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Properties

Label 1-79-79.69-r1-0-0
Degree $1$
Conductor $79$
Sign $0.942 - 0.334i$
Analytic cond. $8.48972$
Root an. cond. $8.48972$
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

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

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.942 - 0.334i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ 0.942 - 0.334i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2820110748 - 0.04856175788i\)
\(L(\frac12)\) \(\approx\) \(0.2820110748 - 0.04856175788i\)
\(L(1)\) \(\approx\) \(0.3805679131 + 0.1961946668i\)
\(L(1)\) \(\approx\) \(0.3805679131 + 0.1961946668i\)

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

−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