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Properties

Label 1-79-79.43-r1-0-0
Degree $1$
Conductor $79$
Sign $0.908 + 0.417i$
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  + (−0.996 − 0.0804i)2-s + (−0.692 − 0.721i)3-s + (0.987 + 0.160i)4-s + (0.948 − 0.316i)5-s + (0.632 + 0.774i)6-s + (−0.278 + 0.960i)7-s + (−0.970 − 0.239i)8-s + (−0.0402 + 0.999i)9-s + (−0.970 + 0.239i)10-s + (−0.200 − 0.979i)11-s + (−0.568 − 0.822i)12-s + (−0.632 + 0.774i)13-s + (0.354 − 0.935i)14-s + (−0.885 − 0.464i)15-s + (0.948 + 0.316i)16-s + (0.354 + 0.935i)17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0804i)2-s + (−0.692 − 0.721i)3-s + (0.987 + 0.160i)4-s + (0.948 − 0.316i)5-s + (0.632 + 0.774i)6-s + (−0.278 + 0.960i)7-s + (−0.970 − 0.239i)8-s + (−0.0402 + 0.999i)9-s + (−0.970 + 0.239i)10-s + (−0.200 − 0.979i)11-s + (−0.568 − 0.822i)12-s + (−0.632 + 0.774i)13-s + (0.354 − 0.935i)14-s + (−0.885 − 0.464i)15-s + (0.948 + 0.316i)16-s + (0.354 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.908 + 0.417i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ 0.908 + 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8221482663 + 0.1798414896i\)
\(L(\frac12)\) \(\approx\) \(0.8221482663 + 0.1798414896i\)
\(L(1)\) \(\approx\) \(0.6635751109 + 0.02973200873i\)
\(L(1)\) \(\approx\) \(0.6635751109 + 0.02973200873i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.996 - 0.0804i)T \)
3 \( 1 + (-0.692 - 0.721i)T \)
5 \( 1 + (0.948 - 0.316i)T \)
7 \( 1 + (-0.278 + 0.960i)T \)
11 \( 1 + (-0.200 - 0.979i)T \)
13 \( 1 + (-0.632 + 0.774i)T \)
17 \( 1 + (0.354 + 0.935i)T \)
19 \( 1 + (0.799 + 0.600i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.845 - 0.534i)T \)
31 \( 1 + (0.428 + 0.903i)T \)
37 \( 1 + (0.919 - 0.391i)T \)
41 \( 1 + (0.748 + 0.663i)T \)
43 \( 1 + (0.200 - 0.979i)T \)
47 \( 1 + (0.919 + 0.391i)T \)
53 \( 1 + (-0.692 + 0.721i)T \)
59 \( 1 + (-0.987 + 0.160i)T \)
61 \( 1 + (-0.120 + 0.992i)T \)
67 \( 1 + (0.568 + 0.822i)T \)
71 \( 1 + (0.970 + 0.239i)T \)
73 \( 1 + (-0.632 - 0.774i)T \)
83 \( 1 + (0.987 + 0.160i)T \)
89 \( 1 + (-0.970 + 0.239i)T \)
97 \( 1 + (0.120 - 0.992i)T \)
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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

−30.25747196984058041228690297333, −29.37869580709861665900957063888, −28.65799324454212230165926937369, −27.530825632363688912639332894, −26.557731341776628253963436646469, −25.82301971424686158317123753792, −24.595433974633158442190875978720, −23.107956100404946400422609206648, −22.12787496824047638193285391786, −20.71570475618201426264188982705, −20.10286512345676354547358877649, −18.23270583061490716692588381387, −17.55173571232957746888577617995, −16.69938085768786704527578875547, −15.57570813812924007122284941034, −14.306313949072640336851561163868, −12.487844629670963663148849012480, −11.00337346925222897946480158935, −9.99080560665270925584828420716, −9.5855414130922218605069972115, −7.468924271915224229958566520947, −6.38725244717796561341881373904, −4.95357403181296319890348223468, −2.79742114497911742208644624339, −0.68502746231380772912826656571, 1.27186506684628998941854090353, 2.54747292850350838914391634779, 5.62483274689596474799494050556, 6.30129871126027579677070879025, 7.9027830414864853141764030298, 9.143222782120017466420839796233, 10.32304375137689785354643864098, 11.72354669199805094817355976725, 12.5608177129964513577603345405, 14.039247347048432575565959299782, 15.945469317794384085853840040894, 16.84097126397500067043517535651, 17.805464328246007047847594352, 18.72603645545810791675274652172, 19.5406481195864855265264178446, 21.357772334431234245447862011228, 21.88971369709258811335610995046, 23.81416200114429044943408685897, 24.728916070083409505270365672245, 25.40017116000326323787398859056, 26.70898287221428061668591092820, 28.102988343007071059248900496762, 28.86156914615433075715350539620, 29.3526371724919998684731363908, 30.52902162610995339293910180610

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