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Properties

Label 1-79-79.44-r0-0-0
Degree $1$
Conductor $79$
Sign $0.509 - 0.860i$
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  + (0.799 − 0.600i)2-s + (0.987 − 0.160i)3-s + (0.278 − 0.960i)4-s + (−0.845 + 0.534i)5-s + (0.692 − 0.721i)6-s + (−0.632 − 0.774i)7-s + (−0.354 − 0.935i)8-s + (0.948 − 0.316i)9-s + (−0.354 + 0.935i)10-s + (−0.0402 + 0.999i)11-s + (0.120 − 0.992i)12-s + (0.692 + 0.721i)13-s + (−0.970 − 0.239i)14-s + (−0.748 + 0.663i)15-s + (−0.845 − 0.534i)16-s + (−0.970 + 0.239i)17-s + ⋯
L(s)  = 1  + (0.799 − 0.600i)2-s + (0.987 − 0.160i)3-s + (0.278 − 0.960i)4-s + (−0.845 + 0.534i)5-s + (0.692 − 0.721i)6-s + (−0.632 − 0.774i)7-s + (−0.354 − 0.935i)8-s + (0.948 − 0.316i)9-s + (−0.354 + 0.935i)10-s + (−0.0402 + 0.999i)11-s + (0.120 − 0.992i)12-s + (0.692 + 0.721i)13-s + (−0.970 − 0.239i)14-s + (−0.748 + 0.663i)15-s + (−0.845 − 0.534i)16-s + (−0.970 + 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.509 - 0.860i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.509 - 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.404441648 - 0.8005239721i\)
\(L(\frac12)\) \(\approx\) \(1.404441648 - 0.8005239721i\)
\(L(1)\) \(\approx\) \(1.532150440 - 0.6184330724i\)
\(L(1)\) \(\approx\) \(1.532150440 - 0.6184330724i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.799 - 0.600i)T \)
3 \( 1 + (0.987 - 0.160i)T \)
5 \( 1 + (-0.845 + 0.534i)T \)
7 \( 1 + (-0.632 - 0.774i)T \)
11 \( 1 + (-0.0402 + 0.999i)T \)
13 \( 1 + (0.692 + 0.721i)T \)
17 \( 1 + (-0.970 + 0.239i)T \)
19 \( 1 + (0.428 + 0.903i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.200 - 0.979i)T \)
31 \( 1 + (-0.919 - 0.391i)T \)
37 \( 1 + (-0.996 - 0.0804i)T \)
41 \( 1 + (0.885 + 0.464i)T \)
43 \( 1 + (-0.0402 - 0.999i)T \)
47 \( 1 + (-0.996 + 0.0804i)T \)
53 \( 1 + (0.987 + 0.160i)T \)
59 \( 1 + (0.278 + 0.960i)T \)
61 \( 1 + (0.568 - 0.822i)T \)
67 \( 1 + (0.120 - 0.992i)T \)
71 \( 1 + (-0.354 - 0.935i)T \)
73 \( 1 + (0.692 - 0.721i)T \)
83 \( 1 + (0.278 - 0.960i)T \)
89 \( 1 + (-0.354 + 0.935i)T \)
97 \( 1 + (0.568 - 0.822i)T \)
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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

−31.34132616081592038643301799897, −30.870767508319646253586448621968, −29.486655410850219607390851031592, −27.92293317908063555974218951866, −26.688355405155955020117601056322, −25.83372000046051827984384838494, −24.653650375641991395341981635385, −24.12734210234989415195155196368, −22.61944092579071954649459243320, −21.67441158043249205745825763735, −20.46490147646721363793154609187, −19.5996398797180774903751769263, −18.20644142184207098137981838831, −16.04119174974728058949176688074, −15.94713997252389869721244468590, −14.720410487148994990381650234153, −13.36311097607550038921635164934, −12.64199986088990067785892520442, −11.16761626160273583172545775519, −8.90671441491877189820610737635, −8.36135611710310315779020323520, −6.900497809169979746731045709615, −5.280651401643437007311732239116, −3.81041217677690884137016455835, −2.83268947982514497026439303253, 1.95403512108937785933967192447, 3.57473104157603073484627829508, 4.16646889754414956652842556442, 6.560114831904724825895274137005, 7.56130339301241663940797467379, 9.42210644455801688639502847627, 10.53787986197569237609241731500, 11.91217600144976672859598261864, 13.132006459385293096746030529678, 14.04792822821065615514546667761, 15.137430539443303104408532121362, 16.00870990959192432312748696254, 18.29782755324611407396993779245, 19.37553847765270836434014745530, 20.01374947068041212339163731935, 20.946076318214162001026736992078, 22.424538264214811833665212012301, 23.29169400069377819256454874474, 24.22009432430748626214700317172, 25.682314796065371680490202947718, 26.56798922857773446352119895297, 27.82160672101631658371744377514, 29.19069107514489516380319462772, 30.23098973766913585760232890150, 31.01026119260604399074036943566

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