Properties

Label 2-648-9.4-c3-0-26
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

\(L(2)\) \(\approx\) \(0.5839785958\)
\(L(\frac12)\) \(\approx\) \(0.5839785958\)
\(L(\frac{5}{2})\) 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

−10.07734512898101215827094319860, −8.910730253010576839127063005515, −8.170942969703112192352863087099, −7.10816937980587080966206184531, −6.50827124155555391377684565239, −5.32915777005245312083302220317, −4.12743395576972270267858943774, −3.20427978439099664126704942335, −1.98102269094703141889105041936, −0.17559031081413359921367915925, 1.20844108707669363024340951387, 2.61916878878028927485398013101, 4.19965576940551018622483026281, 4.58538877682817814365254288177, 6.03187844206410662484068094463, 6.77002661830037225895402839688, 8.051586064153811855690702338897, 8.619656808104442778924849072964, 9.383455861792718990096822784484, 10.48888394060998175956461541500

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