Properties

Degree 1
Conductor 79
Sign $-0.773 + 0.633i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.919 + 0.391i)2-s + (−0.632 + 0.774i)3-s + (0.692 − 0.721i)4-s + (−0.0402 + 0.999i)5-s + (0.278 − 0.960i)6-s + (0.987 + 0.160i)7-s + (−0.354 + 0.935i)8-s + (−0.200 − 0.979i)9-s + (−0.354 − 0.935i)10-s + (−0.845 + 0.534i)11-s + (0.120 + 0.992i)12-s + (0.278 + 0.960i)13-s + (−0.970 + 0.239i)14-s + (−0.748 − 0.663i)15-s + (−0.0402 − 0.999i)16-s + (−0.970 − 0.239i)17-s + ⋯
L(s,χ)  = 1  + (−0.919 + 0.391i)2-s + (−0.632 + 0.774i)3-s + (0.692 − 0.721i)4-s + (−0.0402 + 0.999i)5-s + (0.278 − 0.960i)6-s + (0.987 + 0.160i)7-s + (−0.354 + 0.935i)8-s + (−0.200 − 0.979i)9-s + (−0.354 − 0.935i)10-s + (−0.845 + 0.534i)11-s + (0.120 + 0.992i)12-s + (0.278 + 0.960i)13-s + (−0.970 + 0.239i)14-s + (−0.748 − 0.663i)15-s + (−0.0402 − 0.999i)16-s + (−0.970 − 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.773 + 0.633i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.773 + 0.633i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $-0.773 + 0.633i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (49, \cdot )$
Sato-Tate  :  $\mu(39)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 79,\ (0:\ ),\ -0.773 + 0.633i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1610406868 + 0.4508055810i$
$L(\frac12,\chi)$  $\approx$  $0.1610406868 + 0.4508055810i$
$L(\chi,1)$  $\approx$  0.4454179739 + 0.3624612036i
$L(1,\chi)$  $\approx$  0.4454179739 + 0.3624612036i

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

−30.37667826849631467796935793797, −29.49563646194466610359689817167, −28.42140655917787093708256731299, −27.87358698287362347025494609830, −26.71304761576446602656716105778, −25.17209212203173064242741141463, −24.3888898850124567862532231262, −23.516144603990064756193895914431, −21.75723333602433817719216335374, −20.6754377282311765421762180017, −19.73695720783708517695422073340, −18.41382423574709993463487544981, −17.61453183589604509526587866180, −16.77404840643422925576006891599, −15.58549784841459813082381133363, −13.375478320452206725558971607994, −12.52292684808794998988014748238, −11.32065073489660406372517685481, −10.436665986424934322582498753289, −8.36479120058208516080603997722, −8.06445485484297199222014015906, −6.271631265245346849815181867719, −4.71146283627155183124251704609, −2.24600808969841665738679041368, −0.77663914887470648626135632770, 2.239268167154229217199349283681, 4.51054728367740388094681607844, 5.994066678795298252882136496991, 7.17286647346507897769641360352, 8.64066621789361655326997748227, 10.033939642397471933619083498526, 10.92840274256769238480134797360, 11.70773395710933031502834130476, 14.22030196559637954075574605970, 15.23397492950712923712171141761, 15.98528220312988154199400341068, 17.586010928587508155881696910431, 17.93316170291568632021363669259, 19.25884192252781913507506042542, 20.80194331875483444171150387985, 21.62820627479131549973998576519, 23.17873885378963778926395532548, 23.87869730276697427435150938360, 25.501813764309793615756647949079, 26.4862202395216315879465501820, 27.12330471749758309313703864492, 28.197183942518455294502435340205, 29.012925385292907228623493981380, 30.30530706846233359559938716827, 31.625557999958824128542484219031

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