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Properties

Label 1-26e2-676.567-r0-0-0
Degree $1$
Conductor $676$
Sign $-0.736 - 0.676i$
Analytic cond. $3.13933$
Root an. cond. $3.13933$
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  + (0.354 − 0.935i)3-s + (0.239 − 0.970i)5-s + (0.822 + 0.568i)7-s + (−0.748 − 0.663i)9-s + (−0.663 − 0.748i)11-s + (−0.822 − 0.568i)15-s + (−0.568 + 0.822i)17-s − i·19-s + (0.822 − 0.568i)21-s + 23-s + (−0.885 − 0.464i)25-s + (−0.885 + 0.464i)27-s + (−0.748 − 0.663i)29-s + (−0.464 − 0.885i)31-s + (−0.935 + 0.354i)33-s + ⋯
L(s)  = 1  + (0.354 − 0.935i)3-s + (0.239 − 0.970i)5-s + (0.822 + 0.568i)7-s + (−0.748 − 0.663i)9-s + (−0.663 − 0.748i)11-s + (−0.822 − 0.568i)15-s + (−0.568 + 0.822i)17-s − i·19-s + (0.822 − 0.568i)21-s + 23-s + (−0.885 − 0.464i)25-s + (−0.885 + 0.464i)27-s + (−0.748 − 0.663i)29-s + (−0.464 − 0.885i)31-s + (−0.935 + 0.354i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.736 - 0.676i$
Analytic conductor: \(3.13933\)
Root analytic conductor: \(3.13933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 676,\ (0:\ ),\ -0.736 - 0.676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5344398267 - 1.370641795i\)
\(L(\frac12)\) \(\approx\) \(0.5344398267 - 1.370641795i\)
\(L(1)\) \(\approx\) \(0.9909399798 - 0.6579573389i\)
\(L(1)\) \(\approx\) \(0.9909399798 - 0.6579573389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.354 - 0.935i)T \)
5 \( 1 + (0.239 - 0.970i)T \)
7 \( 1 + (0.822 + 0.568i)T \)
11 \( 1 + (-0.663 - 0.748i)T \)
17 \( 1 + (-0.568 + 0.822i)T \)
19 \( 1 - iT \)
23 \( 1 + T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (-0.464 - 0.885i)T \)
37 \( 1 + (-0.464 - 0.885i)T \)
41 \( 1 + (-0.935 - 0.354i)T \)
43 \( 1 + (0.885 + 0.464i)T \)
47 \( 1 + (0.992 + 0.120i)T \)
53 \( 1 + (0.568 - 0.822i)T \)
59 \( 1 + (0.239 - 0.970i)T \)
61 \( 1 + (0.568 + 0.822i)T \)
67 \( 1 + (-0.992 - 0.120i)T \)
71 \( 1 + (0.935 + 0.354i)T \)
73 \( 1 + (0.663 + 0.748i)T \)
79 \( 1 + (-0.120 + 0.992i)T \)
83 \( 1 + (-0.935 + 0.354i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.239 - 0.970i)T \)
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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

−22.90595505648443446855428504489, −22.243164065731787680711936449933, −21.339116347424165333726794657116, −20.63051308943767195647553997506, −20.132626850622110349691909180048, −18.897173349128840396549474066031, −18.133751009024431902821947811093, −17.31743584157190862214125608066, −16.45951793616748823106656167501, −15.36491176664211064888227550335, −14.88302799172491301666107446979, −14.082070417983617447130244684683, −13.42783993253117916308965511657, −12.00358678145423434168843611488, −10.89046871244950183215302575747, −10.58003166434846919654571368902, −9.69350429999996640428127676022, −8.70655635060682625785760396851, −7.60832948625702327996420405940, −6.982890410367677042693699007781, −5.500366055239515498079546687579, −4.777661970556026654798254858151, −3.74313859195917815299525446643, −2.79683575872035697647523990391, −1.79360656676837583611633297781, 0.65902062409338839807845607651, 1.85179673464184909838505874253, 2.586739668153243875515192974907, 4.03858798872355285937329164908, 5.29878486584398229697247601712, 5.83471440504136374643241839233, 7.10664438485337995602699450568, 8.11652782713655124174345314459, 8.664249264896608081773313206, 9.342277125279820063961948156868, 10.942390743446054718913127020525, 11.57564789195047765671718108811, 12.68283739479901957907426787141, 13.145859530343001540104546350837, 13.9286408208184566819225251505, 14.98349431837006623220434140351, 15.70348724794441337627398393622, 17.00697576733676921905918472929, 17.48499064391820649866421365144, 18.39079188133672118003601868332, 19.134835301288246993233438247678, 19.95436661032839942450686252375, 20.88083771013131119133887720048, 21.32578581634352767473911670325, 22.4278733675389948390166255820

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