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Properties

Label 2-2592-216.43-c0-0-0
Degree $2$
Conductor $2592$
Sign $0.686 - 0.727i$
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  + (−0.326 + 0.118i)11-s + (0.766 + 1.32i)17-s + (−0.939 + 1.62i)19-s + (0.766 + 0.642i)25-s + (1.43 − 1.20i)41-s + (1.43 − 0.524i)43-s + (−0.939 − 0.342i)49-s + (1.76 + 0.642i)59-s + (−0.266 + 0.223i)67-s + (−0.766 + 1.32i)73-s + (−0.766 − 0.642i)83-s + (−0.5 + 0.866i)89-s + (−0.326 + 0.118i)97-s + 1.53·107-s + (−0.939 − 0.342i)113-s + ⋯
L(s)  = 1  + (−0.326 + 0.118i)11-s + (0.766 + 1.32i)17-s + (−0.939 + 1.62i)19-s + (0.766 + 0.642i)25-s + (1.43 − 1.20i)41-s + (1.43 − 0.524i)43-s + (−0.939 − 0.342i)49-s + (1.76 + 0.642i)59-s + (−0.266 + 0.223i)67-s + (−0.766 + 1.32i)73-s + (−0.766 − 0.642i)83-s + (−0.5 + 0.866i)89-s + (−0.326 + 0.118i)97-s + 1.53·107-s + (−0.939 − 0.342i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.686 - 0.727i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.686 - 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.148972689\)
\(L(\frac12)\) \(\approx\) \(1.148972689\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.061125082933734012076760559209, −8.362676828581099276625014162154, −7.73809012371321818110973926684, −6.91635708907360414262187466626, −5.90101537275762690498361867441, −5.51303323676246235273757427086, −4.22144545673409337907756702901, −3.66864028886608599048794510312, −2.45356546337570507460625161434, −1.39632014969099906300913460330, 0.821224091531145128149934751758, 2.45775454327244143439376554693, 3.03935522669636885035998363831, 4.38825405081167792219324991990, 4.92153951003030610557563753914, 5.90195846171961077541587048700, 6.72729801734107997418125481548, 7.43121131952015559497785307804, 8.197316679674322483881255232976, 9.046183539159848155927070928460

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