Properties

Label 2-648-9.7-c3-0-17
Degree $2$
Conductor $648$
Sign $0.939 - 0.342i$
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  + (4.67 + 8.10i)5-s + (−4.17 + 7.23i)7-s + (14.5 − 25.1i)11-s + (−18.3 − 31.8i)13-s + 28.4·17-s + 73.7·19-s + (−15.2 − 26.4i)23-s + (18.7 − 32.4i)25-s + (92.4 − 160. i)29-s + (95.4 + 165. i)31-s − 78.2·35-s + 160.·37-s + (104. + 181. i)41-s + (−58.4 + 101. i)43-s + (−140. + 243. i)47-s + ⋯
L(s)  = 1  + (0.418 + 0.724i)5-s + (−0.225 + 0.390i)7-s + (0.398 − 0.690i)11-s + (−0.392 − 0.679i)13-s + 0.405·17-s + 0.890·19-s + (−0.138 − 0.239i)23-s + (0.149 − 0.259i)25-s + (0.592 − 1.02i)29-s + (0.552 + 0.957i)31-s − 0.377·35-s + 0.713·37-s + (0.399 + 0.691i)41-s + (−0.207 + 0.359i)43-s + (−0.436 + 0.756i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.939 - 0.342i$
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: \((2,\ 648,\ (\ :3/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.205082495\)
\(L(\frac12)\) \(\approx\) \(2.205082495\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.67 - 8.10i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (4.17 - 7.23i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-14.5 + 25.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (18.3 + 31.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 28.4T + 4.91e3T^{2} \)
19 \( 1 - 73.7T + 6.85e3T^{2} \)
23 \( 1 + (15.2 + 26.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-92.4 + 160. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-95.4 - 165. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 160.T + 5.06e4T^{2} \)
41 \( 1 + (-104. - 181. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (58.4 - 101. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (140. - 243. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 397.T + 1.48e5T^{2} \)
59 \( 1 + (-189. - 327. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-148. + 257. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-413. - 717. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 729.T + 3.57e5T^{2} \)
73 \( 1 - 1.10e3T + 3.89e5T^{2} \)
79 \( 1 + (-532. + 921. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-295. + 511. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 227.T + 7.04e5T^{2} \)
97 \( 1 + (642. - 1.11e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.11508203222183043740376279918, −9.531869816351672485935364004146, −8.423687737707536315249780665244, −7.59624143160592967857933390382, −6.44516247025440567188579085319, −5.89557368260111530536584771352, −4.73467184683474604021421761653, −3.27584649677394159734917413949, −2.58670923024984572116775579271, −0.918429456703710330615923380638, 0.879049266342801329050244487834, 2.03060484627319535384120771705, 3.53230893313971585511597178639, 4.63579816145004138540532165432, 5.42937844469274316680780321468, 6.62770499674834878822460436937, 7.38619296656606172647038009706, 8.440838291110118028358672599042, 9.549803598263313050687638595659, 9.720989804961986204904604346801

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