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Properties

Label 2-648-216.139-c0-0-0
Degree $2$
Conductor $648$
Sign $0.835 - 0.549i$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.500 + 0.866i)8-s + (−0.266 − 1.50i)11-s + (0.173 + 0.984i)16-s + (−0.939 + 1.62i)17-s + (−0.173 − 0.300i)19-s + (0.266 − 1.50i)22-s + (−0.939 − 0.342i)25-s + (−0.173 + 0.984i)32-s + (−1.43 + 1.20i)34-s + (−0.0603 − 0.342i)38-s + (0.326 − 0.118i)41-s + (−0.326 − 1.85i)43-s + (0.766 − 1.32i)44-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.500 + 0.866i)8-s + (−0.266 − 1.50i)11-s + (0.173 + 0.984i)16-s + (−0.939 + 1.62i)17-s + (−0.173 − 0.300i)19-s + (0.266 − 1.50i)22-s + (−0.939 − 0.342i)25-s + (−0.173 + 0.984i)32-s + (−1.43 + 1.20i)34-s + (−0.0603 − 0.342i)38-s + (0.326 − 0.118i)41-s + (−0.326 − 1.85i)43-s + (0.766 − 1.32i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(0.323394\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ 0.835 - 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.539018024\)
\(L(\frac12)\) \(\approx\) \(1.539018024\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (-0.173 - 0.984i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
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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

−10.99499839329361191451061371949, −10.31003032364141070256733860982, −8.693412397972193019223976436747, −8.295021265772578485683355549645, −7.11875655881540555242163360819, −6.14076220214897074690398026316, −5.57865849778524076113325276868, −4.27009057692490596875312490315, −3.45490098698584044943874717969, −2.15089482639313799589237408829, 1.89428405712027679101302479928, 2.94297834986848735156757820177, 4.36269865603094444550033539475, 4.90411437313865413709001679114, 6.09002364281092967419911421891, 7.05910181905728487363872681380, 7.72841356606116940592746544978, 9.343509992617560949627938483052, 9.852634747992512280081838869333, 10.89217340528981091442715169834

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