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Properties

Label 1-1352-1352.901-r0-0-0
Degree $1$
Conductor $1352$
Sign $0.360 - 0.932i$
Analytic cond. $6.27866$
Root an. cond. $6.27866$
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  + (0.919 − 0.391i)3-s + (0.568 − 0.822i)5-s + (−0.948 + 0.316i)7-s + (0.692 − 0.721i)9-s + (0.692 + 0.721i)11-s + (0.200 − 0.979i)15-s + (0.948 − 0.316i)17-s + (−0.5 + 0.866i)19-s + (−0.748 + 0.663i)21-s + (−0.5 − 0.866i)23-s + (−0.354 − 0.935i)25-s + (0.354 − 0.935i)27-s + (−0.278 − 0.960i)29-s + (0.354 − 0.935i)31-s + (0.919 + 0.391i)33-s + ⋯
L(s)  = 1  + (0.919 − 0.391i)3-s + (0.568 − 0.822i)5-s + (−0.948 + 0.316i)7-s + (0.692 − 0.721i)9-s + (0.692 + 0.721i)11-s + (0.200 − 0.979i)15-s + (0.948 − 0.316i)17-s + (−0.5 + 0.866i)19-s + (−0.748 + 0.663i)21-s + (−0.5 − 0.866i)23-s + (−0.354 − 0.935i)25-s + (0.354 − 0.935i)27-s + (−0.278 − 0.960i)29-s + (0.354 − 0.935i)31-s + (0.919 + 0.391i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $0.360 - 0.932i$
Analytic conductor: \(6.27866\)
Root analytic conductor: \(6.27866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1352,\ (0:\ ),\ 0.360 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.904315200 - 1.305761434i\)
\(L(\frac12)\) \(\approx\) \(1.904315200 - 1.305761434i\)
\(L(1)\) \(\approx\) \(1.461779929 - 0.4545566840i\)
\(L(1)\) \(\approx\) \(1.461779929 - 0.4545566840i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.919 - 0.391i)T \)
5 \( 1 + (0.568 - 0.822i)T \)
7 \( 1 + (-0.948 + 0.316i)T \)
11 \( 1 + (0.692 + 0.721i)T \)
17 \( 1 + (0.948 - 0.316i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.278 - 0.960i)T \)
31 \( 1 + (0.354 - 0.935i)T \)
37 \( 1 + (0.987 - 0.160i)T \)
41 \( 1 + (0.919 - 0.391i)T \)
43 \( 1 + (-0.987 - 0.160i)T \)
47 \( 1 + (-0.885 + 0.464i)T \)
53 \( 1 + (0.748 + 0.663i)T \)
59 \( 1 + (0.428 + 0.903i)T \)
61 \( 1 + (0.200 + 0.979i)T \)
67 \( 1 + (-0.845 - 0.534i)T \)
71 \( 1 + (-0.799 - 0.600i)T \)
73 \( 1 + (0.970 - 0.239i)T \)
79 \( 1 + (0.885 - 0.464i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.996 - 0.0804i)T \)
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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

−21.32713829499641647034750309564, −20.01410577112564844415612436560, −19.57285715689064330089305516265, −18.96901675857431835924472611107, −18.164050525002716507791240458119, −17.13271281372794492654822406903, −16.375724812159879787349111007173, −15.69321954225919792709871217165, −14.64561919915267103775863030703, −14.339761057679222518008994650, −13.38825652823239487859055040728, −12.97589592222993158135043161830, −11.64087385691349908510153773206, −10.72728286476771873051527036241, −10.014866159966721837948077763032, −9.45286329030644956028197274943, −8.65520040091634874251208826, −7.63666844995698496037919543365, −6.78129645829952868829845349835, −6.125957112967599573182954007482, −5.0043977032564653754173377087, −3.619639646883722999432466125844, −3.38985705467981014644727184782, −2.42635607847468053438955960792, −1.30052200441201973890818468718, 0.84525904958241662473340590005, 1.929329067393566501398446560770, 2.62353178218431455766585543232, 3.79977132122968251604786072568, 4.466790573736473792392978958960, 5.88412329795765673056443849022, 6.34468005655464479452251295597, 7.47011702327194548761308135851, 8.243291510711655693337599039329, 9.11243993648842036250194083910, 9.69871028686133551402199298598, 10.17799592067146354674808731074, 11.96888427278105058626915070096, 12.33276356143606731506162336540, 13.09371810030654420756270881706, 13.730988852093394139208718077871, 14.63328687720665661110585649001, 15.20738734464834737670078164213, 16.39224106419553541801259492101, 16.72576612272657757549218277801, 17.85674394511619569134716186071, 18.55845653215788112770431980497, 19.36492112345875849909615213653, 19.91722217930894841977783116792, 20.80386895653977899198048803546

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