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Properties

Label 2-18e2-12.11-c3-0-37
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

\(L(2)\) \(\approx\) \(1.075140603\)
\(L(\frac12)\) \(\approx\) \(1.075140603\)
\(L(\frac{5}{2})\) 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

−10.87291634457255652558940862602, −10.24291192977387935760907761011, −9.462095766821291031760521982531, −8.033483001787491260535678112026, −7.23284724765781362499680044609, −6.72631953883904068284595364665, −5.52691739495119145969720285831, −3.99444205947813953867810597948, −2.49732658310613800514054322243, −0.59489962670585932755708845728, 1.10258720267077714410567375383, 2.29824186341261123060454873998, 3.87911419833235872898638296589, 5.08794398256336226180867540609, 6.23475452799475562663588862721, 7.974101531137019686076968574128, 8.557449362919925285655932152405, 9.148017835799895371763320680617, 10.14869188945132410741835542566, 11.25850256929629985045313635656

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