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Properties

Label 2-2592-648.43-c0-0-0
Degree $2$
Conductor $2592$
Sign $0.987 - 0.154i$
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  + (0.993 − 0.116i)3-s + (0.973 − 0.230i)9-s + (0.997 + 1.34i)11-s + (−0.238 − 1.35i)17-s + (−0.207 + 1.17i)19-s + (−0.686 + 0.727i)25-s + (0.939 − 0.342i)27-s + (1.14 + 1.21i)33-s + (0.569 − 1.90i)41-s + (−0.569 + 0.0665i)43-s + (−0.993 − 0.116i)49-s + (−0.393 − 1.31i)51-s + (−0.0694 + 1.19i)57-s + (−0.473 + 0.635i)59-s + (0.113 − 0.0268i)67-s + ⋯
L(s)  = 1  + (0.993 − 0.116i)3-s + (0.973 − 0.230i)9-s + (0.997 + 1.34i)11-s + (−0.238 − 1.35i)17-s + (−0.207 + 1.17i)19-s + (−0.686 + 0.727i)25-s + (0.939 − 0.342i)27-s + (1.14 + 1.21i)33-s + (0.569 − 1.90i)41-s + (−0.569 + 0.0665i)43-s + (−0.993 − 0.116i)49-s + (−0.393 − 1.31i)51-s + (−0.0694 + 1.19i)57-s + (−0.473 + 0.635i)59-s + (0.113 − 0.0268i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.987 - 0.154i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.987 - 0.154i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.822011133\)
\(L(\frac12)\) \(\approx\) \(1.822011133\)
\(L(1)\) 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 \( 1 + (-0.993 + 0.116i)T \)
good5 \( 1 + (0.686 - 0.727i)T^{2} \)
7 \( 1 + (0.993 + 0.116i)T^{2} \)
11 \( 1 + (-0.997 - 1.34i)T + (-0.286 + 0.957i)T^{2} \)
13 \( 1 + (0.0581 - 0.998i)T^{2} \)
17 \( 1 + (0.238 + 1.35i)T + (-0.939 + 0.342i)T^{2} \)
19 \( 1 + (0.207 - 1.17i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.993 - 0.116i)T^{2} \)
29 \( 1 + (-0.893 - 0.448i)T^{2} \)
31 \( 1 + (-0.597 - 0.802i)T^{2} \)
37 \( 1 + (-0.766 - 0.642i)T^{2} \)
41 \( 1 + (-0.569 + 1.90i)T + (-0.835 - 0.549i)T^{2} \)
43 \( 1 + (0.569 - 0.0665i)T + (0.973 - 0.230i)T^{2} \)
47 \( 1 + (-0.597 + 0.802i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.473 - 0.635i)T + (-0.286 - 0.957i)T^{2} \)
61 \( 1 + (-0.396 - 0.918i)T^{2} \)
67 \( 1 + (-0.113 + 0.0268i)T + (0.893 - 0.448i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-1.49 - 1.25i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.835 - 0.549i)T^{2} \)
83 \( 1 + (0.539 + 1.80i)T + (-0.835 + 0.549i)T^{2} \)
89 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.707 + 1.64i)T + (-0.686 - 0.727i)T^{2} \)
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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

−9.160623943291090007005933394846, −8.383105161920266141181950503294, −7.38148354284580610452192812966, −7.15787623123410143916090408670, −6.15158833381975292993276058196, −4.99146243533184828163945223420, −4.15670295893799546745709338262, −3.46968098279942649685091779056, −2.29735761717636344661256588093, −1.53272300742316856818235962333, 1.29864136565821150952511504031, 2.44721960608617232424579156293, 3.42500123181355991277787349569, 4.04576380311407486056518544202, 4.95898288902841114599749437418, 6.32926534002660771056654654986, 6.52165444064488284435541143881, 7.88325795205166540152750879113, 8.256972879425790895066015043061, 9.035597674194945010509135879331

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