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Properties

Label 2-18e2-12.11-c3-0-46
Degree $2$
Conductor $324$
Sign $0.0255 + 0.999i$
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  + (−2.82 + 0.0360i)2-s + (7.99 − 0.204i)4-s − 16.9i·5-s + 3.56i·7-s + (−22.6 + 0.866i)8-s + (0.610 + 47.8i)10-s + 50.1·11-s + 37.9·13-s + (−0.128 − 10.0i)14-s + (63.9 − 3.26i)16-s − 84.3i·17-s + 62.9i·19-s + (−3.45 − 135. i)20-s + (−141. + 1.80i)22-s + 75.2·23-s + ⋯
L(s)  = 1  + (−0.999 + 0.0127i)2-s + (0.999 − 0.0255i)4-s − 1.51i·5-s + 0.192i·7-s + (−0.999 + 0.0382i)8-s + (0.0193 + 1.51i)10-s + 1.37·11-s + 0.810·13-s + (−0.00245 − 0.192i)14-s + (0.998 − 0.0510i)16-s − 1.20i·17-s + 0.759i·19-s + (−0.0386 − 1.51i)20-s + (−1.37 + 0.0175i)22-s + 0.681·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0255 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0255 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.0255 + 0.999i$
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: \((2,\ 324,\ (\ :3/2),\ 0.0255 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.241537490\)
\(L(\frac12)\) \(\approx\) \(1.241537490\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.82 - 0.0360i)T \)
3 \( 1 \)
good5 \( 1 + 16.9iT - 125T^{2} \)
7 \( 1 - 3.56iT - 343T^{2} \)
11 \( 1 - 50.1T + 1.33e3T^{2} \)
13 \( 1 - 37.9T + 2.19e3T^{2} \)
17 \( 1 + 84.3iT - 4.91e3T^{2} \)
19 \( 1 - 62.9iT - 6.85e3T^{2} \)
23 \( 1 - 75.2T + 1.21e4T^{2} \)
29 \( 1 - 121. iT - 2.43e4T^{2} \)
31 \( 1 + 19.9iT - 2.97e4T^{2} \)
37 \( 1 - 17.7T + 5.06e4T^{2} \)
41 \( 1 + 345. iT - 6.89e4T^{2} \)
43 \( 1 + 130. iT - 7.95e4T^{2} \)
47 \( 1 + 306.T + 1.03e5T^{2} \)
53 \( 1 + 479. iT - 1.48e5T^{2} \)
59 \( 1 + 491.T + 2.05e5T^{2} \)
61 \( 1 - 99.8T + 2.26e5T^{2} \)
67 \( 1 + 619. iT - 3.00e5T^{2} \)
71 \( 1 + 254.T + 3.57e5T^{2} \)
73 \( 1 - 100.T + 3.89e5T^{2} \)
79 \( 1 + 988. iT - 4.93e5T^{2} \)
83 \( 1 - 503.T + 5.71e5T^{2} \)
89 \( 1 - 1.01e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.00e3T + 9.12e5T^{2} \)
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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

−10.94398202913908009474850361266, −9.646050898032608420525810067700, −8.994052994152721750859108795041, −8.502350827195344837597806515135, −7.28372576384075456172900153864, −6.16893594104536511387723889864, −5.05287441190159023424964766169, −3.59109426649195572088738580896, −1.67875966194531358243365131787, −0.71440530291888701926706538027, 1.33664488884608182368453943685, 2.82154493879282517413422078721, 3.89197606028164606649283304219, 6.18632612428602444266672907302, 6.58631973557988451235188735388, 7.56958336204694764868355779188, 8.660763658067852841561540813966, 9.591747649040254028240002273627, 10.56460311634789837376792769464, 11.13831479389553497243322176931

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