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Properties

Label 1-5616-5616.4091-r0-0-0
Degree $1$
Conductor $5616$
Sign $0.580 - 0.814i$
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  + (0.984 − 0.173i)5-s + (−0.939 + 0.342i)7-s + (−0.342 − 0.939i)11-s − 17-s + (0.866 − 0.5i)19-s + (0.939 + 0.342i)23-s + (0.939 − 0.342i)25-s + (−0.642 + 0.766i)29-s + (−0.766 + 0.642i)31-s + (−0.866 + 0.5i)35-s + (−0.866 − 0.5i)37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (0.766 + 0.642i)47-s + (0.766 − 0.642i)49-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)5-s + (−0.939 + 0.342i)7-s + (−0.342 − 0.939i)11-s − 17-s + (0.866 − 0.5i)19-s + (0.939 + 0.342i)23-s + (0.939 − 0.342i)25-s + (−0.642 + 0.766i)29-s + (−0.766 + 0.642i)31-s + (−0.866 + 0.5i)35-s + (−0.866 − 0.5i)37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (0.766 + 0.642i)47-s + (0.766 − 0.642i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5616\)    =    \(2^{4} \cdot 3^{3} \cdot 13\)
Sign: $0.580 - 0.814i$
Analytic conductor: \(26.0805\)
Root analytic conductor: \(26.0805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5616} (4091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5616,\ (0:\ ),\ 0.580 - 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.456612774 - 0.7507523964i\)
\(L(\frac12)\) \(\approx\) \(1.456612774 - 0.7507523964i\)
\(L(1)\) \(\approx\) \(1.087943524 - 0.1159752081i\)
\(L(1)\) \(\approx\) \(1.087943524 - 0.1159752081i\)

Euler product

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

−17.90635080124202680777078776034, −17.22340742660468930464520663450, −16.794961835581599437870132398373, −15.937495946682896393362574784733, −15.3186258522902659708599246532, −14.667357957144231133436595378596, −13.79355213458045514878534265938, −13.34736025481450854123765903482, −12.78206214112941294918467423729, −12.14184907312886535706284549331, −11.12444316776353650211561032703, −10.46455966389197894557510147323, −9.92777529086037423570186102648, −9.2852466490398294786288130369, −8.82233262529484060104084055134, −7.47303378806656969987592153861, −7.189513973184736925650546616128, −6.33683358501884192519528811122, −5.75154935342117896324835348899, −4.97147765260760665593358718117, −4.15177699952579572235659828616, −3.30367870101677012613186121209, −2.47714687728870206211530372468, −1.91512087970884955107012611506, −0.82897136008551990156499721022, 0.49841842180520776742985921178, 1.50413757702569527449738411410, 2.43155073204781537562778371338, 3.05988457289190392519597413001, 3.72137399662218017765830403220, 4.996124614003632064423568053016, 5.4081698029568534421553803487, 6.12632890500268143192991427018, 6.80081606899575673862169326035, 7.43033186743960900914786390245, 8.64414295832505966989725997514, 9.09617347508810323018520519727, 9.47417682878018317969533067115, 10.51910615353783383349984120249, 10.885802356727866424283990945141, 11.77585364204951073800522765401, 12.67788365648524247451127644667, 13.18233215560563983118672438008, 13.59613200165623486931726031555, 14.341261416557459430311900891183, 15.1527583967420441767783918864, 15.98663821423565154093793028888, 16.29141026065904863222228225484, 17.05915336694181416898936651400, 17.87664567929894022478494993889

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