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Properties

Label 2-18e2-9.4-c3-0-9
Degree $2$
Conductor $324$
Sign $-0.766 + 0.642i$
Analytic cond. $19.1166$
Root an. cond. $4.37225$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9 + 15.5i)5-s + (−4 − 6.92i)7-s + (18 + 31.1i)11-s + (5 − 8.66i)13-s − 18·17-s − 100·19-s + (36 − 62.3i)23-s + (−99.5 − 172. i)25-s + (−117 − 202. i)29-s + (8 − 13.8i)31-s + 144·35-s − 226·37-s + (45 − 77.9i)41-s + (−226 − 391. i)43-s + (216 + 374. i)47-s + ⋯
L(s)  = 1  + (−0.804 + 1.39i)5-s + (−0.215 − 0.374i)7-s + (0.493 + 0.854i)11-s + (0.106 − 0.184i)13-s − 0.256·17-s − 1.20·19-s + (0.326 − 0.565i)23-s + (−0.796 − 1.37i)25-s + (−0.749 − 1.29i)29-s + (0.0463 − 0.0802i)31-s + 0.695·35-s − 1.00·37-s + (0.171 − 0.296i)41-s + (−0.801 − 1.38i)43-s + (0.670 + 1.16i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + (9 - 15.5i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (4 + 6.92i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-18 - 31.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-5 + 8.66i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 18T + 4.91e3T^{2} \)
19 \( 1 + 100T + 6.85e3T^{2} \)
23 \( 1 + (-36 + 62.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (117 + 202. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-8 + 13.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 226T + 5.06e4T^{2} \)
41 \( 1 + (-45 + 77.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (226 + 391. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-216 - 374. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 414T + 1.48e5T^{2} \)
59 \( 1 + (342 - 592. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (211 + 365. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (166 - 287. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 360T + 3.57e5T^{2} \)
73 \( 1 - 26T + 3.89e5T^{2} \)
79 \( 1 + (256 + 443. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (594 + 1.02e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 630T + 7.04e5T^{2} \)
97 \( 1 + (-527 - 912. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.73853287673954617269323677771, −10.19654369423144681302239975724, −8.930998785315700811138168964847, −7.71095089543020084460753617303, −6.98503989643137200029833562190, −6.21448453346420236943158412481, −4.42807257478413394746059578834, −3.56312585870681463522394508650, −2.24295913081658819090944787102, 0, 1.45726786870607816565944993370, 3.41298218900240172364526820560, 4.47560117966489706225804945176, 5.49874255140450193148474317013, 6.68372067474497949575620304304, 8.017185507758760957633677608508, 8.769378165223303546761876182299, 9.307547538123566023498884367750, 10.83424097124165793134393736530

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