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Properties

Label 2-18e2-12.11-c3-0-29
Degree $2$
Conductor $324$
Sign $-0.568 - 0.822i$
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  + (2.50 + 1.31i)2-s + (4.54 + 6.58i)4-s + 19.4i·5-s − 17.4i·7-s + (2.73 + 22.4i)8-s + (−25.5 + 48.6i)10-s + 65.7·11-s − 2.26·13-s + (22.9 − 43.7i)14-s + (−22.6 + 59.8i)16-s + 78.1i·17-s + 15.9i·19-s + (−127. + 88.2i)20-s + (164. + 86.3i)22-s − 180.·23-s + ⋯
L(s)  = 1  + (0.885 + 0.464i)2-s + (0.568 + 0.822i)4-s + 1.73i·5-s − 0.943i·7-s + (0.121 + 0.992i)8-s + (−0.806 + 1.53i)10-s + 1.80·11-s − 0.0483·13-s + (0.438 − 0.835i)14-s + (−0.353 + 0.935i)16-s + 1.11i·17-s + 0.192i·19-s + (−1.42 + 0.986i)20-s + (1.59 + 0.836i)22-s − 1.63·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.568 - 0.822i$
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: \((2,\ 324,\ (\ :3/2),\ -0.568 - 0.822i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.223826708\)
\(L(\frac12)\) \(\approx\) \(3.223826708\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.50 - 1.31i)T \)
3 \( 1 \)
good5 \( 1 - 19.4iT - 125T^{2} \)
7 \( 1 + 17.4iT - 343T^{2} \)
11 \( 1 - 65.7T + 1.33e3T^{2} \)
13 \( 1 + 2.26T + 2.19e3T^{2} \)
17 \( 1 - 78.1iT - 4.91e3T^{2} \)
19 \( 1 - 15.9iT - 6.85e3T^{2} \)
23 \( 1 + 180.T + 1.21e4T^{2} \)
29 \( 1 + 11.6iT - 2.43e4T^{2} \)
31 \( 1 - 30.2iT - 2.97e4T^{2} \)
37 \( 1 + 44.8T + 5.06e4T^{2} \)
41 \( 1 + 307. iT - 6.89e4T^{2} \)
43 \( 1 + 88.3iT - 7.95e4T^{2} \)
47 \( 1 + 44.8T + 1.03e5T^{2} \)
53 \( 1 + 90.6iT - 1.48e5T^{2} \)
59 \( 1 - 605.T + 2.05e5T^{2} \)
61 \( 1 - 283.T + 2.26e5T^{2} \)
67 \( 1 - 622. iT - 3.00e5T^{2} \)
71 \( 1 - 828.T + 3.57e5T^{2} \)
73 \( 1 - 706.T + 3.89e5T^{2} \)
79 \( 1 + 260. iT - 4.93e5T^{2} \)
83 \( 1 - 902.T + 5.71e5T^{2} \)
89 \( 1 + 44.4iT - 7.04e5T^{2} \)
97 \( 1 + 1.35e3T + 9.12e5T^{2} \)
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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

−11.54900392184299358859783382148, −10.75167608030434749665766710961, −9.935684251679867203437328576259, −8.334364430202129464788913259302, −7.22904453617433941972808643594, −6.65731275824638003759389454840, −5.92308089604453892603369153358, −3.91471086925480870420171017864, −3.72505577887812501186400757357, −2.04395734371884224611234120503, 0.901732298716070988181601361045, 2.07698336491174386060740949314, 3.82240389376622998754421472842, 4.74294671572722287671498687026, 5.61876883228196295215332081682, 6.58640767252330437596599889829, 8.203843639496537739514929395936, 9.298748381971074523225366509642, 9.619846738170060793126455641234, 11.40736641812087402389072417492

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