Properties

Degree 1
Conductor 79
Sign $0.942 - 0.334i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.748 + 0.663i)2-s + (−0.568 + 0.822i)3-s + (0.120 − 0.992i)4-s + (−0.970 + 0.239i)5-s + (−0.120 − 0.992i)6-s + (−0.568 + 0.822i)7-s + (0.568 + 0.822i)8-s + (−0.354 − 0.935i)9-s + (0.568 − 0.822i)10-s + (−0.970 − 0.239i)11-s + (0.748 + 0.663i)12-s + (0.120 − 0.992i)13-s + (−0.120 − 0.992i)14-s + (0.354 − 0.935i)15-s + (−0.970 − 0.239i)16-s + (−0.120 + 0.992i)17-s + ⋯
L(s,χ)  = 1  + (−0.748 + 0.663i)2-s + (−0.568 + 0.822i)3-s + (0.120 − 0.992i)4-s + (−0.970 + 0.239i)5-s + (−0.120 − 0.992i)6-s + (−0.568 + 0.822i)7-s + (0.568 + 0.822i)8-s + (−0.354 − 0.935i)9-s + (0.568 − 0.822i)10-s + (−0.970 − 0.239i)11-s + (0.748 + 0.663i)12-s + (0.120 − 0.992i)13-s + (−0.120 − 0.992i)14-s + (0.354 − 0.935i)15-s + (−0.970 − 0.239i)16-s + (−0.120 + 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.942 - 0.334i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.942 - 0.334i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $0.942 - 0.334i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (69, \cdot )$
Sato-Tate  :  $\mu(26)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 79,\ (1:\ ),\ 0.942 - 0.334i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2820110748 - 0.04856175788i$
$L(\frac12,\chi)$  $\approx$  $0.2820110748 - 0.04856175788i$
$L(\chi,1)$  $\approx$  0.3805679131 + 0.1961946668i
$L(1,\chi)$  $\approx$  0.3805679131 + 0.1961946668i

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

−30.94767037100526295622005341596, −29.4675292522385376657415986620, −28.960805634176247401096551575889, −27.920019579672066454172404320439, −26.78338785743670325784899943976, −25.85807155045161869790077362138, −24.36112524477251815624281238767, −23.320091550712195594741110784856, −22.44203321937679960856791964052, −20.7514581678850903934890529397, −19.769676978444262551229022425471, −18.910837066852939417763522864917, −17.97582303691757541782438299832, −16.62812068411011217460798171672, −16.01008130694550030941751101388, −13.603374934778381951262643720377, −12.65705607843698855594804984962, −11.571932556573552695492837585879, −10.72100643730175558544002699399, −9.118504368783636012737474240147, −7.55574200607471852864632905782, −7.04392505529466401157211476698, −4.671624900795304244199913190391, −2.95760829295619675212927762648, −1.00294028475315768430039062144, 0.244835603723904535812164964523, 3.223691371685253413153250890557, 5.12207485306837183711973524525, 6.14222429795289107897934507238, 7.75115269635902619607616855841, 8.91409277339960674175917411589, 10.266331977734274393969129404044, 11.11611107710074318901691123636, 12.5699402928291675758114149049, 14.74775312496614098471195918899, 15.68416759939970551670660746357, 16.0415314330438218328142699140, 17.52487302039007679515297440939, 18.58898000995772709015303720362, 19.62718898531657762314804200818, 20.96312116674077391536460697752, 22.53607004934361255953080848775, 23.17101885045431980136921227040, 24.37646971148439243259893732131, 25.76753654943630147440166313323, 26.64899781527363212275821899723, 27.514743035318708089775668151641, 28.381990380598412794358268386020, 29.18694979347523701179719103411, 31.13493596517722685131129298407

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