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Properties

Label 1-79-79.54-r1-0-0
Degree $1$
Conductor $79$
Sign $0.692 + 0.721i$
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  + (−0.632 + 0.774i)2-s + (0.845 + 0.534i)3-s + (−0.200 − 0.979i)4-s + (−0.919 − 0.391i)5-s + (−0.948 + 0.316i)6-s + (0.0402 − 0.999i)7-s + (0.885 + 0.464i)8-s + (0.428 + 0.903i)9-s + (0.885 − 0.464i)10-s + (0.799 + 0.600i)11-s + (0.354 − 0.935i)12-s + (0.948 + 0.316i)13-s + (0.748 + 0.663i)14-s + (−0.568 − 0.822i)15-s + (−0.919 + 0.391i)16-s + (0.748 − 0.663i)17-s + ⋯
L(s)  = 1  + (−0.632 + 0.774i)2-s + (0.845 + 0.534i)3-s + (−0.200 − 0.979i)4-s + (−0.919 − 0.391i)5-s + (−0.948 + 0.316i)6-s + (0.0402 − 0.999i)7-s + (0.885 + 0.464i)8-s + (0.428 + 0.903i)9-s + (0.885 − 0.464i)10-s + (0.799 + 0.600i)11-s + (0.354 − 0.935i)12-s + (0.948 + 0.316i)13-s + (0.748 + 0.663i)14-s + (−0.568 − 0.822i)15-s + (−0.919 + 0.391i)16-s + (0.748 − 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.692 + 0.721i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ 0.692 + 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.366871502 + 0.5829025858i\)
\(L(\frac12)\) \(\approx\) \(1.366871502 + 0.5829025858i\)
\(L(1)\) \(\approx\) \(0.9794227116 + 0.3414260839i\)
\(L(1)\) \(\approx\) \(0.9794227116 + 0.3414260839i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.632 + 0.774i)T \)
3 \( 1 + (0.845 + 0.534i)T \)
5 \( 1 + (-0.919 - 0.391i)T \)
7 \( 1 + (0.0402 - 0.999i)T \)
11 \( 1 + (0.799 + 0.600i)T \)
13 \( 1 + (0.948 + 0.316i)T \)
17 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 + (0.692 - 0.721i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.996 - 0.0804i)T \)
31 \( 1 + (0.987 + 0.160i)T \)
37 \( 1 + (-0.278 - 0.960i)T \)
41 \( 1 + (-0.120 - 0.992i)T \)
43 \( 1 + (-0.799 + 0.600i)T \)
47 \( 1 + (-0.278 + 0.960i)T \)
53 \( 1 + (0.845 - 0.534i)T \)
59 \( 1 + (0.200 - 0.979i)T \)
61 \( 1 + (0.970 + 0.239i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (-0.885 - 0.464i)T \)
73 \( 1 + (0.948 - 0.316i)T \)
83 \( 1 + (-0.200 - 0.979i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (-0.970 - 0.239i)T \)
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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

−30.58381422866440841256166348128, −29.96478605778510716753493980852, −28.516509907771506278034135118432, −27.50751645525506565541580798890, −26.592929940453828090145295223931, −25.52950889466386649801762717405, −24.59182362588581844633799406233, −23.055068328247172107730242913345, −21.79231369911486781123803402530, −20.63828058692327257964978229217, −19.58959795722456561732012177687, −18.787249086819278463101682953332, −18.17712033987384933810127310157, −16.36279326676223016569220794806, −15.10395595772144457498284474803, −13.80458621490035236791860409969, −12.30172594262285098925366207971, −11.70754643041741049690978172354, −10.104054789204874685320729528074, −8.51248381125913255764489301192, −8.19093714958131865135056835805, −6.50251126806275883240476720528, −3.80447176522316146137540806300, −2.89924906842158661356454350163, −1.190998255358945341114883575637, 1.13749270870445890025824234518, 3.74182724693601253950249728301, 4.82425838680232555471547268019, 6.99942736613368771767996376429, 7.90408478911985154866946228276, 9.03355827802572051216777509016, 10.08282164151632064162168431940, 11.4801870764111631681652942443, 13.57830484261301881086427542450, 14.43538974504697524820929239808, 15.74550353390409597774293496417, 16.2890583983556636301539829215, 17.62007926298742987439392795404, 19.21255394268461813151013099632, 19.8913989740601893750848907731, 20.77724464775905150205746378500, 22.72440171259058462615969170030, 23.62204331501888617476204062796, 24.7709422075395918400217524901, 25.80272422290271573915537205809, 26.72731494399228872369776080875, 27.50731314828331641441558459531, 28.301686107515083429568928703663, 30.14973877239531956255872227423, 31.22297050600896871886268025251

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