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Properties

Label 2-648-9.4-c3-0-10
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  + (−10.6 + 18.3i)5-s + (−14.6 − 25.3i)7-s + (−0.5 − 0.866i)11-s + (26.4 − 45.8i)13-s − 96.9·17-s + 126.·19-s + (11.4 − 19.8i)23-s + (−162. − 282. i)25-s + (66.7 + 115. i)29-s + (−50.9 + 88.1i)31-s + 620.·35-s + 105.·37-s + (−8.16 + 14.1i)41-s + (100. + 174. i)43-s + (125. + 218. i)47-s + ⋯
L(s)  = 1  + (−0.949 + 1.64i)5-s + (−0.789 − 1.36i)7-s + (−0.0137 − 0.0237i)11-s + (0.564 − 0.978i)13-s − 1.38·17-s + 1.52·19-s + (0.103 − 0.180i)23-s + (−1.30 − 2.25i)25-s + (0.427 + 0.740i)29-s + (−0.294 + 0.510i)31-s + 2.99·35-s + 0.466·37-s + (−0.0311 + 0.0538i)41-s + (0.356 + 0.618i)43-s + (0.390 + 0.676i)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

\(L(2)\) \(\approx\) \(1.188008208\)
\(L(\frac12)\) \(\approx\) \(1.188008208\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.6 - 18.3i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (14.6 + 25.3i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-26.4 + 45.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 96.9T + 4.91e3T^{2} \)
19 \( 1 - 126.T + 6.85e3T^{2} \)
23 \( 1 + (-11.4 + 19.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-66.7 - 115. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (50.9 - 88.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 105.T + 5.06e4T^{2} \)
41 \( 1 + (8.16 - 14.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-100. - 174. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-125. - 218. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 148.T + 1.48e5T^{2} \)
59 \( 1 + (-36.8 + 63.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-303. - 526. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (380. - 659. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 701.T + 3.57e5T^{2} \)
73 \( 1 + 287T + 3.89e5T^{2} \)
79 \( 1 + (-64.2 - 111. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (80.2 + 138. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 430.T + 7.04e5T^{2} \)
97 \( 1 + (-15.5 - 26.9i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.53472559016232843155175904464, −9.645104968656907522160138698097, −8.282454640886099310132437692047, −7.29562485301200257167826274957, −6.99314408486071765384476789170, −6.03622818184473690271119833662, −4.38266701258955613491675087711, −3.45172389912122454480350303786, −2.89440328695167018699611879684, −0.71995449724321232194834298790, 0.54781451236649169563561393349, 2.03381291834694034859520708205, 3.53490789971284125806866846430, 4.52940671450424102296275787083, 5.40327791881779998398243276209, 6.36219937903706618567575896250, 7.58413585828725941796760513228, 8.598668070322332507891203322915, 9.072990715534684030962615665968, 9.616283370579816959245781699574

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