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Properties

Label 2-18e2-12.11-c3-0-18
Degree $2$
Conductor $324$
Sign $-0.787 - 0.616i$
Analytic cond. $19.1166$
Root an. cond. $4.37225$
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  + (2.67 + 0.922i)2-s + (6.29 + 4.93i)4-s + 7.80i·5-s + 19.8i·7-s + (12.2 + 18.9i)8-s + (−7.19 + 20.8i)10-s − 35.6·11-s − 71.9·13-s + (−18.2 + 53.0i)14-s + (15.3 + 62.1i)16-s − 27.8i·17-s − 40.9i·19-s + (−38.4 + 49.1i)20-s + (−95.3 − 32.9i)22-s + 114.·23-s + ⋯
L(s)  = 1  + (0.945 + 0.326i)2-s + (0.787 + 0.616i)4-s + 0.697i·5-s + 1.07i·7-s + (0.543 + 0.839i)8-s + (−0.227 + 0.659i)10-s − 0.977·11-s − 1.53·13-s + (−0.349 + 1.01i)14-s + (0.239 + 0.970i)16-s − 0.397i·17-s − 0.494i·19-s + (−0.430 + 0.549i)20-s + (−0.924 − 0.318i)22-s + 1.04·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.787 - 0.616i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ -0.787 - 0.616i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.564967937\)
\(L(\frac12)\) \(\approx\) \(2.564967937\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.67 - 0.922i)T \)
3 \( 1 \)
good5 \( 1 - 7.80iT - 125T^{2} \)
7 \( 1 - 19.8iT - 343T^{2} \)
11 \( 1 + 35.6T + 1.33e3T^{2} \)
13 \( 1 + 71.9T + 2.19e3T^{2} \)
17 \( 1 + 27.8iT - 4.91e3T^{2} \)
19 \( 1 + 40.9iT - 6.85e3T^{2} \)
23 \( 1 - 114.T + 1.21e4T^{2} \)
29 \( 1 - 49.8iT - 2.43e4T^{2} \)
31 \( 1 + 34.2iT - 2.97e4T^{2} \)
37 \( 1 + 309.T + 5.06e4T^{2} \)
41 \( 1 - 405. iT - 6.89e4T^{2} \)
43 \( 1 - 522. iT - 7.95e4T^{2} \)
47 \( 1 - 388.T + 1.03e5T^{2} \)
53 \( 1 + 534. iT - 1.48e5T^{2} \)
59 \( 1 - 709.T + 2.05e5T^{2} \)
61 \( 1 + 441.T + 2.26e5T^{2} \)
67 \( 1 - 953. iT - 3.00e5T^{2} \)
71 \( 1 - 850.T + 3.57e5T^{2} \)
73 \( 1 + 135.T + 3.89e5T^{2} \)
79 \( 1 - 717. iT - 4.93e5T^{2} \)
83 \( 1 - 515.T + 5.71e5T^{2} \)
89 \( 1 + 454. iT - 7.04e5T^{2} \)
97 \( 1 - 1.03e3T + 9.12e5T^{2} \)
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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

−11.68767866520460624812428945851, −10.89387195556175406596111916789, −9.787915872694605693433335102428, −8.526964657643354929666034267957, −7.41354187604707478621394431743, −6.71027566527286949843314949933, −5.41426105361743776746493412250, −4.81745254346949728016085363738, −2.97952368551981084047311082761, −2.45914117020510218698147333182, 0.63298563747701765134819774814, 2.23080318944325783112183166806, 3.65383612867812477624738777273, 4.78972973336250878769573776746, 5.42571019927854247190865228401, 6.98024977254373145228016326783, 7.61198806292236220957341324569, 9.072791864558536360887107022450, 10.45053460650118957829033208176, 10.55273948680507521272362168245

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