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Properties

Label 1-2652-2652.167-r0-0-0
Degree $1$
Conductor $2652$
Sign $0.229 + 0.973i$
Analytic cond. $12.3158$
Root an. cond. $12.3158$
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  + (−0.923 − 0.382i)5-s + (0.793 + 0.608i)7-s + (0.991 + 0.130i)11-s + (−0.258 + 0.965i)19-s + (−0.130 + 0.991i)23-s + (0.707 + 0.707i)25-s + (0.608 + 0.793i)29-s + (0.382 − 0.923i)31-s + (−0.5 − 0.866i)35-s + (−0.608 − 0.793i)37-s + (−0.130 + 0.991i)41-s + (0.258 − 0.965i)43-s + 47-s + (0.258 + 0.965i)49-s + (−0.707 + 0.707i)53-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)5-s + (0.793 + 0.608i)7-s + (0.991 + 0.130i)11-s + (−0.258 + 0.965i)19-s + (−0.130 + 0.991i)23-s + (0.707 + 0.707i)25-s + (0.608 + 0.793i)29-s + (0.382 − 0.923i)31-s + (−0.5 − 0.866i)35-s + (−0.608 − 0.793i)37-s + (−0.130 + 0.991i)41-s + (0.258 − 0.965i)43-s + 47-s + (0.258 + 0.965i)49-s + (−0.707 + 0.707i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2652\)    =    \(2^{2} \cdot 3 \cdot 13 \cdot 17\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(12.3158\)
Root analytic conductor: \(12.3158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2652} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2652,\ (0:\ ),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.103341579 + 0.8730530135i\)
\(L(\frac12)\) \(\approx\) \(1.103341579 + 0.8730530135i\)
\(L(1)\) \(\approx\) \(1.013639452 + 0.1647977919i\)
\(L(1)\) \(\approx\) \(1.013639452 + 0.1647977919i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (0.793 + 0.608i)T \)
11 \( 1 + (0.991 + 0.130i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (-0.130 + 0.991i)T \)
29 \( 1 + (0.608 + 0.793i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (-0.608 - 0.793i)T \)
41 \( 1 + (-0.130 + 0.991i)T \)
43 \( 1 + (0.258 - 0.965i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.965 + 0.258i)T \)
61 \( 1 + (-0.608 + 0.793i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.991 + 0.130i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.130 + 0.991i)T \)
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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

−19.27558023326281168964706343781, −18.57792870765157137820718871139, −17.567357764633448855456930076069, −17.25514808630599722144262714096, −16.28081245018382276074777163649, −15.63427347117798970335674893542, −14.840201657066609663262830546546, −14.27965350962997062515251477559, −13.68654920849584144985363921737, −12.607882137703425757935162120125, −11.85372473895097244044810090228, −11.337597529535456286006614897092, −10.65490423273290738670995268604, −9.94750557289430229255957384504, −8.65575861071248218754567384598, −8.4425581950320101374253135707, −7.33706957346299013598791172104, −6.891533205286150713815906994170, −6.061954473893167523085318012317, −4.67643646449360457800757653597, −4.426853020886457197052175532187, −3.49000824595647171117031342841, −2.62813274211968818840822449947, −1.46653585448749164639712356242, −0.50775384349624738040002694282, 1.12551970691307977520106000441, 1.80790376837598576814036317905, 3.00328561259715059725081089345, 3.953576740949069354514498100095, 4.46628653513394349138217828236, 5.42790210105705169576941116685, 6.13043990965969171146927639477, 7.26748959202738988558426446123, 7.77881639358179262243508071673, 8.7054148348416836056276423909, 9.037644171090237336725314055737, 10.13860585706892986307414210385, 11.02628174517673278757880789919, 11.84049238123515864116454894044, 12.03479965129897868399936797697, 12.89643209021532719560325600486, 13.95821854422351528381008839016, 14.59654166099355753615488540508, 15.22662896700903281330341812349, 15.86546562243686890769629138397, 16.6837181417539156548229792116, 17.31195439358110426142711236684, 18.07086976029287431346021087072, 18.9140608108095869105630612844, 19.40735847642143798103045654307

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