Properties

Degree 1
Conductor 79
Sign $-0.660 - 0.750i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

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

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.660 - 0.750i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.660 - 0.750i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $-0.660 - 0.750i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (56, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 79,\ (1:\ ),\ -0.660 - 0.750i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2591312027 - 0.5729881706i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2591312027 - 0.5729881706i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6931354351 - 0.1388887096i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6931354351 - 0.1388887096i\)

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

−30.96252076928684110155256230744, −30.36950155286839596223439169901, −29.14336969534140401856709403024, −27.68564258309021010181909581466, −27.11024262112550356442490832928, −26.32580117409090072060859484067, −25.36296560573469226645139518398, −23.25743575967097478340717416180, −22.3897349994655473396041977455, −21.28118067298216347707523698219, −20.17044759663811385931795448616, −19.65443494997227104086037022334, −18.14584981345222854045380986294, −17.16680694067640466984517397570, −15.65223698416750872959632290687, −14.56562476992785457132482568409, −13.3134124731165924415395260290, −11.65158341676752793576416263493, −10.47675528369271034102515231360, −10.03514193157784654518694936799, −8.26239063822871945943705273571, −7.41518040024451648459398997378, −4.65931330172815543677484665502, −3.62138650610622922078562051413, −2.28114759555439074710595778466, 0.32320963911797269691310969312, 2.09589542090420501131772349408, 4.607189550204918181256467234402, 6.02761394964115363707221833242, 7.41076780881124883103795742200, 8.62264154285051722398373688307, 8.9826821350950243039836851533, 11.25836445486919501719680259341, 12.624099679762209043731956284, 13.80238736711178171289244435720, 14.98261444515685794361045582098, 16.03711654615577450962069137638, 17.2858509173754003253668408186, 18.42763840040317848507000426371, 19.23591816282825442393284148248, 20.27897087218842664790567053308, 21.8619217339536107681364997159, 23.739505241899892146917854849498, 24.12585589197503258520168525977, 24.87206323042271199908413258211, 26.09600174297423202782468235545, 27.08034424591647756469449486387, 28.36183717618534361476933921529, 29.05517974589062458457829576095, 30.88456813554126898036921975740

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