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Properties

Label 1-5616-5616.2165-r0-0-0
Degree $1$
Conductor $5616$
Sign $0.379 - 0.925i$
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  + (0.766 − 0.642i)5-s + (0.984 + 0.173i)7-s + (−0.173 + 0.984i)11-s + 17-s + (−0.5 − 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.173 − 0.984i)25-s + (−0.342 − 0.939i)29-s + (0.342 − 0.939i)31-s + (0.866 − 0.5i)35-s + (−0.5 + 0.866i)37-s + (0.642 + 0.766i)41-s + (0.342 + 0.939i)43-s + (−0.342 − 0.939i)47-s + (0.939 + 0.342i)49-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)5-s + (0.984 + 0.173i)7-s + (−0.173 + 0.984i)11-s + 17-s + (−0.5 − 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.173 − 0.984i)25-s + (−0.342 − 0.939i)29-s + (0.342 − 0.939i)31-s + (0.866 − 0.5i)35-s + (−0.5 + 0.866i)37-s + (0.642 + 0.766i)41-s + (0.342 + 0.939i)43-s + (−0.342 − 0.939i)47-s + (0.939 + 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5616\)    =    \(2^{4} \cdot 3^{3} \cdot 13\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(26.0805\)
Root analytic conductor: \(26.0805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5616} (2165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5616,\ (0:\ ),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.959697752 - 1.314108697i\)
\(L(\frac12)\) \(\approx\) \(1.959697752 - 1.314108697i\)
\(L(1)\) \(\approx\) \(1.344279601 - 0.2522999061i\)
\(L(1)\) \(\approx\) \(1.344279601 - 0.2522999061i\)

Euler product

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

−17.89541758504189066885334122846, −17.4004559631306086037138204340, −16.738913952059261041114720789525, −16.06075419526890855004626360487, −15.20608413225455304201323870091, −14.44315423885857823108824085101, −14.08387632689587873887415690339, −13.63064257122641518703820561991, −12.61066821811254824584900413360, −12.00743347104150966287720214241, −11.08603283005582131346128306284, −10.67156759845432152135545409277, −10.15192664166841129977273774154, −9.16880400908700587606164911275, −8.61765734329938999614965490816, −7.68546823576049277467853566977, −7.31171131843378401679611162144, −6.25723354239425147448136674260, −5.596111346733998278180113886343, −5.246743244732357055518058844486, −4.04500701192195175426272037954, −3.38999159209715981330354559204, −2.60888432157748193916022161901, −1.663412162072630922611210468724, −1.145184923669403412214096378904, 0.59637078455839935007484472578, 1.59608771986085806484911554754, 2.1407216447292429217992968674, 2.8861450756287015614531776689, 4.24149160726395535663930085982, 4.66891457218744791370036728235, 5.28614156042301625779268759469, 6.074715219898173956769116729572, 6.77772207797610514302794100786, 7.817873785365221203720856505721, 8.18397716559206880927135018123, 9.01580036638320228199512501691, 9.75974745132298906450886283790, 10.18050136589779102064815433211, 11.116839872076722345092497878142, 11.78215831622071134289615631348, 12.48527736182572074466500951725, 13.04630942475741265155523381019, 13.731871580241994499064332082924, 14.52081938865640364827928939242, 14.9471467883241446601810468235, 15.689114741067989035297500832705, 16.597471277483846238310814427066, 17.1484658216481800853025701107, 17.609718956896601378050318468969

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