Properties

Degree 1
Conductor 79
Sign $0.583 + 0.812i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + 10-s + (−0.5 + 0.866i)11-s + 12-s + (−0.5 − 0.866i)13-s + 14-s + 15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + 10-s + (−0.5 + 0.866i)11-s + 12-s + (−0.5 − 0.866i)13-s + 14-s + 15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.583 + 0.812i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.583 + 0.812i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $0.583 + 0.812i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (55, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 79,\ (0:\ ),\ 0.583 + 0.812i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2874179459 + 0.1475055772i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2874179459 + 0.1475055772i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4826010944 - 0.07033186451i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4826010944 - 0.07033186451i\)

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

−31.669111416909676774188057189336, −29.464696044187228826856441587822, −28.48263988132845538385351577632, −27.65928410393172729835496440361, −26.61131210722289671192764745032, −26.040870908498393056261296594226, −24.25870118042971104080809591558, −23.644649903678372365461139315060, −22.63364699466752157233695974812, −21.19236957388260053798279121996, −19.9554462406077410294687780019, −18.867137646662503308965756913118, −17.13825640380322819876099444033, −16.57709433310643564855040319794, −15.88770365933354448279241038400, −14.57110748584187079720453962888, −13.17917781680134408215449376172, −11.45097907538805292264753325505, −10.17135159938582430494486204332, −9.17169071883001652905215720521, −7.907226019013610564679993539242, −6.37930689859119539230799953607, −5.032347625856192697066506484485, −3.974570471460419533351110546985, −0.44354526496645776761009679356, 2.060152358130200889363182764028, 3.28748748524946646448515082989, 5.47832368568664555155953704330, 7.2049851547574999822384476201, 8.077523493541901802585602382368, 9.88532420976448160863132780231, 10.92911684589516751352156820007, 12.29099844855721813125001369291, 12.62252258522131565523879009372, 14.408084569979291718310494134031, 15.971289843960303463618692826946, 17.49544949681144608775423265003, 18.36287354474951425159961008788, 19.061396511977339369999117293386, 20.0283534410676588859348932328, 21.67083885362830241760786691936, 22.6550031214220125960893725459, 23.3245715713070520508236065693, 25.22354898245661120636601611707, 25.84800705883768869475712812061, 27.45916253144070627567057125347, 28.10695566199825629223438158804, 29.35494352164842391907404761154, 29.9620001956546200992548709423, 31.054102518159888721444299967336

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