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Properties

Label 1-26e2-676.555-r1-0-0
Degree $1$
Conductor $676$
Sign $0.940 + 0.340i$
Analytic cond. $72.6462$
Root an. cond. $72.6462$
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)3-s + (−0.748 + 0.663i)5-s + (−0.278 + 0.960i)7-s + (−0.996 + 0.0804i)9-s + (0.996 + 0.0804i)11-s + (−0.692 − 0.721i)15-s + (0.278 − 0.960i)17-s + (0.5 − 0.866i)19-s + (−0.970 − 0.239i)21-s + (0.5 + 0.866i)23-s + (0.120 − 0.992i)25-s + (−0.120 − 0.992i)27-s + (0.428 − 0.903i)29-s + (−0.120 − 0.992i)31-s + (−0.0402 + 0.999i)33-s + ⋯
L(s)  = 1  + (0.0402 + 0.999i)3-s + (−0.748 + 0.663i)5-s + (−0.278 + 0.960i)7-s + (−0.996 + 0.0804i)9-s + (0.996 + 0.0804i)11-s + (−0.692 − 0.721i)15-s + (0.278 − 0.960i)17-s + (0.5 − 0.866i)19-s + (−0.970 − 0.239i)21-s + (0.5 + 0.866i)23-s + (0.120 − 0.992i)25-s + (−0.120 − 0.992i)27-s + (0.428 − 0.903i)29-s + (−0.120 − 0.992i)31-s + (−0.0402 + 0.999i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.940 + 0.340i$
Analytic conductor: \(72.6462\)
Root analytic conductor: \(72.6462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (555, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 676,\ (1:\ ),\ 0.940 + 0.340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.441298667 + 0.2526053567i\)
\(L(\frac12)\) \(\approx\) \(1.441298667 + 0.2526053567i\)
\(L(1)\) \(\approx\) \(0.8879588645 + 0.3578489849i\)
\(L(1)\) \(\approx\) \(0.8879588645 + 0.3578489849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.0402 + 0.999i)T \)
5 \( 1 + (-0.748 + 0.663i)T \)
7 \( 1 + (-0.278 + 0.960i)T \)
11 \( 1 + (0.996 + 0.0804i)T \)
17 \( 1 + (0.278 - 0.960i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.428 - 0.903i)T \)
31 \( 1 + (-0.120 - 0.992i)T \)
37 \( 1 + (0.799 - 0.600i)T \)
41 \( 1 + (-0.0402 - 0.999i)T \)
43 \( 1 + (-0.799 - 0.600i)T \)
47 \( 1 + (0.354 + 0.935i)T \)
53 \( 1 + (-0.970 + 0.239i)T \)
59 \( 1 + (0.200 + 0.979i)T \)
61 \( 1 + (0.692 - 0.721i)T \)
67 \( 1 + (0.632 - 0.774i)T \)
71 \( 1 + (0.845 - 0.534i)T \)
73 \( 1 + (0.568 - 0.822i)T \)
79 \( 1 + (0.354 + 0.935i)T \)
83 \( 1 + (-0.885 + 0.464i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.948 - 0.316i)T \)
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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

−22.8888505051904614713034820335, −21.75885080278454987046104781796, −20.420957601628562412221853707862, −20.01695588472205190589907928059, −19.33800876921263416103057089335, −18.58344795164844288158216315028, −17.45561392122727546962277115090, −16.74229202183363660028968335896, −16.25729957822724467038640737643, −14.75996022556292783711612182721, −14.23327767211576097755010682489, −13.12178788695582803199728111103, −12.56316815041546299428752160420, −11.779767603972807981960616281093, −10.918921202160505910876333250322, −9.75503157030528227874022910321, −8.55527578359405234791466931641, −8.05368632409235841501276915683, −6.99526042845306787752261233051, −6.4018210558747758708553606791, −5.09900023905581567510222412839, −3.9492674371406999941417745850, −3.18828745941547015547873994652, −1.43947950954538312298231740354, −0.93630382802159805412541276844, 0.44218298706706596132333428431, 2.44127959439097420071794260572, 3.23054715292771084537225755489, 4.10048614736199493867405587281, 5.12690600632712642991132015162, 6.108012074385722006344948624823, 7.13616083902597773097809534317, 8.20260099749901663459190617898, 9.32653438794521281137348344092, 9.611911031062486369631377497101, 11.0252061599560508892013094037, 11.54083988998498895144478836543, 12.217562354673312036208116269581, 13.70619436857820474014300162201, 14.50474866547869387756495926504, 15.41857310202869505409483529659, 15.645921322969352289318514580039, 16.68349555276072632336701996968, 17.62018406082371826004703993387, 18.62982450360990708947935915828, 19.42275614771232042007935256297, 20.06482190996638909186939091540, 21.08129634380814854913591739768, 21.96552008585745230719409773890, 22.44918459974135340440601491767

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