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Properties

Label 2-648-9.4-c3-0-28
Degree $2$
Conductor $648$
Sign $-0.342 + 0.939i$
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  + (5.26 − 9.11i)5-s + (0.177 + 0.307i)7-s + (−4.30 − 7.46i)11-s + (17.3 − 30.1i)13-s + 72.5·17-s − 78.3·19-s + (24.6 − 42.6i)23-s + (7.15 + 12.3i)25-s + (−69.4 − 120. i)29-s + (31.2 − 54.0i)31-s + 3.73·35-s + 36.0·37-s + (−95.4 + 165. i)41-s + (−266. − 461. i)43-s + (35.3 + 61.2i)47-s + ⋯
L(s)  = 1  + (0.470 − 0.814i)5-s + (0.00958 + 0.0165i)7-s + (−0.118 − 0.204i)11-s + (0.371 − 0.642i)13-s + 1.03·17-s − 0.946·19-s + (0.223 − 0.386i)23-s + (0.0572 + 0.0991i)25-s + (−0.444 − 0.769i)29-s + (0.180 − 0.313i)31-s + 0.0180·35-s + 0.160·37-s + (−0.363 + 0.630i)41-s + (−0.944 − 1.63i)43-s + (0.109 + 0.189i)47-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(1.759185479\)
\(L(\frac12)\) \(\approx\) \(1.759185479\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.26 + 9.11i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-0.177 - 0.307i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (4.30 + 7.46i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-17.3 + 30.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 72.5T + 4.91e3T^{2} \)
19 \( 1 + 78.3T + 6.85e3T^{2} \)
23 \( 1 + (-24.6 + 42.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (69.4 + 120. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-31.2 + 54.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 36.0T + 5.06e4T^{2} \)
41 \( 1 + (95.4 - 165. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (266. + 461. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-35.3 - 61.2i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 245.T + 1.48e5T^{2} \)
59 \( 1 + (417. - 723. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (240. + 416. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (55.2 - 95.7i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 58.4T + 3.57e5T^{2} \)
73 \( 1 + 313.T + 3.89e5T^{2} \)
79 \( 1 + (497. + 861. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (366. + 635. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 994.T + 7.04e5T^{2} \)
97 \( 1 + (529. + 916. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.953379852122065175509919178522, −8.908419365896068376720045666473, −8.325736558582798178174308741548, −7.31877713873825340841662433827, −6.06009399961022764517004888693, −5.42324381373964660602005079190, −4.36342223987168559892731444388, −3.13767047597355984370778350442, −1.74444643082687577671229240010, −0.50530567120273383673385495752, 1.44993464511246985129412115651, 2.66551327601651172190024736689, 3.73531476088355184780261517786, 4.96825354856421125050361905389, 6.08263370107097271915357690278, 6.77745814163484711934449353541, 7.71358838495399061590545952701, 8.747057426922645863130695335835, 9.661806887619352271071836250990, 10.44740037535785299253597384819

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