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Properties

Label 1-5616-5616.2461-r0-0-0
Degree $1$
Conductor $5616$
Sign $-0.114 + 0.993i$
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  + (0.342 − 0.939i)5-s + (0.766 + 0.642i)7-s + (−0.642 + 0.766i)11-s + 17-s + (−0.866 + 0.5i)19-s + (−0.766 + 0.642i)23-s + (−0.766 − 0.642i)25-s + (0.984 + 0.173i)29-s + (−0.173 − 0.984i)31-s + (0.866 − 0.5i)35-s + (−0.866 − 0.5i)37-s + (−0.939 − 0.342i)41-s + (−0.984 − 0.173i)43-s + (−0.173 + 0.984i)47-s + (0.173 + 0.984i)49-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)5-s + (0.766 + 0.642i)7-s + (−0.642 + 0.766i)11-s + 17-s + (−0.866 + 0.5i)19-s + (−0.766 + 0.642i)23-s + (−0.766 − 0.642i)25-s + (0.984 + 0.173i)29-s + (−0.173 − 0.984i)31-s + (0.866 − 0.5i)35-s + (−0.866 − 0.5i)37-s + (−0.939 − 0.342i)41-s + (−0.984 − 0.173i)43-s + (−0.173 + 0.984i)47-s + (0.173 + 0.984i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5616\)    =    \(2^{4} \cdot 3^{3} \cdot 13\)
Sign: $-0.114 + 0.993i$
Analytic conductor: \(26.0805\)
Root analytic conductor: \(26.0805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5616} (2461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5616,\ (0:\ ),\ -0.114 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8380278849 + 0.9400544721i\)
\(L(\frac12)\) \(\approx\) \(0.8380278849 + 0.9400544721i\)
\(L(1)\) \(\approx\) \(1.057946141 + 0.07770729733i\)
\(L(1)\) \(\approx\) \(1.057946141 + 0.07770729733i\)

Euler product

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

−17.82306093678525101577834154784, −16.94652616990256574936041750460, −16.53676551702642850524294324654, −15.52913391957952727235199815025, −15.0206544201900280635835586381, −14.14773630490613998710700373069, −13.938556114484091908170320333189, −13.23798947887969976648385150510, −12.26175285538538204028039637954, −11.60665008510752262742908333134, −10.80454636672216176243863307996, −10.40737409047294323082238714115, −9.95874631627844848025973075397, −8.74749172669348966148826393504, −8.170995254394520115874781619888, −7.58576152505701923818561017581, −6.67229913441687208281231936410, −6.27783012774996052388696400104, −5.230738668721719053869568853913, −4.75007945137716653990895239873, −3.61401062002109201813208902316, −3.149240423601663030118569176172, −2.21003885057574930293885628882, −1.48880352199805590191513542576, −0.311478386876408066970428366521, 1.10826447596537718966932923988, 1.89772827596966978994298253858, 2.36379178825827921934158877230, 3.57461084908481142145051039897, 4.397868089465288665088137996334, 5.104593613290248996048883906248, 5.54785333973589504571102419832, 6.2822078481821739582662052167, 7.37249758551265753468625018118, 8.07932041283116564581590647476, 8.46695320888878794557963813174, 9.29323034014011115692433687506, 10.03421220048222019824729352875, 10.47840782309675696981754519242, 11.63950704930057999529933700231, 12.07887725047194898154895346637, 12.63507005646282744171011693245, 13.326938637567721332786030287468, 14.07458875016194213191938933695, 14.771724697813857209362354121751, 15.3621688058669966766787989062, 16.08941701080537741187958127429, 16.681555777622058138198311262657, 17.575211166678038922869498936304, 17.74496517355685151382981327244

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