Properties

Label 2-648-648.187-c0-0-0
Degree $2$
Conductor $648$
Sign $0.952 + 0.305i$
Analytic cond. $0.323394$
Root an. cond. $0.568677$
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  + (0.893 − 0.448i)2-s + (−0.686 + 0.727i)3-s + (0.597 − 0.802i)4-s + (−0.286 + 0.957i)6-s + (0.173 − 0.984i)8-s + (−0.0581 − 0.998i)9-s + (1.16 + 0.275i)11-s + (0.173 + 0.984i)12-s + (−0.286 − 0.957i)16-s + (1.57 + 0.571i)17-s + (−0.500 − 0.866i)18-s + (−1.82 + 0.665i)19-s + (1.16 − 0.275i)22-s + (0.597 + 0.802i)24-s + (−0.835 − 0.549i)25-s + ⋯
L(s)  = 1  + (0.893 − 0.448i)2-s + (−0.686 + 0.727i)3-s + (0.597 − 0.802i)4-s + (−0.286 + 0.957i)6-s + (0.173 − 0.984i)8-s + (−0.0581 − 0.998i)9-s + (1.16 + 0.275i)11-s + (0.173 + 0.984i)12-s + (−0.286 − 0.957i)16-s + (1.57 + 0.571i)17-s + (−0.500 − 0.866i)18-s + (−1.82 + 0.665i)19-s + (1.16 − 0.275i)22-s + (0.597 + 0.802i)24-s + (−0.835 − 0.549i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.952 + 0.305i$
Analytic conductor: \(0.323394\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ 0.952 + 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.291380429\)
\(L(\frac12)\) \(\approx\) \(1.291380429\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.893 + 0.448i)T \)
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

−10.70716598974547888179567643079, −10.14733746166584988226188216206, −9.418827049596235342658372039413, −8.137404354408620170645329107161, −6.59679554498683474347203896074, −6.15008737905165369608162086921, −5.10874679156340899095832734268, −4.10921421989493223474440663100, −3.48875568387725635237668670730, −1.66458176566064431164403609055, 1.80172039958653432120378480743, 3.31296644192748182659080611566, 4.51351549924167581399763489475, 5.50090746785284466708217928665, 6.33503169881069216898373759701, 6.97538724980102166591150841236, 7.900944363436951773435078820956, 8.782673305750210132345062960983, 10.16848579141557827292856247454, 11.24488381351012770589532151385

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