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Properties

Label 1-79-79.2-r0-0-0
Degree $1$
Conductor $79$
Sign $-0.159 + 0.987i$
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  + (0.278 + 0.960i)2-s + (0.948 + 0.316i)3-s + (−0.845 + 0.534i)4-s + (0.428 + 0.903i)5-s + (−0.0402 + 0.999i)6-s + (−0.200 − 0.979i)7-s + (−0.748 − 0.663i)8-s + (0.799 + 0.600i)9-s + (−0.748 + 0.663i)10-s + (−0.996 + 0.0804i)11-s + (−0.970 + 0.239i)12-s + (−0.0402 − 0.999i)13-s + (0.885 − 0.464i)14-s + (0.120 + 0.992i)15-s + (0.428 − 0.903i)16-s + (0.885 + 0.464i)17-s + ⋯
L(s)  = 1  + (0.278 + 0.960i)2-s + (0.948 + 0.316i)3-s + (−0.845 + 0.534i)4-s + (0.428 + 0.903i)5-s + (−0.0402 + 0.999i)6-s + (−0.200 − 0.979i)7-s + (−0.748 − 0.663i)8-s + (0.799 + 0.600i)9-s + (−0.748 + 0.663i)10-s + (−0.996 + 0.0804i)11-s + (−0.970 + 0.239i)12-s + (−0.0402 − 0.999i)13-s + (0.885 − 0.464i)14-s + (0.120 + 0.992i)15-s + (0.428 − 0.903i)16-s + (0.885 + 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.159 + 0.987i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ -0.159 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8616503674 + 1.011911401i\)
\(L(\frac12)\) \(\approx\) \(0.8616503674 + 1.011911401i\)
\(L(1)\) \(\approx\) \(1.106394885 + 0.8168107412i\)
\(L(1)\) \(\approx\) \(1.106394885 + 0.8168107412i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.278 + 0.960i)T \)
3 \( 1 + (0.948 + 0.316i)T \)
5 \( 1 + (0.428 + 0.903i)T \)
7 \( 1 + (-0.200 - 0.979i)T \)
11 \( 1 + (-0.996 + 0.0804i)T \)
13 \( 1 + (-0.0402 - 0.999i)T \)
17 \( 1 + (0.885 + 0.464i)T \)
19 \( 1 + (-0.632 - 0.774i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.919 - 0.391i)T \)
31 \( 1 + (0.692 - 0.721i)T \)
37 \( 1 + (0.987 - 0.160i)T \)
41 \( 1 + (0.568 - 0.822i)T \)
43 \( 1 + (-0.996 - 0.0804i)T \)
47 \( 1 + (0.987 + 0.160i)T \)
53 \( 1 + (0.948 - 0.316i)T \)
59 \( 1 + (-0.845 - 0.534i)T \)
61 \( 1 + (-0.354 + 0.935i)T \)
67 \( 1 + (-0.970 + 0.239i)T \)
71 \( 1 + (-0.748 - 0.663i)T \)
73 \( 1 + (-0.0402 + 0.999i)T \)
83 \( 1 + (-0.845 + 0.534i)T \)
89 \( 1 + (-0.748 + 0.663i)T \)
97 \( 1 + (-0.354 + 0.935i)T \)
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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

−31.056855442007407908217082418476, −29.733469364059097444765387724523, −28.82925830034413149420197517274, −27.991664908474576396037597603154, −26.602203088040684015210198994807, −25.40863034611656022855090863746, −24.36928321208964852726782088493, −23.35266807732717754047003545188, −21.57156166230747813627473992144, −21.08207619297145910813154237488, −20.10233566254985761823755143956, −18.828601403023364140133883305008, −18.29427374620668070308251175609, −16.37602856282933707777421313810, −14.867575464686852861498989548547, −13.783482714855448923058628526793, −12.75162112134057392147926545970, −12.044150595268332678260712412764, −10.07434707556612277427905348486, −9.11582363700111310687684263353, −8.20514911873065308512140732581, −5.899371105196305322850657797982, −4.49321050665415495321827938079, −2.81926820059397211407621569952, −1.720602323940253625815867944164, 2.84503090284633206137366870575, 4.0179888748980211263969466538, 5.691925090436431836826456001968, 7.2913653944145055168430979705, 7.96780529061631276693763832497, 9.687768792207466203685071569993, 10.535512378282568035258418609372, 13.07187677576216849732879064429, 13.66673033759115236486132194400, 14.84784105847165634606823552946, 15.54610255720792822712567617976, 16.97259326233436706518997911512, 18.127180791647898619816103791836, 19.29959617746370553001512370012, 20.765163991145882987910081170616, 21.76903049846450647464696849709, 22.91187749288867786187733350892, 23.91924322546731216519685894225, 25.366664119484418850020925592, 25.99329521820789546209695400011, 26.62277383687178229405552598006, 27.756785996701971190010119854056, 29.909965992567650549343481201123, 30.36953946095586937719892524753, 31.74964683452648604814898592564

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