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Properties

Label 1-79-79.14-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.859 - 0.511i$
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  + (0.885 − 0.464i)2-s + (−0.120 − 0.992i)3-s + (0.568 − 0.822i)4-s + (−0.354 + 0.935i)5-s + (−0.568 − 0.822i)6-s + (−0.120 − 0.992i)7-s + (0.120 − 0.992i)8-s + (−0.970 + 0.239i)9-s + (0.120 + 0.992i)10-s + (−0.354 − 0.935i)11-s + (−0.885 − 0.464i)12-s + (0.568 − 0.822i)13-s + (−0.568 − 0.822i)14-s + (0.970 + 0.239i)15-s + (−0.354 − 0.935i)16-s + (−0.568 + 0.822i)17-s + ⋯
L(s)  = 1  + (0.885 − 0.464i)2-s + (−0.120 − 0.992i)3-s + (0.568 − 0.822i)4-s + (−0.354 + 0.935i)5-s + (−0.568 − 0.822i)6-s + (−0.120 − 0.992i)7-s + (0.120 − 0.992i)8-s + (−0.970 + 0.239i)9-s + (0.120 + 0.992i)10-s + (−0.354 − 0.935i)11-s + (−0.885 − 0.464i)12-s + (0.568 − 0.822i)13-s + (−0.568 − 0.822i)14-s + (0.970 + 0.239i)15-s + (−0.354 − 0.935i)16-s + (−0.568 + 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5612068684 - 2.039536300i\)
\(L(\frac12)\) \(\approx\) \(0.5612068684 - 2.039536300i\)
\(L(1)\) \(\approx\) \(1.104499913 - 1.021646591i\)
\(L(1)\) \(\approx\) \(1.104499913 - 1.021646591i\)

Euler product

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

−31.46854815364634057694605782066, −30.88755907701501451875432021878, −28.84925563669231633072584264342, −28.30835720051622963315961672544, −26.97560163423097217355824607515, −25.71203323734876701008742020056, −24.89976103848198017112249755267, −23.55674683031853955706124401677, −22.748937081678477907450732731668, −21.44633452028571660572800680354, −20.92428412684022966287568621883, −19.73019039526404990624969699986, −17.69628181006109865894210517932, −16.47820043355407844517573196156, −15.66386064790822457308609372211, −14.98547899435421466091367418592, −13.39233000610770171608601164643, −12.176535950207968552034223517752, −11.27841364621891328681779485725, −9.31748739990092793764621283824, −8.38022024779671346984472723482, −6.489281868028272444040881099454, −5.00106491587728609474839278812, −4.42773436523390784769805034581, −2.67163598312254079472667555896, 0.786017106850271251267711722136, 2.65403853672115540256299823366, 3.8502545455924228388848493021, 5.88956680328123690105557329146, 6.79547673557466885048807462275, 8.076403022142696051084843665678, 10.64163124785256965836433765904, 11.00968064355094446134981220322, 12.56747482583813816445646693457, 13.51481808674851324223743738979, 14.37108822183808873969519992811, 15.695880671395589461406374567994, 17.31636755249273886003943779894, 18.7822426272580446346698167376, 19.3893808157436144384240631721, 20.54978275516112878851845273726, 21.97746070843421432309343035047, 23.21629643072581279018518918410, 23.43240613025119346085608966888, 24.756885172524476212625180439152, 25.97551854834123137452314747758, 27.31213572322773352066563635430, 28.85035023262381229886390493332, 29.70070304527318823157832516822, 30.32675683693911385416688471664

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