Properties

Label 1-79-79.51-r0-0-0
Degree $1$
Conductor $79$
Sign $-0.689 + 0.724i$
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  + (0.692 + 0.721i)2-s + (−0.200 + 0.979i)3-s + (−0.0402 + 0.999i)4-s + (−0.996 + 0.0804i)5-s + (−0.845 + 0.534i)6-s + (0.948 − 0.316i)7-s + (−0.748 + 0.663i)8-s + (−0.919 − 0.391i)9-s + (−0.748 − 0.663i)10-s + (0.428 + 0.903i)11-s + (−0.970 − 0.239i)12-s + (−0.845 − 0.534i)13-s + (0.885 + 0.464i)14-s + (0.120 − 0.992i)15-s + (−0.996 − 0.0804i)16-s + (0.885 − 0.464i)17-s + ⋯
L(s)  = 1  + (0.692 + 0.721i)2-s + (−0.200 + 0.979i)3-s + (−0.0402 + 0.999i)4-s + (−0.996 + 0.0804i)5-s + (−0.845 + 0.534i)6-s + (0.948 − 0.316i)7-s + (−0.748 + 0.663i)8-s + (−0.919 − 0.391i)9-s + (−0.748 − 0.663i)10-s + (0.428 + 0.903i)11-s + (−0.970 − 0.239i)12-s + (−0.845 − 0.534i)13-s + (0.885 + 0.464i)14-s + (0.120 − 0.992i)15-s + (−0.996 − 0.0804i)16-s + (0.885 − 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.689 + 0.724i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ -0.689 + 0.724i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4409559662 + 1.028376086i\)
\(L(\frac12)\) \(\approx\) \(0.4409559662 + 1.028376086i\)
\(L(1)\) \(\approx\) \(0.8429379693 + 0.8481669066i\)
\(L(1)\) \(\approx\) \(0.8429379693 + 0.8481669066i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.692 + 0.721i)T \)
3 \( 1 + (-0.200 + 0.979i)T \)
5 \( 1 + (-0.996 + 0.0804i)T \)
7 \( 1 + (0.948 - 0.316i)T \)
11 \( 1 + (0.428 + 0.903i)T \)
13 \( 1 + (-0.845 - 0.534i)T \)
17 \( 1 + (0.885 - 0.464i)T \)
19 \( 1 + (0.987 + 0.160i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.799 + 0.600i)T \)
31 \( 1 + (0.278 - 0.960i)T \)
37 \( 1 + (-0.632 + 0.774i)T \)
41 \( 1 + (0.568 + 0.822i)T \)
43 \( 1 + (0.428 - 0.903i)T \)
47 \( 1 + (-0.632 - 0.774i)T \)
53 \( 1 + (-0.200 - 0.979i)T \)
59 \( 1 + (-0.0402 - 0.999i)T \)
61 \( 1 + (-0.354 - 0.935i)T \)
67 \( 1 + (-0.970 - 0.239i)T \)
71 \( 1 + (-0.748 + 0.663i)T \)
73 \( 1 + (-0.845 + 0.534i)T \)
83 \( 1 + (-0.0402 + 0.999i)T \)
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

−30.70049473989492146912334690126, −29.94793045401204421774483555339, −28.7752470556416063044726936134, −27.847971584067447896069590095696, −26.7882004907783136570019282385, −24.56064315247580819046848580316, −24.28529092882957967012823111557, −23.24627653693888434793431359199, −22.177751971220690253547895183025, −20.94480232631704033353857283168, −19.5836049472812685465100917897, −19.07820188240244554210640055385, −17.872657216783163629549564050488, −16.25819972901581838587888906508, −14.62303646959732751208144076440, −13.962310610262470126168311979646, −12.23416549469694924576591416809, −11.91017170501810709238917487224, −10.80448454689575015988842281721, −8.73450410848447085142511697595, −7.45365354483290944391659592000, −5.89347621321117521753353734742, −4.553600776197009970350031613100, −2.88792004467878319222504409278, −1.21693449347865705201792520500, 3.27450801892356215011944204527, 4.46717429566661469509537265378, 5.286399937034310814842515018446, 7.240673324439791048812877536247, 8.16083996144526598846281765978, 9.83990311914595515734868510314, 11.512049440983456245298442717544, 12.15430588839749434335458627233, 14.18154568172215496169336571980, 14.927126563056735140517581617121, 15.79929904885389312027281959562, 16.92269944199611119502450576411, 17.87492054282222986416734628205, 20.02147789349156221247391673184, 20.76674104184895718852829307123, 22.09210909336089717974211620925, 22.89525178405718509680442284669, 23.7772371199433913532793189990, 24.991452259529085752452451452155, 26.28570656271178925464084327839, 27.27758850048327624563659895091, 27.75931160231464373940792602677, 29.70524355037927344714868407580, 30.87963173743320706260999607847, 31.60234573046014778173041199698

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