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Properties

Label 1-4560-4560.4253-r0-0-0
Degree $1$
Conductor $4560$
Sign $-0.880 + 0.474i$
Analytic cond. $21.1765$
Root an. cond. $21.1765$
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  + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)11-s + (−0.766 − 0.642i)13-s + (−0.984 + 0.173i)17-s + (−0.342 − 0.939i)23-s + (0.984 + 0.173i)29-s + (−0.5 + 0.866i)31-s − 37-s + (0.766 − 0.642i)41-s + (−0.939 − 0.342i)43-s + (0.984 + 0.173i)47-s + (0.5 + 0.866i)49-s + (0.939 − 0.342i)53-s + (0.984 − 0.173i)59-s + (−0.342 − 0.939i)61-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)11-s + (−0.766 − 0.642i)13-s + (−0.984 + 0.173i)17-s + (−0.342 − 0.939i)23-s + (0.984 + 0.173i)29-s + (−0.5 + 0.866i)31-s − 37-s + (0.766 − 0.642i)41-s + (−0.939 − 0.342i)43-s + (0.984 + 0.173i)47-s + (0.5 + 0.866i)49-s + (0.939 − 0.342i)53-s + (0.984 − 0.173i)59-s + (−0.342 − 0.939i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.880 + 0.474i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (4253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ -0.880 + 0.474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04139197770 - 0.1638845146i\)
\(L(\frac12)\) \(\approx\) \(-0.04139197770 - 0.1638845146i\)
\(L(1)\) \(\approx\) \(0.7860319279 - 0.1476669026i\)
\(L(1)\) \(\approx\) \(0.7860319279 - 0.1476669026i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (0.984 + 0.173i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.984 + 0.173i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (-0.342 - 0.939i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.642 - 0.766i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (0.984 - 0.173i)T \)
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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

−18.739641327461954040725675644594, −17.79410209924698006791985424322, −17.37919196545147980772818760547, −16.52942154190799860220879027091, −15.987192996154052923522171283420, −15.21607147553357699931426996488, −14.7172525660631516863166030185, −13.83649828819589090533411837916, −13.24948139691060728288699073973, −12.47014955590820831471750214873, −11.80828154397474988815243520581, −11.42758695001557923817080277083, −10.201189419270831324446019719489, −9.750220543361788989288375445482, −9.053418087620484189507775282383, −8.556831547557367834809834662104, −7.28001503819136320768469643642, −6.96888391086931676416115802304, −6.141577997610500427647595131273, −5.46231960263148157208777510817, −4.40831625891852811245749706488, −3.97113473616379468767337566482, −2.87560705753181623099671814515, −2.250727981591216472108900553893, −1.36145864080448912588281804567, 0.0507505135901881036811764494, 0.96744001106297712318418029328, 2.10757334480698157736131019668, 2.94154024384451117875083405055, 3.67996662772440085555418658849, 4.37179262415235866543606126813, 5.24050970447854093983350707852, 6.13878744165779027696682052432, 6.77470229780563096032639682000, 7.25033886478303070767464412867, 8.36964691911631956055816143788, 8.872933844302763619531297361711, 9.6584785460298295911584755912, 10.454396820217594119107115217663, 10.796756548394274489069360648464, 11.9295335150539497654018738092, 12.39506144981513452796364100180, 13.09896706650060697978190634994, 13.86328093163406714402694585080, 14.36242919331660358679054969677, 15.2026876226813623054387794461, 15.933537632394951113318317960626, 16.45395968889204587355204666466, 17.249929273685831033022569921830, 17.63503986174393595849109074668

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