Properties

Degree 1
Conductor 79
Sign $0.514 - 0.857i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.514 - 0.857i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.514 - 0.857i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $0.514 - 0.857i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (10, \cdot )$
Sato-Tate  :  $\mu(13)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 79,\ (0:\ ),\ 0.514 - 0.857i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5799010961 - 0.3281845837i$
$L(\frac12,\chi)$  $\approx$  $0.5799010961 - 0.3281845837i$
$L(\chi,1)$  $\approx$  0.7361725503 - 0.1550179494i
$L(1,\chi)$  $\approx$  0.7361725503 - 0.1550179494i

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

−31.07287904678130373071032360263, −30.67402261087247477512152382507, −28.542766037303269634765761506, −28.234420712591708622503606423341, −27.00898865759246741139509319932, −26.39432091007136045969525833669, −25.23502091088261994625329793954, −23.85267169112765859359537935211, −22.20196155285481922141720670786, −21.19788688468857908445028504184, −20.4778770928172849254950749071, −19.31192622244712179982713054652, −18.50693613001686110319210088889, −16.89954921190771506779661739657, −15.815007227064226536378970003900, −14.96633563685064850489548493462, −13.14022417656966486746054195316, −11.72678373979762764119580699304, −10.902863564077409044268004995960, −9.442591249172754295217038870734, −8.4964390506879475239096629471, −7.59602995332507826456835438211, −4.914204672943874867780426001120, −3.64381593445458535755736190141, −2.21602141823185114703109526411, 0.957882976748462904925991603734, 3.10168186608807756610617966120, 5.226454677971722580345502234660, 7.146266147094480814512858217128, 7.647131327582937848422279150912, 8.631165206398353255542534513510, 10.36801981044794877478141713716, 11.52861831512952643775810669066, 13.24702834827388605002347795439, 14.41688704332441521028251564036, 15.38256006205122480084130295201, 16.60539706693239931121090942916, 18.130121051551806592536841817168, 18.55833041602712599809777187038, 20.01441507371305016410369331970, 20.428996739250586233738467067501, 23.0325451738581637497524043641, 23.58331301693421234582099713670, 24.574720458054029231808736476132, 25.599350343054146584505399111640, 26.83968959505559183334561149867, 27.203398470546886118664493128219, 28.84670846214286419959948390061, 29.85805924987371565944618026304, 31.02767791943940663415865964304

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