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Properties

Label 1-79-79.75-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.582 - 0.813i$
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  + (−0.845 + 0.534i)2-s + (−0.799 − 0.600i)3-s + (0.428 − 0.903i)4-s + (−0.632 + 0.774i)5-s + (0.996 + 0.0804i)6-s + (0.919 − 0.391i)7-s + (0.120 + 0.992i)8-s + (0.278 + 0.960i)9-s + (0.120 − 0.992i)10-s + (0.987 − 0.160i)11-s + (−0.885 + 0.464i)12-s + (−0.996 + 0.0804i)13-s + (−0.568 + 0.822i)14-s + (0.970 − 0.239i)15-s + (−0.632 − 0.774i)16-s + (−0.568 − 0.822i)17-s + ⋯
L(s)  = 1  + (−0.845 + 0.534i)2-s + (−0.799 − 0.600i)3-s + (0.428 − 0.903i)4-s + (−0.632 + 0.774i)5-s + (0.996 + 0.0804i)6-s + (0.919 − 0.391i)7-s + (0.120 + 0.992i)8-s + (0.278 + 0.960i)9-s + (0.120 − 0.992i)10-s + (0.987 − 0.160i)11-s + (−0.885 + 0.464i)12-s + (−0.996 + 0.0804i)13-s + (−0.568 + 0.822i)14-s + (0.970 − 0.239i)15-s + (−0.632 − 0.774i)16-s + (−0.568 − 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.582 - 0.813i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.582 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1241111287 - 0.2414528542i\)
\(L(\frac12)\) \(\approx\) \(0.1241111287 - 0.2414528542i\)
\(L(1)\) \(\approx\) \(0.4664573875 + 0.009403598448i\)
\(L(1)\) \(\approx\) \(0.4664573875 + 0.009403598448i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.845 + 0.534i)T \)
3 \( 1 + (-0.799 - 0.600i)T \)
5 \( 1 + (-0.632 + 0.774i)T \)
7 \( 1 + (0.919 - 0.391i)T \)
11 \( 1 + (0.987 - 0.160i)T \)
13 \( 1 + (-0.996 + 0.0804i)T \)
17 \( 1 + (-0.568 - 0.822i)T \)
19 \( 1 + (-0.200 + 0.979i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.692 - 0.721i)T \)
31 \( 1 + (-0.0402 - 0.999i)T \)
37 \( 1 + (-0.948 + 0.316i)T \)
41 \( 1 + (0.354 + 0.935i)T \)
43 \( 1 + (-0.987 - 0.160i)T \)
47 \( 1 + (-0.948 - 0.316i)T \)
53 \( 1 + (-0.799 + 0.600i)T \)
59 \( 1 + (-0.428 - 0.903i)T \)
61 \( 1 + (0.748 + 0.663i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (-0.120 - 0.992i)T \)
73 \( 1 + (-0.996 - 0.0804i)T \)
83 \( 1 + (0.428 - 0.903i)T \)
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

−31.07284669880009183869203185811, −29.945472541533847670507168272742, −28.75992438840075620201968591930, −27.72789643449515306869649153240, −27.56529862577906024775242245854, −26.350248631451303666071035282527, −24.713694426697792429087638951228, −23.86754736642126702448722902527, −22.15915542228263297295023242581, −21.46262994673260531134520521298, −20.24406782593541825575191991375, −19.38245424999392041534966528043, −17.63859502956277754240968470219, −17.25214435479758072236577411321, −16.001481838862194736928606729081, −14.95849677553143576530010146771, −12.58542591874311658854298243218, −11.78241985748283836854195853851, −10.95919765159156498641566761618, −9.460636301244584612131699878065, −8.54124865864756755110971833464, −7.02300187048061535414603088733, −5.04733528469081663098721235517, −3.8748454066539234458272605278, −1.53117053412417794153845511859, 0.19652122373473538458009893951, 1.93866304663944842706045396405, 4.62658929211868552601029304570, 6.276696465490354138883864152538, 7.25156487109556241842816405914, 8.13048353132095166460521321809, 10.02068028991111117960626073367, 11.22790040288752140420378647083, 11.86678593077653080726098288554, 14.058288882320072759980699535319, 14.93773535405066222392010816946, 16.46003461020083037111837077525, 17.29896922693703728178048423416, 18.31320185749774770278795656300, 19.14210014479516798678604783750, 20.26309311651867595840271238784, 22.23950853941071291954777741515, 23.12557098868949825150646479123, 24.291414515284288709805571331073, 24.81279252017890917980613404296, 26.55263972966741507791602424620, 27.25708994376828821825256951552, 28.060508423881909817053637720591, 29.54950008259297305077006894177, 30.05036177142924100504510376521

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