↑

Properties

Label 2-18e2-12.11-c3-0-56
Degree $2$
Conductor $324$
Sign $0.440 + 0.897i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.75 − 0.639i)2-s + (7.18 − 3.52i)4-s + 5.44i·5-s − 24.1i·7-s + (17.5 − 14.3i)8-s + (3.48 + 14.9i)10-s + 50.7·11-s − 50.1·13-s + (−15.4 − 66.6i)14-s + (39.1 − 50.6i)16-s − 51.7i·17-s + 27.9i·19-s + (19.1 + 39.0i)20-s + (139. − 32.4i)22-s − 7.87·23-s + ⋯
L(s)  = 1  + (0.974 − 0.226i)2-s + (0.897 − 0.440i)4-s + 0.486i·5-s − 1.30i·7-s + (0.774 − 0.632i)8-s + (0.110 + 0.474i)10-s + 1.39·11-s − 1.07·13-s + (−0.295 − 1.27i)14-s + (0.611 − 0.790i)16-s − 0.737i·17-s + 0.337i·19-s + (0.214 + 0.437i)20-s + (1.35 − 0.314i)22-s − 0.0713·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.440 + 0.897i$
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: \((2,\ 324,\ (\ :3/2),\ 0.440 + 0.897i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.601678273\)
\(L(\frac12)\) \(\approx\) \(3.601678273\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.75 + 0.639i)T \)
3 \( 1 \)
good5 \( 1 - 5.44iT - 125T^{2} \)
7 \( 1 + 24.1iT - 343T^{2} \)
11 \( 1 - 50.7T + 1.33e3T^{2} \)
13 \( 1 + 50.1T + 2.19e3T^{2} \)
17 \( 1 + 51.7iT - 4.91e3T^{2} \)
19 \( 1 - 27.9iT - 6.85e3T^{2} \)
23 \( 1 + 7.87T + 1.21e4T^{2} \)
29 \( 1 + 245. iT - 2.43e4T^{2} \)
31 \( 1 + 59.3iT - 2.97e4T^{2} \)
37 \( 1 - 295.T + 5.06e4T^{2} \)
41 \( 1 - 169. iT - 6.89e4T^{2} \)
43 \( 1 - 329. iT - 7.95e4T^{2} \)
47 \( 1 + 95.9T + 1.03e5T^{2} \)
53 \( 1 + 300. iT - 1.48e5T^{2} \)
59 \( 1 - 226.T + 2.05e5T^{2} \)
61 \( 1 + 347.T + 2.26e5T^{2} \)
67 \( 1 - 1.04e3iT - 3.00e5T^{2} \)
71 \( 1 + 243.T + 3.57e5T^{2} \)
73 \( 1 + 1.09e3T + 3.89e5T^{2} \)
79 \( 1 - 612. iT - 4.93e5T^{2} \)
83 \( 1 - 566.T + 5.71e5T^{2} \)
89 \( 1 - 212. iT - 7.04e5T^{2} \)
97 \( 1 + 468.T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.31884473059470149596797816117, −10.17872695679485814118032236964, −9.585266183555939955158667076823, −7.72361683192476007167399561985, −6.97358545543978671878807025192, −6.18279093474746437799984470198, −4.65788306139962105417880694471, −3.92975319953579437311677227951, −2.66342011884050117740333477155, −1.02215580272244498543596392683, 1.74047132524037387801074126382, 3.04451980334134463575398348384, 4.41137186574559929912245344317, 5.33290981150388958948434966672, 6.28296201708629399095744711542, 7.25826418463044558308280215391, 8.600520321391976253573442990886, 9.229217542211529294673385957610, 10.66995978091349163872268596425, 11.81200832728811070721211860064

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