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Properties

Label 2-18e2-12.11-c3-0-19
Degree $2$
Conductor $324$
Sign $-0.285 - 0.958i$
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  + (−2.26 − 1.69i)2-s + (2.28 + 7.66i)4-s + 16.3i·5-s + 28.7i·7-s + (7.77 − 21.2i)8-s + (27.6 − 37.0i)10-s + 25.2·11-s + 74.9·13-s + (48.5 − 65.1i)14-s + (−53.5 + 35.0i)16-s − 28.4i·17-s + 132. i·19-s + (−125. + 37.3i)20-s + (−57.1 − 42.6i)22-s − 13.9·23-s + ⋯
L(s)  = 1  + (−0.801 − 0.597i)2-s + (0.285 + 0.958i)4-s + 1.46i·5-s + 1.55i·7-s + (0.343 − 0.939i)8-s + (0.872 − 1.17i)10-s + 0.690·11-s + 1.59·13-s + (0.927 − 1.24i)14-s + (−0.836 + 0.547i)16-s − 0.405i·17-s + 1.60i·19-s + (−1.39 + 0.417i)20-s + (−0.554 − 0.412i)22-s − 0.126·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.285 - 0.958i$
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: \((2,\ 324,\ (\ :3/2),\ -0.285 - 0.958i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.251485815\)
\(L(\frac12)\) \(\approx\) \(1.251485815\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.26 + 1.69i)T \)
3 \( 1 \)
good5 \( 1 - 16.3iT - 125T^{2} \)
7 \( 1 - 28.7iT - 343T^{2} \)
11 \( 1 - 25.2T + 1.33e3T^{2} \)
13 \( 1 - 74.9T + 2.19e3T^{2} \)
17 \( 1 + 28.4iT - 4.91e3T^{2} \)
19 \( 1 - 132. iT - 6.85e3T^{2} \)
23 \( 1 + 13.9T + 1.21e4T^{2} \)
29 \( 1 + 134. iT - 2.43e4T^{2} \)
31 \( 1 - 117. iT - 2.97e4T^{2} \)
37 \( 1 - 164.T + 5.06e4T^{2} \)
41 \( 1 - 83.5iT - 6.89e4T^{2} \)
43 \( 1 + 43.4iT - 7.95e4T^{2} \)
47 \( 1 + 449.T + 1.03e5T^{2} \)
53 \( 1 - 5.67iT - 1.48e5T^{2} \)
59 \( 1 - 836.T + 2.05e5T^{2} \)
61 \( 1 + 446.T + 2.26e5T^{2} \)
67 \( 1 - 130. iT - 3.00e5T^{2} \)
71 \( 1 + 209.T + 3.57e5T^{2} \)
73 \( 1 - 163.T + 3.89e5T^{2} \)
79 \( 1 + 1.29e3iT - 4.93e5T^{2} \)
83 \( 1 + 444.T + 5.71e5T^{2} \)
89 \( 1 - 1.17e3iT - 7.04e5T^{2} \)
97 \( 1 - 523.T + 9.12e5T^{2} \)
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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

−11.49690102458951457855886125710, −10.54012217809774050100568334975, −9.673468922725917692696323911021, −8.715732941760254240250688869137, −7.924915212957662004894083656237, −6.58585824020252516745025088593, −5.96001140998758031251095735968, −3.77027056394052131274256253816, −2.88539174891027090553960300677, −1.70014028881732775400288044571, 0.67507602843881209810342780895, 1.35736580092500609009730444514, 3.96040653670403936772035544732, 4.90819080830847193030311461742, 6.21149589667027542868531942891, 7.13161259706345020971381144986, 8.248105497497399762768135535146, 8.878710021229657764592442917336, 9.727914437311034208747229370696, 10.84389714040961704400653824792

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