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Properties

Label 2-2592-648.547-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  + (0.835 + 0.549i)3-s + (0.396 + 0.918i)9-s + (0.113 − 1.94i)11-s + (0.914 − 0.767i)17-s + (0.0890 + 0.0747i)19-s + (0.597 + 0.802i)25-s + (−0.173 + 0.984i)27-s + (1.16 − 1.56i)33-s + (1.65 + 0.193i)41-s + (−1.65 − 1.09i)43-s + (−0.835 + 0.549i)49-s + (1.18 − 0.138i)51-s + (0.0333 + 0.111i)57-s + (0.103 + 1.78i)59-s + (0.227 + 0.526i)67-s + ⋯
L(s)  = 1  + (0.835 + 0.549i)3-s + (0.396 + 0.918i)9-s + (0.113 − 1.94i)11-s + (0.914 − 0.767i)17-s + (0.0890 + 0.0747i)19-s + (0.597 + 0.802i)25-s + (−0.173 + 0.984i)27-s + (1.16 − 1.56i)33-s + (1.65 + 0.193i)41-s + (−1.65 − 1.09i)43-s + (−0.835 + 0.549i)49-s + (1.18 − 0.138i)51-s + (0.0333 + 0.111i)57-s + (0.103 + 1.78i)59-s + (0.227 + 0.526i)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: $\chi_{2592} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.987 - 0.154i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.691608465\)
\(L(\frac12)\) \(\approx\) \(1.691608465\)
\(L(1)\) 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 \( 1 + (-0.835 - 0.549i)T \)
good5 \( 1 + (-0.597 - 0.802i)T^{2} \)
7 \( 1 + (0.835 - 0.549i)T^{2} \)
11 \( 1 + (-0.113 + 1.94i)T + (-0.993 - 0.116i)T^{2} \)
13 \( 1 + (0.286 + 0.957i)T^{2} \)
17 \( 1 + (-0.914 + 0.767i)T + (0.173 - 0.984i)T^{2} \)
19 \( 1 + (-0.0890 - 0.0747i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.835 + 0.549i)T^{2} \)
29 \( 1 + (0.686 + 0.727i)T^{2} \)
31 \( 1 + (0.0581 - 0.998i)T^{2} \)
37 \( 1 + (0.939 - 0.342i)T^{2} \)
41 \( 1 + (-1.65 - 0.193i)T + (0.973 + 0.230i)T^{2} \)
43 \( 1 + (1.65 + 1.09i)T + (0.396 + 0.918i)T^{2} \)
47 \( 1 + (0.0581 + 0.998i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.103 - 1.78i)T + (-0.993 + 0.116i)T^{2} \)
61 \( 1 + (-0.893 - 0.448i)T^{2} \)
67 \( 1 + (-0.227 - 0.526i)T + (-0.686 + 0.727i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (0.744 - 0.270i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.973 + 0.230i)T^{2} \)
83 \( 1 + (-0.344 + 0.0403i)T + (0.973 - 0.230i)T^{2} \)
89 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (1.22 - 0.615i)T + (0.597 - 0.802i)T^{2} \)
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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

−9.009262479226104447965609710745, −8.441776161158421625474163501763, −7.75653467746855270546870865227, −6.93986938720824538754816497181, −5.79058379371644757625381785713, −5.24198381468973920853421470399, −4.13072973351909721536989977840, −3.27096452839676575814169574695, −2.78532305494518740494445847094, −1.23055239076061700600208774757, 1.43360390172849836559938629800, 2.24700006768492972712325922841, 3.26679812428407508322523587268, 4.21726100341029354861997950041, 4.98085855590067322451276290913, 6.23869871029465080242449689329, 6.85359473061078542499916748955, 7.63547670806693175525108006809, 8.135165934950096076203573578559, 9.015284657005773384571993036315

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