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Properties

Label 2-2592-648.499-c0-0-0
Degree $2$
Conductor $2592$
Sign $0.323 - 0.946i$
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  + (0.686 + 0.727i)3-s + (−0.0581 + 0.998i)9-s + (−1.16 + 0.275i)11-s + (1.57 − 0.571i)17-s + (1.82 + 0.665i)19-s + (−0.835 + 0.549i)25-s + (−0.766 + 0.642i)27-s + (−0.997 − 0.656i)33-s + (−1.22 + 0.615i)41-s + (1.22 + 1.30i)43-s + (−0.686 + 0.727i)49-s + (1.49 + 0.749i)51-s + (0.770 + 1.78i)57-s + (0.558 + 0.132i)59-s + (0.0460 − 0.790i)67-s + ⋯
L(s)  = 1  + (0.686 + 0.727i)3-s + (−0.0581 + 0.998i)9-s + (−1.16 + 0.275i)11-s + (1.57 − 0.571i)17-s + (1.82 + 0.665i)19-s + (−0.835 + 0.549i)25-s + (−0.766 + 0.642i)27-s + (−0.997 − 0.656i)33-s + (−1.22 + 0.615i)41-s + (1.22 + 1.30i)43-s + (−0.686 + 0.727i)49-s + (1.49 + 0.749i)51-s + (0.770 + 1.78i)57-s + (0.558 + 0.132i)59-s + (0.0460 − 0.790i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.323 - 0.946i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.323 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.490956825\)
\(L(\frac12)\) \(\approx\) \(1.490956825\)
\(L(1)\) 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 \( 1 + (-0.686 - 0.727i)T \)
good5 \( 1 + (0.835 - 0.549i)T^{2} \)
7 \( 1 + (0.686 - 0.727i)T^{2} \)
11 \( 1 + (1.16 - 0.275i)T + (0.893 - 0.448i)T^{2} \)
13 \( 1 + (-0.396 - 0.918i)T^{2} \)
17 \( 1 + (-1.57 + 0.571i)T + (0.766 - 0.642i)T^{2} \)
19 \( 1 + (-1.82 - 0.665i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (0.686 + 0.727i)T^{2} \)
29 \( 1 + (0.993 - 0.116i)T^{2} \)
31 \( 1 + (-0.973 + 0.230i)T^{2} \)
37 \( 1 + (-0.173 - 0.984i)T^{2} \)
41 \( 1 + (1.22 - 0.615i)T + (0.597 - 0.802i)T^{2} \)
43 \( 1 + (-1.22 - 1.30i)T + (-0.0581 + 0.998i)T^{2} \)
47 \( 1 + (-0.973 - 0.230i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.558 - 0.132i)T + (0.893 + 0.448i)T^{2} \)
61 \( 1 + (0.286 + 0.957i)T^{2} \)
67 \( 1 + (-0.0460 + 0.790i)T + (-0.993 - 0.116i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.0201 + 0.114i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (-0.597 - 0.802i)T^{2} \)
83 \( 1 + (1.36 + 0.687i)T + (0.597 + 0.802i)T^{2} \)
89 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.569 + 1.90i)T + (-0.835 - 0.549i)T^{2} \)
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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

−9.528705116886015520467122398601, −8.353929783477977953453315303874, −7.66498355434438140523490867627, −7.40875647683821946253533203738, −5.79470867047011991299695949380, −5.30772295372463821877257326043, −4.49906101954500903348923367865, −3.29870435627552255020741481576, −2.95769571327057761950084349778, −1.57271122656731010308753591351, 0.976825832805072159265396401704, 2.23686943242191202854800703026, 3.12348967431904607159113531540, 3.80465260985551774022268713404, 5.28365087066958312572161868459, 5.69449800480866289272153146222, 6.84671299108292432046031517010, 7.55213328141704051505681434769, 8.032003976820636994998901749773, 8.730861765001367375679657990226

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