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Properties

Label 2-18e2-9.7-c3-0-8
Degree $2$
Conductor $324$
Sign $-0.173 + 0.984i$
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  + (−1.5 − 2.59i)5-s + (2 − 3.46i)7-s + (12 − 20.7i)11-s + (12.5 + 21.6i)13-s − 21·17-s − 52·19-s + (−84 − 145. i)23-s + (58 − 100. i)25-s + (88.5 − 153. i)29-s + (62 + 107. i)31-s − 12·35-s − 265·37-s + (−213 − 368. i)41-s + (80 − 138. i)43-s + (270 − 467. i)47-s + ⋯
L(s)  = 1  + (−0.134 − 0.232i)5-s + (0.107 − 0.187i)7-s + (0.328 − 0.569i)11-s + (0.266 + 0.461i)13-s − 0.299·17-s − 0.627·19-s + (−0.761 − 1.31i)23-s + (0.464 − 0.803i)25-s + (0.566 − 0.981i)29-s + (0.359 + 0.622i)31-s − 0.0579·35-s − 1.17·37-s + (−0.811 − 1.40i)41-s + (0.283 − 0.491i)43-s + (0.837 − 1.45i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.343187825\)
\(L(\frac12)\) \(\approx\) \(1.343187825\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-12 + 20.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-12.5 - 21.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 21T + 4.91e3T^{2} \)
19 \( 1 + 52T + 6.85e3T^{2} \)
23 \( 1 + (84 + 145. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-88.5 + 153. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-62 - 107. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 265T + 5.06e4T^{2} \)
41 \( 1 + (213 + 368. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-80 + 138. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-270 + 467. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 258T + 1.48e5T^{2} \)
59 \( 1 + (264 + 457. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-252.5 + 437. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-122 - 211. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 204T + 3.57e5T^{2} \)
73 \( 1 + 397T + 3.89e5T^{2} \)
79 \( 1 + (100 - 173. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-270 + 467. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 453T + 7.04e5T^{2} \)
97 \( 1 + (145 - 251. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.84611000424966160724525866539, −10.15265706925924009527018765735, −8.772520913133446563615335350715, −8.363476396434094498605581186543, −6.92983882324330148520787007670, −6.12696750957581783072254795827, −4.73961624072525861003486273148, −3.77809750531836244476227595412, −2.18176607861530931626572138183, −0.49249243989808514936793690198, 1.53701907886629696013771176405, 3.06682374657034306688268259310, 4.31583716744626583491764209087, 5.52407729899431169942267919761, 6.63954365576572072827632667093, 7.60829363830556501942699795630, 8.625999041060918191249177280798, 9.593923613501455549061917756939, 10.55287851910520646583074969246, 11.42984413616213853092117343169

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