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Properties

Label 2-648-9.4-c3-0-30
Degree $2$
Conductor $648$
Sign $-0.766 + 0.642i$
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  + (0.5 − 0.866i)5-s + (4.5 + 7.79i)7-s + (−8.5 − 14.7i)11-s + (22 − 38.1i)13-s − 56·17-s − 94·19-s + (−25 + 43.3i)23-s + (62 + 107. i)25-s + (−15 − 25.9i)29-s + (69.5 − 120. i)31-s + 9·35-s − 174·37-s + (159 − 275. i)41-s + (121 + 209. i)43-s + (−315 − 545. i)47-s + ⋯
L(s)  = 1  + (0.0447 − 0.0774i)5-s + (0.242 + 0.420i)7-s + (−0.232 − 0.403i)11-s + (0.469 − 0.812i)13-s − 0.798·17-s − 1.13·19-s + (−0.226 + 0.392i)23-s + (0.495 + 0.859i)25-s + (−0.0960 − 0.166i)29-s + (0.402 − 0.697i)31-s + 0.0434·35-s − 0.773·37-s + (0.605 − 1.04i)41-s + (0.429 + 0.743i)43-s + (−0.977 − 1.69i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(38.2332\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7085456059\)
\(L(\frac12)\) \(\approx\) \(0.7085456059\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.5 + 0.866i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-4.5 - 7.79i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (8.5 + 14.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-22 + 38.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 56T + 4.91e3T^{2} \)
19 \( 1 + 94T + 6.85e3T^{2} \)
23 \( 1 + (25 - 43.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (15 + 25.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-69.5 + 120. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 174T + 5.06e4T^{2} \)
41 \( 1 + (-159 + 275. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-121 - 209. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (315 + 545. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 547T + 1.48e5T^{2} \)
59 \( 1 + (118 - 204. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (164 + 284. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (307 - 531. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 296T + 3.57e5T^{2} \)
73 \( 1 - 433T + 3.89e5T^{2} \)
79 \( 1 + (-28 - 48.4i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (612.5 + 1.06e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + (695.5 + 1.20e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.837147302139991608711255304318, −8.761544845304094734242110554217, −8.286133586009239283131128188087, −7.16881321204474761490963590497, −6.10062025859361580805018850192, −5.33631781586267697438110369147, −4.19258400790481336278605493758, −3.00621394938612184427228711668, −1.78974512782785708124823126552, −0.19403357732587673744841873986, 1.48664089640745571680540529741, 2.67962106120890090989119212410, 4.16624452393139676092523149421, 4.74581425578722237631561091653, 6.24687155032318875850501273850, 6.80090186525685319524403585925, 7.945961442248365591546744696329, 8.729580655473882668707042761947, 9.613229516949614003355887643910, 10.70988383238445188806050539623

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