Properties

Label 2-18e2-12.11-c3-0-23
Degree $2$
Conductor $324$
Sign $-0.356 - 0.934i$
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  + (0.513 + 2.78i)2-s + (−7.47 + 2.85i)4-s − 2.40i·5-s − 2.65i·7-s + (−11.7 − 19.3i)8-s + (6.70 − 1.23i)10-s + 48.2·11-s + 40.7·13-s + (7.39 − 1.36i)14-s + (47.6 − 42.6i)16-s + 36.3i·17-s + 125. i·19-s + (6.87 + 18.0i)20-s + (24.7 + 134. i)22-s − 194.·23-s + ⋯
L(s)  = 1  + (0.181 + 0.983i)2-s + (−0.934 + 0.356i)4-s − 0.215i·5-s − 0.143i·7-s + (−0.520 − 0.853i)8-s + (0.211 − 0.0391i)10-s + 1.32·11-s + 0.868·13-s + (0.141 − 0.0260i)14-s + (0.745 − 0.666i)16-s + 0.518i·17-s + 1.51i·19-s + (0.0769 + 0.201i)20-s + (0.240 + 1.30i)22-s − 1.75·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.356 - 0.934i$
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: \((2,\ 324,\ (\ :3/2),\ -0.356 - 0.934i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.845842020\)
\(L(\frac12)\) \(\approx\) \(1.845842020\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.513 - 2.78i)T \)
3 \( 1 \)
good5 \( 1 + 2.40iT - 125T^{2} \)
7 \( 1 + 2.65iT - 343T^{2} \)
11 \( 1 - 48.2T + 1.33e3T^{2} \)
13 \( 1 - 40.7T + 2.19e3T^{2} \)
17 \( 1 - 36.3iT - 4.91e3T^{2} \)
19 \( 1 - 125. iT - 6.85e3T^{2} \)
23 \( 1 + 194.T + 1.21e4T^{2} \)
29 \( 1 - 177. iT - 2.43e4T^{2} \)
31 \( 1 + 175. iT - 2.97e4T^{2} \)
37 \( 1 - 199.T + 5.06e4T^{2} \)
41 \( 1 - 233. iT - 6.89e4T^{2} \)
43 \( 1 - 291. iT - 7.95e4T^{2} \)
47 \( 1 + 61.8T + 1.03e5T^{2} \)
53 \( 1 - 352. iT - 1.48e5T^{2} \)
59 \( 1 - 141.T + 2.05e5T^{2} \)
61 \( 1 - 15.4T + 2.26e5T^{2} \)
67 \( 1 - 152. iT - 3.00e5T^{2} \)
71 \( 1 + 28.0T + 3.57e5T^{2} \)
73 \( 1 - 124.T + 3.89e5T^{2} \)
79 \( 1 + 748. iT - 4.93e5T^{2} \)
83 \( 1 + 348.T + 5.71e5T^{2} \)
89 \( 1 + 416. iT - 7.04e5T^{2} \)
97 \( 1 - 1.50e3T + 9.12e5T^{2} \)
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

−11.68530284848850518155846659456, −10.33996913371115381515524808249, −9.361894351889647957170356248508, −8.484025124408220648750141306349, −7.69439201612487070610764708186, −6.38139627187857872278603891384, −5.89993652261902059051029438803, −4.35678596922442963672341785204, −3.62958155783184590405366833243, −1.26938629402932056024796356638, 0.75598928085712460918537129922, 2.22531172187213723127903786493, 3.55386899246269021964876044584, 4.50575952736159761610702950658, 5.83546413146800133873819062549, 6.88541912577441794963254659822, 8.454821697395534294414855233963, 9.145762725638194270689087955939, 10.06216774239612074429162179747, 11.10114737649566903096253564103

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