Properties

Label 2-648-648.131-c1-0-47
Degree $2$
Conductor $648$
Sign $0.508 - 0.860i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.14 + 0.832i)2-s + (1.73 + 0.0244i)3-s + (0.613 − 1.90i)4-s + (0.0414 − 0.711i)5-s + (−2.00 + 1.41i)6-s + (0.543 + 2.29i)7-s + (0.883 + 2.68i)8-s + (2.99 + 0.0848i)9-s + (0.545 + 0.847i)10-s + (0.518 + 0.788i)11-s + (1.10 − 3.28i)12-s + (0.389 + 3.33i)13-s + (−2.53 − 2.17i)14-s + (0.0892 − 1.23i)15-s + (−3.24 − 2.33i)16-s + (0.293 − 0.349i)17-s + ⋯
L(s)  = 1  + (−0.808 + 0.588i)2-s + (0.999 + 0.0141i)3-s + (0.306 − 0.951i)4-s + (0.0185 − 0.318i)5-s + (−0.816 + 0.577i)6-s + (0.205 + 0.867i)7-s + (0.312 + 0.949i)8-s + (0.999 + 0.0282i)9-s + (0.172 + 0.268i)10-s + (0.156 + 0.237i)11-s + (0.320 − 0.947i)12-s + (0.108 + 0.924i)13-s + (−0.676 − 0.580i)14-s + (0.0230 − 0.317i)15-s + (−0.811 − 0.583i)16-s + (0.0711 − 0.0847i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.508 - 0.860i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.508 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31677 + 0.751245i\)
\(L(\frac12)\) \(\approx\) \(1.31677 + 0.751245i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 - 0.832i)T \)
3 \( 1 + (-1.73 - 0.0244i)T \)
good5 \( 1 + (-0.0414 + 0.711i)T + (-4.96 - 0.580i)T^{2} \)
7 \( 1 + (-0.543 - 2.29i)T + (-6.25 + 3.14i)T^{2} \)
11 \( 1 + (-0.518 - 0.788i)T + (-4.35 + 10.1i)T^{2} \)
13 \( 1 + (-0.389 - 3.33i)T + (-12.6 + 2.99i)T^{2} \)
17 \( 1 + (-0.293 + 0.349i)T + (-2.95 - 16.7i)T^{2} \)
19 \( 1 + (-1.44 + 1.21i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (5.49 + 1.30i)T + (20.5 + 10.3i)T^{2} \)
29 \( 1 + (4.29 - 5.76i)T + (-8.31 - 27.7i)T^{2} \)
31 \( 1 + (-4.14 + 1.24i)T + (25.9 - 17.0i)T^{2} \)
37 \( 1 + (-6.12 - 1.07i)T + (34.7 + 12.6i)T^{2} \)
41 \( 1 + (1.75 + 0.755i)T + (28.1 + 29.8i)T^{2} \)
43 \( 1 + (-2.02 - 1.01i)T + (25.6 + 34.4i)T^{2} \)
47 \( 1 + (-1.07 + 3.59i)T + (-39.2 - 25.8i)T^{2} \)
53 \( 1 + (-4.06 + 7.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.35 - 3.58i)T + (-23.3 - 54.1i)T^{2} \)
61 \( 1 + (3.59 - 3.39i)T + (3.54 - 60.8i)T^{2} \)
67 \( 1 + (2.22 + 2.99i)T + (-19.2 + 64.1i)T^{2} \)
71 \( 1 + (-0.189 + 0.0690i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-2.37 - 0.863i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (12.5 - 5.41i)T + (54.2 - 57.4i)T^{2} \)
83 \( 1 + (1.86 - 0.806i)T + (56.9 - 60.3i)T^{2} \)
89 \( 1 + (-1.88 + 5.18i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.615 - 10.5i)T + (-96.3 + 11.2i)T^{2} \)
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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

−10.32164254204416134172060696246, −9.464235122682349867532040118114, −8.912372309695322712694059568482, −8.298086836384821273888744603850, −7.35266717006168681039678447782, −6.50621657958830169637080126904, −5.33221981173744505019453550621, −4.27178605025177881169097407969, −2.60441642662567237660304440852, −1.55499982094118034397774774116, 1.10116037362891642861341506984, 2.51346771169641551099999302732, 3.49885375772841435752603210945, 4.33714790345711103129494046494, 6.20315715159742580831252208936, 7.44568256059677806965540830282, 7.81129176680764716837614336735, 8.682336456622614774485553669510, 9.676171146849528510470664147917, 10.27138067421678979903447100961

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