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Properties

Label 2-648-648.427-c0-0-0
Degree $2$
Conductor $648$
Sign $-0.211 + 0.977i$
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  + (−0.686 − 0.727i)2-s + (0.597 − 0.802i)3-s + (−0.0581 + 0.998i)4-s + (−0.993 + 0.116i)6-s + (0.766 − 0.642i)8-s + (−0.286 − 0.957i)9-s + (−0.0460 − 0.106i)11-s + (0.766 + 0.642i)12-s + (−0.993 − 0.116i)16-s + (0.337 − 1.91i)17-s + (−0.500 + 0.866i)18-s + (0.137 + 0.780i)19-s + (−0.0460 + 0.106i)22-s + (−0.0581 − 0.998i)24-s + (0.973 − 0.230i)25-s + ⋯
L(s)  = 1  + (−0.686 − 0.727i)2-s + (0.597 − 0.802i)3-s + (−0.0581 + 0.998i)4-s + (−0.993 + 0.116i)6-s + (0.766 − 0.642i)8-s + (−0.286 − 0.957i)9-s + (−0.0460 − 0.106i)11-s + (0.766 + 0.642i)12-s + (−0.993 − 0.116i)16-s + (0.337 − 1.91i)17-s + (−0.500 + 0.866i)18-s + (0.137 + 0.780i)19-s + (−0.0460 + 0.106i)22-s + (−0.0581 − 0.998i)24-s + (0.973 − 0.230i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.211 + 0.977i$
Analytic conductor: \(0.323394\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ -0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7914801250\)
\(L(\frac12)\) \(\approx\) \(0.7914801250\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.686 + 0.727i)T \)
3 \( 1 + (-0.597 + 0.802i)T \)
good5 \( 1 + (-0.973 + 0.230i)T^{2} \)
7 \( 1 + (-0.597 - 0.802i)T^{2} \)
11 \( 1 + (0.0460 + 0.106i)T + (-0.686 + 0.727i)T^{2} \)
13 \( 1 + (-0.893 - 0.448i)T^{2} \)
17 \( 1 + (-0.337 + 1.91i)T + (-0.939 - 0.342i)T^{2} \)
19 \( 1 + (-0.137 - 0.780i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.597 + 0.802i)T^{2} \)
29 \( 1 + (0.835 + 0.549i)T^{2} \)
31 \( 1 + (-0.396 - 0.918i)T^{2} \)
37 \( 1 + (-0.766 + 0.642i)T^{2} \)
41 \( 1 + (0.819 - 0.868i)T + (-0.0581 - 0.998i)T^{2} \)
43 \( 1 + (0.819 - 1.10i)T + (-0.286 - 0.957i)T^{2} \)
47 \( 1 + (-0.396 + 0.918i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.786 - 1.82i)T + (-0.686 - 0.727i)T^{2} \)
61 \( 1 + (0.993 - 0.116i)T^{2} \)
67 \( 1 + (0.512 + 1.71i)T + (-0.835 + 0.549i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.439 - 0.368i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (0.0581 - 0.998i)T^{2} \)
83 \( 1 + (-1.28 - 1.36i)T + (-0.0581 + 0.998i)T^{2} \)
89 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-1.65 - 0.193i)T + (0.973 + 0.230i)T^{2} \)
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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

−10.42261711646253025309794381910, −9.516072226693409049906744010507, −8.900408160436850079694509048485, −7.933872298852693664507230723008, −7.34508908679926416971536262260, −6.37490551991027724729027819247, −4.80832192965778853144308670945, −3.36276082695495321447019622914, −2.60807273551117587801918806412, −1.19277327915257926647068187499, 1.94171306769174556875580755088, 3.52396726306414145871935528108, 4.71629305523329904731433404693, 5.59463353390529508232630017436, 6.71221249139882609614951667793, 7.70834799402477478340171557210, 8.595491441131312720328377392701, 9.019360033771816685679335269459, 10.21693579602008771493739847712, 10.46612903538974459009079675638

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