Properties

Label 2-2592-648.139-c0-0-0
Degree $2$
Conductor $2592$
Sign $-0.360 - 0.932i$
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  + (0.0581 + 0.998i)3-s + (−0.993 + 0.116i)9-s + (0.512 + 0.257i)11-s + (0.606 + 0.509i)17-s + (−1.36 + 1.14i)19-s + (0.396 + 0.918i)25-s + (−0.173 − 0.984i)27-s + (−0.227 + 0.526i)33-s + (−0.0694 − 0.0932i)41-s + (0.0694 + 1.19i)43-s + (−0.0581 + 0.998i)49-s + (−0.473 + 0.635i)51-s + (−1.22 − 1.30i)57-s + (1.49 − 0.749i)59-s + (−1.36 + 0.159i)67-s + ⋯
L(s)  = 1  + (0.0581 + 0.998i)3-s + (−0.993 + 0.116i)9-s + (0.512 + 0.257i)11-s + (0.606 + 0.509i)17-s + (−1.36 + 1.14i)19-s + (0.396 + 0.918i)25-s + (−0.173 − 0.984i)27-s + (−0.227 + 0.526i)33-s + (−0.0694 − 0.0932i)41-s + (0.0694 + 1.19i)43-s + (−0.0581 + 0.998i)49-s + (−0.473 + 0.635i)51-s + (−1.22 − 1.30i)57-s + (1.49 − 0.749i)59-s + (−1.36 + 0.159i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.360 - 0.932i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ -0.360 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.093355170\)
\(L(\frac12)\) \(\approx\) \(1.093355170\)
\(L(1)\) 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 \( 1 + (-0.0581 - 0.998i)T \)
good5 \( 1 + (-0.396 - 0.918i)T^{2} \)
7 \( 1 + (0.0581 - 0.998i)T^{2} \)
11 \( 1 + (-0.512 - 0.257i)T + (0.597 + 0.802i)T^{2} \)
13 \( 1 + (0.686 + 0.727i)T^{2} \)
17 \( 1 + (-0.606 - 0.509i)T + (0.173 + 0.984i)T^{2} \)
19 \( 1 + (1.36 - 1.14i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.0581 + 0.998i)T^{2} \)
29 \( 1 + (-0.973 - 0.230i)T^{2} \)
31 \( 1 + (-0.893 - 0.448i)T^{2} \)
37 \( 1 + (0.939 + 0.342i)T^{2} \)
41 \( 1 + (0.0694 + 0.0932i)T + (-0.286 + 0.957i)T^{2} \)
43 \( 1 + (-0.0694 - 1.19i)T + (-0.993 + 0.116i)T^{2} \)
47 \( 1 + (-0.893 + 0.448i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1.49 + 0.749i)T + (0.597 - 0.802i)T^{2} \)
61 \( 1 + (0.835 + 0.549i)T^{2} \)
67 \( 1 + (1.36 - 0.159i)T + (0.973 - 0.230i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.86 - 0.679i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.286 + 0.957i)T^{2} \)
83 \( 1 + (0.207 - 0.278i)T + (-0.286 - 0.957i)T^{2} \)
89 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (1.62 - 1.06i)T + (0.396 - 0.918i)T^{2} \)
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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

−9.497039141185297856847730715494, −8.492469104090376150610010197706, −8.090546181671619063883237065814, −6.94429638108302266226993898190, −6.06299393958797455643168659870, −5.41851386374420909962467915842, −4.38326386250914940531706650616, −3.84105893832032263948564883079, −2.89964744644242775316032688290, −1.63083498498791637130970180426, 0.72090489308195568551623401066, 2.05287816733723600769647254039, 2.86586714759134692146189568969, 3.96047613101445157795766612847, 5.01306507663102748662740817360, 5.89571551627122371420062667942, 6.71559172177529225292265720247, 7.11221561619176325646487938161, 8.170969566948614450032692426850, 8.657133263895251515132185532640

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