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Properties

Label 1-79-79.36-r0-0-0
Degree $1$
Conductor $79$
Sign $0.986 + 0.162i$
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  + (−0.996 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.986 + 0.162i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.986 + 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8838277525 + 0.07226680029i\)
\(L(\frac12)\) \(\approx\) \(0.8838277525 + 0.07226680029i\)
\(L(1)\) \(\approx\) \(0.9351449117 + 0.06343973476i\)
\(L(1)\) \(\approx\) \(0.9351449117 + 0.06343973476i\)

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

−30.70565966539271564128758256740, −30.04229465720158418878930975011, −28.84005165375465485438269682266, −28.09648831537636997098540088613, −26.46688317997022427013190509490, −25.790481091925300693823552109751, −24.84283785599122708393011956071, −24.27120974977049056248243834996, −22.30502108380030361594561358226, −20.94333125633928950732439469216, −20.04665290584702016551788372907, −18.81171438528769723336091314648, −17.969129493569126765803372135216, −17.343886774836228381935324177650, −15.31664407020446889641773802419, −14.704821917972936676152062231158, −13.0368043186848717663638151423, −11.92590503249254502606069106384, −10.205712633327968154476149861629, −9.24677103803615943618991244748, −8.10139358255991738072482248094, −6.909258582585036045747190901, −5.673573533108817346924328915922, −2.659568554790573749053352268194, −1.88538305571252779566224561621, 1.73312358123638359529092089175, 3.322243977572936474156937888407, 5.21469366531022068520920417501, 7.068077752497055037877557089790, 8.38916474670706222014283936409, 9.49722359593726023104610687840, 10.25173050643684099023319478286, 11.49132819149069195175566171711, 13.54186860807326195645014129271, 14.36989637686752059450069226110, 16.09966171771379737416090183602, 16.69957660581886310043756172650, 17.896610848926564259650755295563, 19.249837498608394390567456261742, 20.3296774323648019389736066359, 21.03802927765830820276942480219, 22.00617696470694280160975461917, 24.1481691039335844242201230081, 24.95651558944160113449258083992, 26.18445480955758106076225035465, 26.72895869116857246381631114999, 27.69663363659780341744958993127, 29.11611089956319969635806046288, 29.65988803137997685161127831797, 31.16688303423492406170859478786

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