Properties

Degree 1
Conductor 79
Sign $0.944 - 0.328i$
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

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.944 - 0.328i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.944 - 0.328i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $0.944 - 0.328i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (26, \cdot )$
Sato-Tate  :  $\mu(39)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 79,\ (0:\ ),\ 0.944 - 0.328i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.310862636 - 0.2217081046i$
$L(\frac12,\chi)$  $\approx$  $1.310862636 - 0.2217081046i$
$L(\chi,1)$  $\approx$  1.369294340 - 0.1766153088i
$L(1,\chi)$  $\approx$  1.369294340 - 0.1766153088i

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

−31.59948715973941998966559532498, −29.92909358056389036481738498539, −29.29067095019980255312827938850, −28.170713000538587999386343403844, −27.148172750062714448790001697842, −25.218210257138906223517619816606, −24.48399688614748994193588829246, −23.83368987611298911828282346721, −22.44103297854839021931987750596, −21.65729727736459325118222497626, −20.838717679754821093086433842001, −19.177352571063123390501936518199, −17.53439138469023810127265320956, −16.7103165218087269669233642290, −15.80701251047149217210792788220, −14.40929787847206161526424075125, −13.06040170232532005097821803769, −12.02549253102257288249986800552, −11.42385872445042117917091694186, −9.4172359593038585442806606258, −7.79236130944077129365190831128, −6.15971627053376947116629159229, −5.35978074281895904602274954774, −4.2719917877821442588440375374, −1.94909690305608978653562166493, 1.84544500757720920501603815431, 3.82729927941831135229941462618, 4.97859555783026935438727815350, 6.47450258386241233798362931051, 7.195138109379113301155990144809, 10.17487561730313384556641812017, 10.64459928616803950818563990737, 11.87779709181975138050538759144, 12.96784910615332748966884932543, 14.42584331110439679102480691038, 15.14063167822986140536333661548, 16.81996758927689319083540283118, 17.68288029151621084467866789466, 19.18155493057713262087893193119, 20.39203951856967294339728196065, 21.792544238656226206953264804445, 22.316020178834918388205198305977, 23.351118332790141855076212743662, 24.13738727602897240977832501984, 25.57200043175608655042423429412, 27.018437755260570933301962581890, 28.08064201520689899821872585957, 29.313335122722049802044484905448, 30.11780685430721190780361168644, 30.58885349252142180426186951895

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