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Properties

Label 2-648-648.259-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  + (0.973 + 0.230i)2-s + (0.396 − 0.918i)3-s + (0.893 + 0.448i)4-s + (0.597 − 0.802i)6-s + (0.766 + 0.642i)8-s + (−0.686 − 0.727i)9-s + (−1.77 + 0.207i)11-s + (0.766 − 0.642i)12-s + (0.597 + 0.802i)16-s + (−0.0996 − 0.564i)17-s + (−0.499 − 0.866i)18-s + (−0.344 + 1.95i)19-s + (−1.77 − 0.207i)22-s + (0.893 − 0.448i)24-s + (−0.286 − 0.957i)25-s + ⋯
L(s)  = 1  + (0.973 + 0.230i)2-s + (0.396 − 0.918i)3-s + (0.893 + 0.448i)4-s + (0.597 − 0.802i)6-s + (0.766 + 0.642i)8-s + (−0.686 − 0.727i)9-s + (−1.77 + 0.207i)11-s + (0.766 − 0.642i)12-s + (0.597 + 0.802i)16-s + (−0.0996 − 0.564i)17-s + (−0.499 − 0.866i)18-s + (−0.344 + 1.95i)19-s + (−1.77 − 0.207i)22-s + (0.893 − 0.448i)24-s + (−0.286 − 0.957i)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: $\chi_{648} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ 0.952 + 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.684012409\)
\(L(\frac12)\) \(\approx\) \(1.684012409\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.973 - 0.230i)T \)
3 \( 1 + (-0.396 + 0.918i)T \)
good5 \( 1 + (0.286 + 0.957i)T^{2} \)
7 \( 1 + (-0.396 - 0.918i)T^{2} \)
11 \( 1 + (1.77 - 0.207i)T + (0.973 - 0.230i)T^{2} \)
13 \( 1 + (0.835 + 0.549i)T^{2} \)
17 \( 1 + (0.0996 + 0.564i)T + (-0.939 + 0.342i)T^{2} \)
19 \( 1 + (0.344 - 1.95i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.396 + 0.918i)T^{2} \)
29 \( 1 + (0.0581 + 0.998i)T^{2} \)
31 \( 1 + (0.993 - 0.116i)T^{2} \)
37 \( 1 + (-0.766 - 0.642i)T^{2} \)
41 \( 1 + (-0.770 + 0.182i)T + (0.893 - 0.448i)T^{2} \)
43 \( 1 + (-0.770 + 1.78i)T + (-0.686 - 0.727i)T^{2} \)
47 \( 1 + (0.993 + 0.116i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (1.18 + 0.138i)T + (0.973 + 0.230i)T^{2} \)
61 \( 1 + (-0.597 + 0.802i)T^{2} \)
67 \( 1 + (-1.14 - 1.21i)T + (-0.0581 + 0.998i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (1.05 + 0.882i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.893 - 0.448i)T^{2} \)
83 \( 1 + (1.82 + 0.433i)T + (0.893 + 0.448i)T^{2} \)
89 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.0694 + 0.0932i)T + (-0.286 + 0.957i)T^{2} \)
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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

−10.84694119521260685209305965307, −10.07834226195873862312700150338, −8.572374553047652374990089247827, −7.80015805332667759677353774600, −7.29844345694904213112679058817, −6.08760341941194724538275343967, −5.47666154247412098884667916527, −4.15894393494304601657223555750, −2.91612837821210557776232741429, −2.05103618923782202105829402726, 2.41066466162424057981024329576, 3.13910019911263792920910597940, 4.39919871447247528732958302391, 5.09406544190842561276151585235, 5.94225432801587414605392931864, 7.27540858023427113901615104662, 8.153293467746806051618116051176, 9.270904545901900236332692364183, 10.17336475903316580885344658527, 10.99671084549663412471991975577

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