Properties

Label 1-5520-5520.3323-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.520 - 0.853i$
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  + (−0.989 − 0.142i)7-s + (0.909 − 0.415i)11-s + (−0.142 − 0.989i)13-s + (0.755 + 0.654i)17-s + (0.755 − 0.654i)19-s + (0.755 + 0.654i)29-s + (0.959 − 0.281i)31-s + (−0.841 − 0.540i)37-s + (0.841 − 0.540i)41-s + (0.959 + 0.281i)43-s i·47-s + (0.959 + 0.281i)49-s + (0.142 − 0.989i)53-s + (−0.989 + 0.142i)59-s + (−0.281 − 0.959i)61-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)7-s + (0.909 − 0.415i)11-s + (−0.142 − 0.989i)13-s + (0.755 + 0.654i)17-s + (0.755 − 0.654i)19-s + (0.755 + 0.654i)29-s + (0.959 − 0.281i)31-s + (−0.841 − 0.540i)37-s + (0.841 − 0.540i)41-s + (0.959 + 0.281i)43-s i·47-s + (0.959 + 0.281i)49-s + (0.142 − 0.989i)53-s + (−0.989 + 0.142i)59-s + (−0.281 − 0.959i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.520 - 0.853i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (3323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.520 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.520470202 - 0.8540145426i\)
\(L(\frac12)\) \(\approx\) \(1.520470202 - 0.8540145426i\)
\(L(1)\) \(\approx\) \(1.073328782 - 0.1690651961i\)
\(L(1)\) \(\approx\) \(1.073328782 - 0.1690651961i\)

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

−18.04681530684111495963766311389, −17.13380592377881092415369853775, −16.70235521175685671559145885967, −15.98425915444175210556494696606, −15.521545871523496866237178958240, −14.53206512510331063896085641045, −13.9942273784037427906829169147, −13.538688662417762909392671899944, −12.34363493527268321399787608665, −12.160637082826907037240558339, −11.54521501579101524544208698290, −10.47384385159797107688607462240, −9.80958441712668341752932646288, −9.34892516979856066015959219984, −8.75513234454848450942825872291, −7.63942140353921784276945815467, −7.17351673595631240469339096684, −6.25278602472343764366467127075, −5.970150583377790723345464291080, −4.75445439898934821178833598130, −4.24020221184785813173106729785, −3.28657541989790550358000944838, −2.76283771636035044857866857299, −1.67218450535669819583180372733, −0.90085234399938058079808912642, 0.59054127968083431828679509554, 1.2511922689993828209629994863, 2.5120579794288364585018102777, 3.23378811928268834749777155493, 3.71765848351960274100884078922, 4.65750982484037342057414433253, 5.63990334311133140367910819774, 6.05827273970793838995107240009, 6.94332670370700010703600112797, 7.466980091220564326968126024763, 8.4565066948909643666301781042, 8.97088869193965311356980006601, 9.87872488883014265575425595313, 10.22488119991133908862342839134, 11.08691150600662886287786804525, 11.853369608026101205813274327249, 12.56257839092191444729234275547, 12.97640374346100006754981518357, 13.949012693552205760052415501181, 14.27281150300458066196292982642, 15.27721360634858224119799895453, 15.79504313186075726443019511872, 16.40389235938975785759604085065, 17.175439975622677704173681122056, 17.59432994052344591830099020814

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