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Properties

Label 2-648-648.619-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  + (−0.835 − 0.549i)2-s + (0.973 + 0.230i)3-s + (0.396 + 0.918i)4-s + (−0.686 − 0.727i)6-s + (0.173 − 0.984i)8-s + (0.893 + 0.448i)9-s + (−0.227 − 0.758i)11-s + (0.173 + 0.984i)12-s + (−0.686 + 0.727i)16-s + (0.109 + 0.0397i)17-s + (−0.499 − 0.866i)18-s + (0.539 − 0.196i)19-s + (−0.227 + 0.758i)22-s + (0.396 − 0.918i)24-s + (−0.0581 + 0.998i)25-s + ⋯
L(s)  = 1  + (−0.835 − 0.549i)2-s + (0.973 + 0.230i)3-s + (0.396 + 0.918i)4-s + (−0.686 − 0.727i)6-s + (0.173 − 0.984i)8-s + (0.893 + 0.448i)9-s + (−0.227 − 0.758i)11-s + (0.173 + 0.984i)12-s + (−0.686 + 0.727i)16-s + (0.109 + 0.0397i)17-s + (−0.499 − 0.866i)18-s + (0.539 − 0.196i)19-s + (−0.227 + 0.758i)22-s + (0.396 − 0.918i)24-s + (−0.0581 + 0.998i)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: $\chi_{648} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ 0.952 + 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8840058431\)
\(L(\frac12)\) \(\approx\) \(0.8840058431\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.835 + 0.549i)T \)
3 \( 1 + (-0.973 - 0.230i)T \)
good5 \( 1 + (0.0581 - 0.998i)T^{2} \)
7 \( 1 + (-0.973 + 0.230i)T^{2} \)
11 \( 1 + (0.227 + 0.758i)T + (-0.835 + 0.549i)T^{2} \)
13 \( 1 + (0.993 - 0.116i)T^{2} \)
17 \( 1 + (-0.109 - 0.0397i)T + (0.766 + 0.642i)T^{2} \)
19 \( 1 + (-0.539 + 0.196i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.973 - 0.230i)T^{2} \)
29 \( 1 + (-0.597 + 0.802i)T^{2} \)
31 \( 1 + (0.286 + 0.957i)T^{2} \)
37 \( 1 + (-0.173 + 0.984i)T^{2} \)
41 \( 1 + (1.62 - 1.06i)T + (0.396 - 0.918i)T^{2} \)
43 \( 1 + (1.62 + 0.385i)T + (0.893 + 0.448i)T^{2} \)
47 \( 1 + (0.286 - 0.957i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.393 + 1.31i)T + (-0.835 - 0.549i)T^{2} \)
61 \( 1 + (0.686 + 0.727i)T^{2} \)
67 \( 1 + (1.77 + 0.891i)T + (0.597 + 0.802i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.310 + 1.76i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (-0.396 - 0.918i)T^{2} \)
83 \( 1 + (1.28 + 0.841i)T + (0.396 + 0.918i)T^{2} \)
89 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.819 - 0.868i)T + (-0.0581 - 0.998i)T^{2} \)
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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

−10.52028314302954469050532094733, −9.786379636895055593710768410146, −9.017926989531276790830733026409, −8.302909250005651854433082013501, −7.58455753829830154652018205865, −6.62401873579710065623808000357, −5.04351899938963909760166427321, −3.64758248090820471619726462084, −2.98171378947512783007168620984, −1.63438328148246247318544547598, 1.62119570828251856248690417829, 2.80059202278079678370494757819, 4.32834115717736976024749010718, 5.54809311855952725675199140986, 6.80444039504943201603201544976, 7.35547564091018071915714978991, 8.295334321166935172529876851481, 8.864544284636239313845996398488, 9.996377236206970391058707720124, 10.18279637291063025365927915627

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