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Properties

Label 1-5520-5520.1757-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.997 - 0.0763i$
Analytic cond. $25.6347$
Root an. cond. $25.6347$
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  + (−0.755 + 0.654i)7-s + (−0.540 + 0.841i)11-s + (−0.654 + 0.755i)13-s + (0.909 − 0.415i)17-s + (−0.909 − 0.415i)19-s + (−0.909 + 0.415i)29-s + (−0.142 − 0.989i)31-s + (−0.959 − 0.281i)37-s + (−0.959 + 0.281i)41-s + (0.142 − 0.989i)43-s − i·47-s + (0.142 − 0.989i)49-s + (−0.654 − 0.755i)53-s + (0.755 + 0.654i)59-s + (0.989 − 0.142i)61-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)7-s + (−0.540 + 0.841i)11-s + (−0.654 + 0.755i)13-s + (0.909 − 0.415i)17-s + (−0.909 − 0.415i)19-s + (−0.909 + 0.415i)29-s + (−0.142 − 0.989i)31-s + (−0.959 − 0.281i)37-s + (−0.959 + 0.281i)41-s + (0.142 − 0.989i)43-s − i·47-s + (0.142 − 0.989i)49-s + (−0.654 − 0.755i)53-s + (0.755 + 0.654i)59-s + (0.989 − 0.142i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.997 - 0.0763i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (1757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.997 - 0.0763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9522619863 - 0.03639171848i\)
\(L(\frac12)\) \(\approx\) \(0.9522619863 - 0.03639171848i\)
\(L(1)\) \(\approx\) \(0.8169704152 + 0.08366988936i\)
\(L(1)\) \(\approx\) \(0.8169704152 + 0.08366988936i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (-0.755 + 0.654i)T \)
11 \( 1 + (-0.540 + 0.841i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (-0.909 - 0.415i)T \)
29 \( 1 + (-0.909 + 0.415i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (0.755 + 0.654i)T \)
61 \( 1 + (0.989 - 0.142i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.281 - 0.959i)T \)
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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

−17.783064126884776953489539120660, −17.078998660535758452010412750817, −16.6421503046423615462697481352, −15.96521220568041213095892797051, −15.28321464931181724012039325052, −14.53752999400189740041086183908, −13.9091018071598202387366280611, −13.16336577475684181696230192123, −12.64140683691025180939613657045, −12.08849120720993128016284813606, −10.94245850320821508239874863335, −10.57245880026764464276967224055, −9.92759038368171292415878400114, −9.260822807221543590293868822590, −8.20001465104250344067404817490, −7.899444786796756644152892280788, −6.98465131813209039066925425446, −6.291247461581116059635578808298, −5.57617542547374609572773916059, −4.91992764139880882512336840881, −3.83227219656020890150009554914, −3.351255652801584956859328333689, −2.627758869774993747871819558519, −1.55311550810743285917745077069, −0.56375362406547067280807882833, 0.40064084075401278186803453246, 1.99621355406725352101250928527, 2.21223788383794065752861669712, 3.27272562398293166731935289904, 3.96080447078502874496497007563, 4.98837132245792709612698242665, 5.38572388267205193847849571918, 6.360136742538428830134495316011, 7.02597575322590327551759446209, 7.55698041989011492488672940778, 8.56681470019788930703441683410, 9.15709127593060575432645868944, 9.88542597051314525628466353885, 10.26844807320095213029178898168, 11.33448709341513430301907767239, 11.971072004313974020479297239650, 12.56316842782605525386640252794, 13.09622298868374881113357300880, 13.86649912862239480209474363735, 14.79424661708948451748649973490, 15.12178866169611002337928204106, 15.863252908433151791154707306201, 16.63382634193207286433597551856, 17.03664985613309869567919929598, 17.950213816955892060498966686983

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