167369, 26.24479404437068864221510001088, 27.46397948658630723281659590485, 29.31447360042910112185981338458, 29.85382003294739447687195213092, 31.19810605857676645221879743837

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