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Properties

Label 2-18e2-12.11-c3-0-38
Degree $2$
Conductor $324$
Sign $0.916 + 0.400i$
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  + (2.36 − 1.54i)2-s + (3.20 − 7.32i)4-s + 1.43i·5-s + 27.5i·7-s + (−3.76 − 22.3i)8-s + (2.21 + 3.38i)10-s + 22.2·11-s + 69.1·13-s + (42.6 + 65.2i)14-s + (−43.4 − 46.9i)16-s + 31.4i·17-s − 11.4i·19-s + (10.4 + 4.58i)20-s + (52.5 − 34.3i)22-s + 145.·23-s + ⋯
L(s)  = 1  + (0.836 − 0.547i)2-s + (0.400 − 0.916i)4-s + 0.127i·5-s + 1.48i·7-s + (−0.166 − 0.986i)8-s + (0.0700 + 0.107i)10-s + 0.608·11-s + 1.47·13-s + (0.815 + 1.24i)14-s + (−0.678 − 0.734i)16-s + 0.448i·17-s − 0.138i·19-s + (0.117 + 0.0512i)20-s + (0.509 − 0.333i)22-s + 1.31·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.916 + 0.400i$
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: \((2,\ 324,\ (\ :3/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.344706134\)
\(L(\frac12)\) \(\approx\) \(3.344706134\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.36 + 1.54i)T \)
3 \( 1 \)
good5 \( 1 - 1.43iT - 125T^{2} \)
7 \( 1 - 27.5iT - 343T^{2} \)
11 \( 1 - 22.2T + 1.33e3T^{2} \)
13 \( 1 - 69.1T + 2.19e3T^{2} \)
17 \( 1 - 31.4iT - 4.91e3T^{2} \)
19 \( 1 + 11.4iT - 6.85e3T^{2} \)
23 \( 1 - 145.T + 1.21e4T^{2} \)
29 \( 1 + 108. iT - 2.43e4T^{2} \)
31 \( 1 - 118. iT - 2.97e4T^{2} \)
37 \( 1 + 300.T + 5.06e4T^{2} \)
41 \( 1 + 398. iT - 6.89e4T^{2} \)
43 \( 1 - 200. iT - 7.95e4T^{2} \)
47 \( 1 - 303.T + 1.03e5T^{2} \)
53 \( 1 - 243. iT - 1.48e5T^{2} \)
59 \( 1 - 83.8T + 2.05e5T^{2} \)
61 \( 1 + 398.T + 2.26e5T^{2} \)
67 \( 1 - 355. iT - 3.00e5T^{2} \)
71 \( 1 - 866.T + 3.57e5T^{2} \)
73 \( 1 - 64.6T + 3.89e5T^{2} \)
79 \( 1 - 409. iT - 4.93e5T^{2} \)
83 \( 1 + 159.T + 5.71e5T^{2} \)
89 \( 1 + 1.49e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.40e3T + 9.12e5T^{2} \)
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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

−11.23084188361154283426360823426, −10.55837296053137042962353198696, −9.151729061017908126442050221009, −8.665572301666962316129763069028, −6.85507780258756236082490753130, −6.00369684355986345857195235704, −5.15272273528354825397912237935, −3.77612312438022450873627910616, −2.70468421742681687866187353449, −1.35344158497461447223642431707, 1.16367271415762390994403779742, 3.28245726980162742154119198924, 4.10173699108962996084667399746, 5.17247588772453002493650037884, 6.53470816386785019757896268256, 7.08472102228889632331975603295, 8.209269705526057103927792256250, 9.168549064712854763800426959829, 10.70052327002637214472328899383, 11.18341660687113144489985548044

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