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Properties

Label 2-648-9.7-c3-0-4
Degree $2$
Conductor $648$
Sign $0.173 - 0.984i$
Analytic cond. $38.2332$
Root an. cond. $6.18330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8 − 13.8i)5-s + (6 − 10.3i)7-s + (−32 + 55.4i)11-s + (−29 − 50.2i)13-s + 32·17-s − 136·19-s + (64 + 110. i)23-s + (−65.4 + 113. i)25-s + (72 − 124. i)29-s + (−10 − 17.3i)31-s − 192·35-s − 18·37-s + (144 + 249. i)41-s + (100 − 173. i)43-s + (−192 + 332. i)47-s + ⋯
L(s)  = 1  + (−0.715 − 1.23i)5-s + (0.323 − 0.561i)7-s + (−0.877 + 1.51i)11-s + (−0.618 − 1.07i)13-s + 0.456·17-s − 1.64·19-s + (0.580 + 1.00i)23-s + (−0.523 + 0.907i)25-s + (0.461 − 0.798i)29-s + (−0.0579 − 0.100i)31-s − 0.927·35-s − 0.0799·37-s + (0.548 + 0.950i)41-s + (0.354 − 0.614i)43-s + (−0.595 + 1.03i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(38.2332\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5982580079\)
\(L(\frac12)\) \(\approx\) \(0.5982580079\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (8 + 13.8i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-6 + 10.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (32 - 55.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (29 + 50.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 32T + 4.91e3T^{2} \)
19 \( 1 + 136T + 6.85e3T^{2} \)
23 \( 1 + (-64 - 110. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-72 + 124. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (10 + 17.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 18T + 5.06e4T^{2} \)
41 \( 1 + (-144 - 249. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-100 + 173. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (192 - 332. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 496T + 1.48e5T^{2} \)
59 \( 1 + (-64 - 110. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-229 + 396. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-248 - 429. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 512T + 3.57e5T^{2} \)
73 \( 1 + 602T + 3.89e5T^{2} \)
79 \( 1 + (554 - 959. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (352 - 609. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 960T + 7.04e5T^{2} \)
97 \( 1 + (103 - 178. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.25512477292168568504778283837, −9.584023040577093149018666384182, −8.362702465725537388302163764349, −7.83603816522834038350724006730, −7.11121683836829037290699900243, −5.53494014480294392281538811907, −4.72189585694596276762886029599, −4.10119908136202563379676472327, −2.47280668123143754883312515921, −1.02902535160579133908424864288, 0.19631785940515196948637598578, 2.29206550522509430282683462076, 3.11219841865055026499303501665, 4.25394546862651660033030232145, 5.47239983871650745135704576087, 6.51589047155888908406421378589, 7.22092105204450207650920847172, 8.341819193560368765286063947817, 8.804070318353789023205797094267, 10.30317300752612981603751666371

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