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Properties

Label 2-18e2-12.11-c3-0-22
Degree $2$
Conductor $324$
Sign $-0.998 + 0.0495i$
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  + (2.04 + 1.94i)2-s + (0.396 + 7.99i)4-s + 16.5i·5-s + 22.2i·7-s + (−14.7 + 17.1i)8-s + (−32.1 + 33.8i)10-s + 12.7·11-s + 22.3·13-s + (−43.3 + 45.5i)14-s + (−63.6 + 6.33i)16-s − 117. i·17-s − 27.7i·19-s + (−131. + 6.54i)20-s + (26.1 + 24.8i)22-s + 35.1·23-s + ⋯
L(s)  = 1  + (0.724 + 0.689i)2-s + (0.0495 + 0.998i)4-s + 1.47i·5-s + 1.20i·7-s + (−0.652 + 0.757i)8-s + (−1.01 + 1.06i)10-s + 0.349·11-s + 0.476·13-s + (−0.827 + 0.869i)14-s + (−0.995 + 0.0989i)16-s − 1.67i·17-s − 0.334i·19-s + (−1.47 + 0.0731i)20-s + (0.253 + 0.240i)22-s + 0.318·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0495i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.998 + 0.0495i$
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: \((2,\ 324,\ (\ :3/2),\ -0.998 + 0.0495i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.478293557\)
\(L(\frac12)\) \(\approx\) \(2.478293557\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.04 - 1.94i)T \)
3 \( 1 \)
good5 \( 1 - 16.5iT - 125T^{2} \)
7 \( 1 - 22.2iT - 343T^{2} \)
11 \( 1 - 12.7T + 1.33e3T^{2} \)
13 \( 1 - 22.3T + 2.19e3T^{2} \)
17 \( 1 + 117. iT - 4.91e3T^{2} \)
19 \( 1 + 27.7iT - 6.85e3T^{2} \)
23 \( 1 - 35.1T + 1.21e4T^{2} \)
29 \( 1 - 1.16iT - 2.43e4T^{2} \)
31 \( 1 - 137. iT - 2.97e4T^{2} \)
37 \( 1 - 233.T + 5.06e4T^{2} \)
41 \( 1 + 15.3iT - 6.89e4T^{2} \)
43 \( 1 - 417. iT - 7.95e4T^{2} \)
47 \( 1 + 232.T + 1.03e5T^{2} \)
53 \( 1 - 180. iT - 1.48e5T^{2} \)
59 \( 1 + 627.T + 2.05e5T^{2} \)
61 \( 1 - 764.T + 2.26e5T^{2} \)
67 \( 1 - 131. iT - 3.00e5T^{2} \)
71 \( 1 + 22.6T + 3.57e5T^{2} \)
73 \( 1 - 387.T + 3.89e5T^{2} \)
79 \( 1 + 561. iT - 4.93e5T^{2} \)
83 \( 1 + 684.T + 5.71e5T^{2} \)
89 \( 1 - 278. iT - 7.04e5T^{2} \)
97 \( 1 - 528.T + 9.12e5T^{2} \)
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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

−11.56968919555851623719829574261, −11.17218736912709797827531873108, −9.644305060459383601439307078200, −8.715451956274160573440725112456, −7.51913168315726897846322801265, −6.69345522472751536977434715141, −5.93501390583910661921954190105, −4.79204213009306965546132728228, −3.23085444776069070478918655347, −2.56625150525059664631885318989, 0.72630389906258282534413590399, 1.67290889017054765545907713126, 3.76755964565019974331504424011, 4.31349372645567272752943017428, 5.47634987766531458873780342983, 6.51373296257439025655552114427, 7.989464410343668390749011259946, 8.977404642586718876892022449489, 9.978865766043257605007661046377, 10.79819863319098016716289822779

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