Properties

Label 2-2592-72.29-c0-0-0
Degree $2$
Conductor $2592$
Sign $-0.342 - 0.939i$
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  + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−1 + 1.73i)29-s + (−0.5 − 0.866i)31-s − 0.999·35-s − 53-s − 0.999·55-s + (1 + 1.73i)59-s − 73-s + (−0.499 − 0.866i)77-s + (1 − 1.73i)79-s + (−0.5 + 0.866i)83-s + (0.5 − 0.866i)97-s + (0.5 − 0.866i)101-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−1 + 1.73i)29-s + (−0.5 − 0.866i)31-s − 0.999·35-s − 53-s − 0.999·55-s + (1 + 1.73i)59-s − 73-s + (−0.499 − 0.866i)77-s + (1 − 1.73i)79-s + (−0.5 + 0.866i)83-s + (0.5 − 0.866i)97-s + (0.5 − 0.866i)101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.033274016\)
\(L(\frac12)\) \(\approx\) \(1.033274016\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.308421185835438886048758043886, −8.720451435882704175613158966602, −7.56761468696955505465284696320, −7.05521667031257861121735904743, −6.16677892930053194207332949836, −5.58076918878965521926601969280, −4.66280258585824677256178483758, −3.45246061285854891360932999537, −2.64368893532343021020690490710, −1.87892607903539875032199093502, 0.64995692051352294619609908117, 1.92278553653261447378687378450, 3.19846511782465732387835815939, 4.00139006917924059981251685928, 4.99308920425133232554012307526, 5.68832144500077914025905953717, 6.45859589274929203428130007560, 7.37340318901150542981047551625, 8.123739647569303933861357751369, 8.856196251765182350681314374375

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