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Properties

Label 2-648-648.43-c0-0-0
Degree $2$
Conductor $648$
Sign $-0.740 + 0.672i$
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  + (−0.286 − 0.957i)2-s + (−0.993 + 0.116i)3-s + (−0.835 + 0.549i)4-s + (0.396 + 0.918i)6-s + (0.766 + 0.642i)8-s + (0.973 − 0.230i)9-s + (−0.997 − 1.34i)11-s + (0.766 − 0.642i)12-s + (0.396 − 0.918i)16-s + (−0.238 − 1.35i)17-s + (−0.5 − 0.866i)18-s + (0.207 − 1.17i)19-s + (−0.997 + 1.34i)22-s + (−0.835 − 0.549i)24-s + (−0.686 + 0.727i)25-s + ⋯
L(s)  = 1  + (−0.286 − 0.957i)2-s + (−0.993 + 0.116i)3-s + (−0.835 + 0.549i)4-s + (0.396 + 0.918i)6-s + (0.766 + 0.642i)8-s + (0.973 − 0.230i)9-s + (−0.997 − 1.34i)11-s + (0.766 − 0.642i)12-s + (0.396 − 0.918i)16-s + (−0.238 − 1.35i)17-s + (−0.5 − 0.866i)18-s + (0.207 − 1.17i)19-s + (−0.997 + 1.34i)22-s + (−0.835 − 0.549i)24-s + (−0.686 + 0.727i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.740 + 0.672i$
Analytic conductor: \(0.323394\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ -0.740 + 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4318204157\)
\(L(\frac12)\) \(\approx\) \(0.4318204157\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.286 + 0.957i)T \)
3 \( 1 + (0.993 - 0.116i)T \)
good5 \( 1 + (0.686 - 0.727i)T^{2} \)
7 \( 1 + (0.993 + 0.116i)T^{2} \)
11 \( 1 + (0.997 + 1.34i)T + (-0.286 + 0.957i)T^{2} \)
13 \( 1 + (0.0581 - 0.998i)T^{2} \)
17 \( 1 + (0.238 + 1.35i)T + (-0.939 + 0.342i)T^{2} \)
19 \( 1 + (-0.207 + 1.17i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.993 - 0.116i)T^{2} \)
29 \( 1 + (-0.893 - 0.448i)T^{2} \)
31 \( 1 + (-0.597 - 0.802i)T^{2} \)
37 \( 1 + (-0.766 - 0.642i)T^{2} \)
41 \( 1 + (-0.569 + 1.90i)T + (-0.835 - 0.549i)T^{2} \)
43 \( 1 + (-0.569 + 0.0665i)T + (0.973 - 0.230i)T^{2} \)
47 \( 1 + (-0.597 + 0.802i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.473 + 0.635i)T + (-0.286 - 0.957i)T^{2} \)
61 \( 1 + (-0.396 - 0.918i)T^{2} \)
67 \( 1 + (0.113 - 0.0268i)T + (0.893 - 0.448i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-1.49 - 1.25i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.835 - 0.549i)T^{2} \)
83 \( 1 + (-0.539 - 1.80i)T + (-0.835 + 0.549i)T^{2} \)
89 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.707 + 1.64i)T + (-0.686 - 0.727i)T^{2} \)
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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

−10.76985864999207701504297430127, −9.705595278068985872881638656183, −9.024427206799810229496956457131, −7.888635918099886105623473765316, −6.97067795395557442954982241356, −5.54597904473303078961871093160, −4.98320483625301576221437985625, −3.70123218045441493555309483749, −2.51804787754036702263103337850, −0.62189711512920072736902268774, 1.74783078472615069124938133200, 4.11060736154609892271839807176, 4.90167574874983706826591980455, 5.89176190516388027900080021208, 6.52248981204108535061590371393, 7.66347298618233021669208465352, 8.057055701176003733676424267959, 9.544341837689455692358411166574, 10.18026142693038014333872928101, 10.76573795485368442963299315732

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