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Properties

Label 1-26e2-676.139-r1-0-0
Degree $1$
Conductor $676$
Sign $0.00929 + 0.999i$
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  + (0.996 + 0.0804i)3-s + (0.120 + 0.992i)5-s + (0.845 − 0.534i)7-s + (0.987 + 0.160i)9-s + (−0.987 + 0.160i)11-s + (0.0402 + 0.999i)15-s + (−0.845 + 0.534i)17-s + (0.5 − 0.866i)19-s + (0.885 − 0.464i)21-s + (0.5 + 0.866i)23-s + (−0.970 + 0.239i)25-s + (0.970 + 0.239i)27-s + (−0.632 + 0.774i)29-s + (0.970 + 0.239i)31-s + (−0.996 + 0.0804i)33-s + ⋯
L(s)  = 1  + (0.996 + 0.0804i)3-s + (0.120 + 0.992i)5-s + (0.845 − 0.534i)7-s + (0.987 + 0.160i)9-s + (−0.987 + 0.160i)11-s + (0.0402 + 0.999i)15-s + (−0.845 + 0.534i)17-s + (0.5 − 0.866i)19-s + (0.885 − 0.464i)21-s + (0.5 + 0.866i)23-s + (−0.970 + 0.239i)25-s + (0.970 + 0.239i)27-s + (−0.632 + 0.774i)29-s + (0.970 + 0.239i)31-s + (−0.996 + 0.0804i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00929 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00929 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.00929 + 0.999i$
Analytic conductor: \(72.6462\)
Root analytic conductor: \(72.6462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 676,\ (1:\ ),\ 0.00929 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.200935455 + 2.180573008i\)
\(L(\frac12)\) \(\approx\) \(2.200935455 + 2.180573008i\)
\(L(1)\) \(\approx\) \(1.540428853 + 0.4624810773i\)
\(L(1)\) \(\approx\) \(1.540428853 + 0.4624810773i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.996 + 0.0804i)T \)
5 \( 1 + (0.120 + 0.992i)T \)
7 \( 1 + (0.845 - 0.534i)T \)
11 \( 1 + (-0.987 + 0.160i)T \)
17 \( 1 + (-0.845 + 0.534i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.632 + 0.774i)T \)
31 \( 1 + (0.970 + 0.239i)T \)
37 \( 1 + (0.278 + 0.960i)T \)
41 \( 1 + (-0.996 - 0.0804i)T \)
43 \( 1 + (-0.278 + 0.960i)T \)
47 \( 1 + (0.748 + 0.663i)T \)
53 \( 1 + (0.885 + 0.464i)T \)
59 \( 1 + (0.919 + 0.391i)T \)
61 \( 1 + (-0.0402 + 0.999i)T \)
67 \( 1 + (0.200 - 0.979i)T \)
71 \( 1 + (-0.428 - 0.903i)T \)
73 \( 1 + (-0.354 + 0.935i)T \)
79 \( 1 + (0.748 + 0.663i)T \)
83 \( 1 + (-0.568 - 0.822i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.799 + 0.600i)T \)
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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

−22.12981138673453852735632544959, −21.155480438687257139682051409037, −20.692887365857469539215352937715, −20.21566858894761725993825735497, −18.995415769058226686998536909844, −18.398693699099119643133151110001, −17.539819236409637062864319812359, −16.422784177507128580500264079, −15.61847436302830354451350260088, −14.96378211933187982885486500757, −13.89902538283719103969294293572, −13.30330126218198312670843366363, −12.450904249735886106388185253269, −11.58630206988275295607142740491, −10.354637694643949823023282434664, −9.437116488096037213035653304347, −8.519354891937445063762781792521, −8.16322277945194598065200312602, −7.13540942258449636566799429980, −5.661276140112675502045781806887, −4.86899624570459491316493680474, −3.96681813522476931280521559654, −2.56496620634430749306792368777, −1.89838876689806737403417258412, −0.607824049859939153000939842029, 1.35180110209143336704685410819, 2.42448456907116660228457128465, 3.18541737775277919604223320209, 4.28752671149847877448480396553, 5.22423547006596579483815012423, 6.72755656639405201835695235439, 7.422072191023638110783446344363, 8.13446355557008289288601916302, 9.14897404127666900314011933352, 10.21271766637852847814220653131, 10.772960208587085639690216248007, 11.68491597740282316058939989549, 13.28946593853785766201714734659, 13.495494668308201867318483777889, 14.55356315890633626610120031892, 15.18700358769542370046528321347, 15.77195247044976672223760301212, 17.20650731416380289505310399945, 18.00731435889728865382965364873, 18.60576851858068771238379467078, 19.61082169814881069136015544921, 20.23475461916091156091089812537, 21.180629462597826090760785298037, 21.68632077131175833427503053516, 22.68846173372079943190334656006

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