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Properties

Label 2-2592-648.619-c0-0-0
Degree $2$
Conductor $2592$
Sign $0.657 - 0.753i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
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  + (−0.973 − 0.230i)3-s + (0.893 + 0.448i)9-s + (0.227 + 0.758i)11-s + (0.109 + 0.0397i)17-s + (−0.539 + 0.196i)19-s + (−0.0581 + 0.998i)25-s + (−0.766 − 0.642i)27-s + (−0.0460 − 0.790i)33-s + (−1.62 + 1.06i)41-s + (1.62 + 0.385i)43-s + (0.973 − 0.230i)49-s + (−0.0971 − 0.0639i)51-s + (0.569 − 0.0665i)57-s + (−0.393 + 1.31i)59-s + (1.77 + 0.891i)67-s + ⋯
L(s)  = 1  + (−0.973 − 0.230i)3-s + (0.893 + 0.448i)9-s + (0.227 + 0.758i)11-s + (0.109 + 0.0397i)17-s + (−0.539 + 0.196i)19-s + (−0.0581 + 0.998i)25-s + (−0.766 − 0.642i)27-s + (−0.0460 − 0.790i)33-s + (−1.62 + 1.06i)41-s + (1.62 + 0.385i)43-s + (0.973 − 0.230i)49-s + (−0.0971 − 0.0639i)51-s + (0.569 − 0.0665i)57-s + (−0.393 + 1.31i)59-s + (1.77 + 0.891i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.657 - 0.753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8026538609\)
\(L(\frac12)\) \(\approx\) \(0.8026538609\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.973 + 0.230i)T \)
good5 \( 1 + (0.0581 - 0.998i)T^{2} \)
7 \( 1 + (-0.973 + 0.230i)T^{2} \)
11 \( 1 + (-0.227 - 0.758i)T + (-0.835 + 0.549i)T^{2} \)
13 \( 1 + (0.993 - 0.116i)T^{2} \)
17 \( 1 + (-0.109 - 0.0397i)T + (0.766 + 0.642i)T^{2} \)
19 \( 1 + (0.539 - 0.196i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.973 - 0.230i)T^{2} \)
29 \( 1 + (-0.597 + 0.802i)T^{2} \)
31 \( 1 + (0.286 + 0.957i)T^{2} \)
37 \( 1 + (-0.173 + 0.984i)T^{2} \)
41 \( 1 + (1.62 - 1.06i)T + (0.396 - 0.918i)T^{2} \)
43 \( 1 + (-1.62 - 0.385i)T + (0.893 + 0.448i)T^{2} \)
47 \( 1 + (0.286 - 0.957i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.393 - 1.31i)T + (-0.835 - 0.549i)T^{2} \)
61 \( 1 + (0.686 + 0.727i)T^{2} \)
67 \( 1 + (-1.77 - 0.891i)T + (0.597 + 0.802i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.310 + 1.76i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (-0.396 - 0.918i)T^{2} \)
83 \( 1 + (-1.28 - 0.841i)T + (0.396 + 0.918i)T^{2} \)
89 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.819 - 0.868i)T + (-0.0581 - 0.998i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.335179909696692870548086443576, −8.301689105370467693037687643126, −7.46287680543103673185714322421, −6.86077232562219547016292595490, −6.10277961589951080785395161312, −5.31002801397328813143308992159, −4.55563714071502623454906265937, −3.70957147269024605863936249371, −2.29829216124685130204376396363, −1.25214524331529426382373102222, 0.67179267947645672608439054894, 2.11715768179747037850304357078, 3.46650392023588962024550059983, 4.25017119331950697802576677639, 5.13021664420774738997939363463, 5.88545662195190116664048884915, 6.53308892470418652961792881932, 7.26802920403820258113407206832, 8.277605077097936673761356778012, 8.980584559811754613248933165308

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