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Properties

Label 2-2592-8.3-c0-0-0
Degree $2$
Conductor $2592$
Sign $1$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11-s + 17-s + 19-s + 25-s + 41-s + 43-s + 49-s − 59-s + 67-s − 73-s + 2·83-s − 2·89-s − 97-s − 107-s − 2·113-s + ⋯
L(s)  = 1  − 11-s + 17-s + 19-s + 25-s + 41-s + 43-s + 49-s − 59-s + 67-s − 73-s + 2·83-s − 2·89-s − 97-s − 107-s − 2·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2592} (1135, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.218384032\)
\(L(\frac12)\) \(\approx\) \(1.218384032\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 - T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 - T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )^{2} \)
89 \( ( 1 + T )^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.145318799826479579052233812887, −8.168805835383482348609806393647, −7.61363438611478048295289800987, −6.90599420915071353460926827190, −5.78111058350556883127017715838, −5.29889641531184509412533275245, −4.34448377308584237939201257616, −3.25619531516544206269276004073, −2.52344962985019302672484341414, −1.08287994296300083238138786986, 1.08287994296300083238138786986, 2.52344962985019302672484341414, 3.25619531516544206269276004073, 4.34448377308584237939201257616, 5.29889641531184509412533275245, 5.78111058350556883127017715838, 6.90599420915071353460926827190, 7.61363438611478048295289800987, 8.168805835383482348609806393647, 9.145318799826479579052233812887

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