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Properties

Label 1-79-79.8-r0-0-0
Degree $1$
Conductor $79$
Sign $0.514 + 0.857i$
Analytic cond. $0.366874$
Root an. cond. $0.366874$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5799010961 + 0.3281845837i\)
\(L(\frac12)\) \(\approx\) \(0.5799010961 + 0.3281845837i\)
\(L(1)\) \(\approx\) \(0.7361725503 + 0.1550179494i\)
\(L(1)\) \(\approx\) \(0.7361725503 + 0.1550179494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.748 - 0.663i)T \)
3 \( 1 + (0.568 + 0.822i)T \)
5 \( 1 + (-0.970 - 0.239i)T \)
7 \( 1 + (0.568 + 0.822i)T \)
11 \( 1 + (-0.970 + 0.239i)T \)
13 \( 1 + (0.120 + 0.992i)T \)
17 \( 1 + (0.120 + 0.992i)T \)
19 \( 1 + (0.885 - 0.464i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.354 - 0.935i)T \)
31 \( 1 + (-0.748 - 0.663i)T \)
37 \( 1 + (0.885 - 0.464i)T \)
41 \( 1 + (-0.970 - 0.239i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (0.885 + 0.464i)T \)
53 \( 1 + (0.568 - 0.822i)T \)
59 \( 1 + (0.120 - 0.992i)T \)
61 \( 1 + (0.885 - 0.464i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (0.568 - 0.822i)T \)
73 \( 1 + (0.120 - 0.992i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (0.885 - 0.464i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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