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Properties

Label 1-5616-5616.5027-r0-0-0
Degree $1$
Conductor $5616$
Sign $0.636 + 0.771i$
Analytic cond. $26.0805$
Root an. cond. $26.0805$
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  + (0.642 + 0.766i)5-s + (0.173 − 0.984i)7-s + (−0.984 − 0.173i)11-s − 17-s + (−0.866 + 0.5i)19-s + (−0.173 − 0.984i)23-s + (−0.173 + 0.984i)25-s + (0.342 + 0.939i)29-s + (0.939 + 0.342i)31-s + (0.866 − 0.5i)35-s + (0.866 + 0.5i)37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (−0.939 + 0.342i)47-s + (−0.939 − 0.342i)49-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)5-s + (0.173 − 0.984i)7-s + (−0.984 − 0.173i)11-s − 17-s + (−0.866 + 0.5i)19-s + (−0.173 − 0.984i)23-s + (−0.173 + 0.984i)25-s + (0.342 + 0.939i)29-s + (0.939 + 0.342i)31-s + (0.866 − 0.5i)35-s + (0.866 + 0.5i)37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (−0.939 + 0.342i)47-s + (−0.939 − 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5616\)    =    \(2^{4} \cdot 3^{3} \cdot 13\)
Sign: $0.636 + 0.771i$
Analytic conductor: \(26.0805\)
Root analytic conductor: \(26.0805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5616} (5027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5616,\ (0:\ ),\ 0.636 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.367633357 + 0.6445212371i\)
\(L(\frac12)\) \(\approx\) \(1.367633357 + 0.6445212371i\)
\(L(1)\) \(\approx\) \(1.055132500 + 0.08580949644i\)
\(L(1)\) \(\approx\) \(1.055132500 + 0.08580949644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.642 + 0.766i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (-0.984 - 0.173i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.342 + 0.939i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.984 + 0.173i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−17.8122180593930058250809896463, −17.30739731729801754990305209393, −16.220596340041693383501939790561, −15.89608073981780434694288847278, −15.12033633827517360495748584663, −14.61023857677217302745684106243, −13.42237899385780618405636940652, −13.210141535229256584749437837125, −12.64304206265478039829986661772, −11.66652198182327625977668809816, −11.29569595324816584429965421566, −10.23240522783389607484890643569, −9.667581963161821791524320222315, −9.0377166280973696637071392731, −8.2943611067977316077400989196, −7.91346279605934159953474680507, −6.702812885613694472735765608975, −6.06652350914038861463037392430, −5.43970518886596601570612020229, −4.73844016272996062123031437587, −4.205217029011526668857536492525, −2.81232642314253004845568456160, −2.34204751336445437165708409597, −1.67326266769825188782069103585, −0.46271573241921924801839599166, 0.783599204482829707935220421873, 1.890819842334692917583979003699, 2.52491297179603917274423683239, 3.27440831000529672315817632414, 4.214191643511609874978970346789, 4.819139386435517706861234839575, 5.72529059109818276308919724078, 6.56878547758811821629865901555, 6.87564911213112260502008313808, 7.84605999636814516071319072819, 8.38981575180759809683167452898, 9.29274131293206337216054708667, 10.192973172834081623302508138303, 10.62598317745868672995203848511, 10.905876016480437671254211599924, 11.93338408209887922745777610563, 12.87780404235733152828921189501, 13.34114066029902922040214396309, 13.96559472563852870772496771292, 14.57783920572449706241998390127, 15.177980463573085033827607243366, 16.04137418834077582353511707251, 16.62595313306424325650277317097, 17.43222525753910395434292914212, 17.85503204679518780712818879674

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