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Properties

Label 1-5520-5520.4259-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.940 + 0.339i$
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  + (0.959 − 0.281i)7-s + (−0.755 + 0.654i)11-s + (−0.281 + 0.959i)13-s + (−0.142 + 0.989i)17-s + (−0.989 + 0.142i)19-s + (0.989 + 0.142i)29-s + (−0.841 − 0.540i)31-s + (0.909 + 0.415i)37-s + (0.415 + 0.909i)41-s + (−0.540 − 0.841i)43-s − 47-s + (0.841 − 0.540i)49-s + (0.281 + 0.959i)53-s + (−0.281 + 0.959i)59-s + (−0.540 + 0.841i)61-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)7-s + (−0.755 + 0.654i)11-s + (−0.281 + 0.959i)13-s + (−0.142 + 0.989i)17-s + (−0.989 + 0.142i)19-s + (0.989 + 0.142i)29-s + (−0.841 − 0.540i)31-s + (0.909 + 0.415i)37-s + (0.415 + 0.909i)41-s + (−0.540 − 0.841i)43-s − 47-s + (0.841 − 0.540i)49-s + (0.281 + 0.959i)53-s + (−0.281 + 0.959i)59-s + (−0.540 + 0.841i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.940 + 0.339i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (4259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.940 + 0.339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1255427217 + 0.7176674211i\)
\(L(\frac12)\) \(\approx\) \(0.1255427217 + 0.7176674211i\)
\(L(1)\) \(\approx\) \(0.9308023743 + 0.1762207289i\)
\(L(1)\) \(\approx\) \(0.9308023743 + 0.1762207289i\)

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 \( 1 + (0.959 - 0.281i)T \)
11 \( 1 + (-0.755 + 0.654i)T \)
13 \( 1 + (-0.281 + 0.959i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (-0.989 + 0.142i)T \)
29 \( 1 + (0.989 + 0.142i)T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + (0.909 + 0.415i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 - T \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (-0.281 + 0.959i)T \)
61 \( 1 + (-0.540 + 0.841i)T \)
67 \( 1 + (-0.755 - 0.654i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (-0.909 - 0.415i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−17.82425685237004341673577544003, −16.9703473478927097636061481189, −16.137620866705652503365247738910, −15.66027062714086584873027881411, −14.865390782792991290684072037487, −14.37952306618187069389708741799, −13.619130336738390137648971637788, −12.89734135243848492346892513674, −12.35417872107246976307671990123, −11.39322022025953990573415237249, −11.008658129848797860155051593477, −10.32478141329771424153113166512, −9.52852525359571505361314420871, −8.63378496678738003298305083029, −8.15965407790389126797853980166, −7.57078333983472517758116586663, −6.69542124524023484445767506726, −5.830089848033070943495038130489, −5.12737291260371992768896702007, −4.717606595611466721866338180864, −3.65562832957635808966344578073, −2.74805531732404684402094013671, −2.2659137781994066052878870680, −1.15993136257748994254285028171, −0.18349985336169071160574840925, 1.3181785034515283920702702405, 1.956305031446198235538525558469, 2.6250677247355818582886305764, 3.81328219177023600387642683813, 4.518354108805239829737910636739, 4.87565515755747988395247493189, 5.95444817935339184650164989309, 6.58017423871179396745859908886, 7.48276509178338778085411612742, 7.95724140937561906821115210316, 8.695603980081396725141615958073, 9.40082524026256188728099307219, 10.41886315966424199359862430213, 10.6253778014486834565444889267, 11.56034092337712327098690127162, 12.1241396992439208092569193958, 12.925044748029816927274953457080, 13.48706482202222986534711259364, 14.3240018696132615767803056300, 14.95069984402021979539887100736, 15.24528512745399303485305089113, 16.367163622571350560229220358116, 16.871626895982280672969007257131, 17.45835752306327000749429119405, 18.236651678747270044681787522150

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