Properties

Degree 1
Conductor 79
Sign $-0.689 - 0.724i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.692 − 0.721i)2-s + (−0.200 − 0.979i)3-s + (−0.0402 − 0.999i)4-s + (−0.996 − 0.0804i)5-s + (−0.845 − 0.534i)6-s + (0.948 + 0.316i)7-s + (−0.748 − 0.663i)8-s + (−0.919 + 0.391i)9-s + (−0.748 + 0.663i)10-s + (0.428 − 0.903i)11-s + (−0.970 + 0.239i)12-s + (−0.845 + 0.534i)13-s + (0.885 − 0.464i)14-s + (0.120 + 0.992i)15-s + (−0.996 + 0.0804i)16-s + (0.885 + 0.464i)17-s + ⋯
L(s,χ)  = 1  + (0.692 − 0.721i)2-s + (−0.200 − 0.979i)3-s + (−0.0402 − 0.999i)4-s + (−0.996 − 0.0804i)5-s + (−0.845 − 0.534i)6-s + (0.948 + 0.316i)7-s + (−0.748 − 0.663i)8-s + (−0.919 + 0.391i)9-s + (−0.748 + 0.663i)10-s + (0.428 − 0.903i)11-s + (−0.970 + 0.239i)12-s + (−0.845 + 0.534i)13-s + (0.885 − 0.464i)14-s + (0.120 + 0.992i)15-s + (−0.996 + 0.0804i)16-s + (0.885 + 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.689 - 0.724i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.689 - 0.724i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $-0.689 - 0.724i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (31, \cdot )$
Sato-Tate  :  $\mu(39)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 79,\ (0:\ ),\ -0.689 - 0.724i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4409559662 - 1.028376086i$
$L(\frac12,\chi)$  $\approx$  $0.4409559662 - 1.028376086i$
$L(\chi,1)$  $\approx$  0.8429379693 - 0.8481669066i
$L(1,\chi)$  $\approx$  0.8429379693 - 0.8481669066i

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

−31.60234573046014778173041199698, −30.87963173743320706260999607847, −29.70524355037927344714868407580, −27.75931160231464373940792602677, −27.27758850048327624563659895091, −26.28570656271178925464084327839, −24.991452259529085752452451452155, −23.7772371199433913532793189990, −22.89525178405718509680442284669, −22.09210909336089717974211620925, −20.76674104184895718852829307123, −20.02147789349156221247391673184, −17.87492054282222986416734628205, −16.92269944199611119502450576411, −15.79929904885389312027281959562, −14.927126563056735140517581617121, −14.18154568172215496169336571980, −12.15430588839749434335458627233, −11.512049440983456245298442717544, −9.83990311914595515734868510314, −8.16083996144526598846281765978, −7.240673324439791048812877536247, −5.286399937034310814842515018446, −4.46717429566661469509537265378, −3.27450801892356215011944204527, 1.21693449347865705201792520500, 2.88792004467878319222504409278, 4.553600776197009970350031613100, 5.89347621321117521753353734742, 7.45365354483290944391659592000, 8.73450410848447085142511697595, 10.80448454689575015988842281721, 11.91017170501810709238917487224, 12.23416549469694924576591416809, 13.962310610262470126168311979646, 14.62303646959732751208144076440, 16.25819972901581838587888906508, 17.872657216783163629549564050488, 19.07820188240244554210640055385, 19.5836049472812685465100917897, 20.94480232631704033353857283168, 22.177751971220690253547895183025, 23.24627653693888434793431359199, 24.28529092882957967012823111557, 24.56064315247580819046848580316, 26.7882004907783136570019282385, 27.847971584067447896069590095696, 28.7752470556416063044726936134, 29.94793045401204421774483555339, 30.70049473989492146912334690126

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