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Properties

Label 1-79-79.13-r0-0-0
Degree $1$
Conductor $79$
Sign $0.713 - 0.700i$
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  + (−0.0402 − 0.999i)2-s + (−0.919 + 0.391i)3-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (0.428 + 0.903i)6-s + (0.799 + 0.600i)7-s + (0.120 + 0.992i)8-s + (0.692 − 0.721i)9-s + (0.120 − 0.992i)10-s + (−0.632 − 0.774i)11-s + (0.885 − 0.464i)12-s + (0.428 − 0.903i)13-s + (0.568 − 0.822i)14-s + (−0.970 + 0.239i)15-s + (0.987 − 0.160i)16-s + (0.568 + 0.822i)17-s + ⋯
L(s)  = 1  + (−0.0402 − 0.999i)2-s + (−0.919 + 0.391i)3-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (0.428 + 0.903i)6-s + (0.799 + 0.600i)7-s + (0.120 + 0.992i)8-s + (0.692 − 0.721i)9-s + (0.120 − 0.992i)10-s + (−0.632 − 0.774i)11-s + (0.885 − 0.464i)12-s + (0.428 − 0.903i)13-s + (0.568 − 0.822i)14-s + (−0.970 + 0.239i)15-s + (0.987 − 0.160i)16-s + (0.568 + 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7558042557 - 0.3092119683i\)
\(L(\frac12)\) \(\approx\) \(0.7558042557 - 0.3092119683i\)
\(L(1)\) \(\approx\) \(0.8403133271 - 0.2755294297i\)
\(L(1)\) \(\approx\) \(0.8403133271 - 0.2755294297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.0402 - 0.999i)T \)
3 \( 1 + (-0.919 + 0.391i)T \)
5 \( 1 + (0.987 + 0.160i)T \)
7 \( 1 + (0.799 + 0.600i)T \)
11 \( 1 + (-0.632 - 0.774i)T \)
13 \( 1 + (0.428 - 0.903i)T \)
17 \( 1 + (0.568 + 0.822i)T \)
19 \( 1 + (0.948 - 0.316i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.278 - 0.960i)T \)
31 \( 1 + (-0.845 + 0.534i)T \)
37 \( 1 + (-0.200 + 0.979i)T \)
41 \( 1 + (-0.354 - 0.935i)T \)
43 \( 1 + (-0.632 + 0.774i)T \)
47 \( 1 + (-0.200 - 0.979i)T \)
53 \( 1 + (-0.919 - 0.391i)T \)
59 \( 1 + (-0.996 - 0.0804i)T \)
61 \( 1 + (-0.748 - 0.663i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (0.120 + 0.992i)T \)
73 \( 1 + (0.428 + 0.903i)T \)
83 \( 1 + (-0.996 + 0.0804i)T \)
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

−31.28143721178879262631271366290, −30.2315940476834433139225882149, −28.93988996336647101920774657419, −28.097070133140089968539441263777, −26.90732357365563139849214443018, −25.7415473606990927003284351809, −24.66722626561265633342616272294, −23.82582647496526411778492523401, −22.94456634775464540292696695886, −21.80261278635585399325169307067, −20.60931477887813210681051657339, −18.344700845256613510576718479989, −18.07751180330815639171688412547, −16.87464527822636468548760987320, −16.17477239601947276719031846722, −14.374624723556434142627465889884, −13.579147445002112306685345690828, −12.33392332920330597460142083192, −10.64959487017388572964484609301, −9.49914324578252813459161328502, −7.77139362311440783748960808762, −6.77174556431516402783525996522, −5.47370612185570607686899352924, −4.61542331392477199208643162678, −1.48305488266479815750823358718, 1.48227914225945312863530221369, 3.26211328932648937996316736749, 5.17687012786206973006139596274, 5.772791727713834491966598833101, 8.23127974638593742913678599573, 9.68941236725762686189816610196, 10.629659978386983102494257505573, 11.53880643990746183571918581414, 12.79064935219788873253699951023, 13.94867126909770865814230428799, 15.43776495352708517475598120825, 17.09767303959279774676109351260, 17.95951758177449622519193407298, 18.61497750041326171878257053256, 20.4840378026045029250774958953, 21.49202173395756066038961266317, 21.85491098434872231363087043728, 23.14571732565825901371260498726, 24.31561311765981055921086222981, 25.93896543536157610238677790167, 27.134500422002131078879551618294, 28.049783911192863063757060991604, 28.84433372061131864631032626300, 29.744711903167143353536425463099, 30.66203406989300387436590157945

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