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Properties

Label 2-18e2-12.11-c3-0-39
Degree $2$
Conductor $324$
Sign $-0.440 + 0.897i$
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  + (−2.75 − 0.639i)2-s + (7.18 + 3.52i)4-s + 5.44i·5-s + 24.1i·7-s + (−17.5 − 14.3i)8-s + (3.48 − 14.9i)10-s − 50.7·11-s − 50.1·13-s + (15.4 − 66.6i)14-s + (39.1 + 50.6i)16-s − 51.7i·17-s − 27.9i·19-s + (−19.1 + 39.0i)20-s + (139. + 32.4i)22-s + 7.87·23-s + ⋯
L(s)  = 1  + (−0.974 − 0.226i)2-s + (0.897 + 0.440i)4-s + 0.486i·5-s + 1.30i·7-s + (−0.774 − 0.632i)8-s + (0.110 − 0.474i)10-s − 1.39·11-s − 1.07·13-s + (0.295 − 1.27i)14-s + (0.611 + 0.790i)16-s − 0.737i·17-s − 0.337i·19-s + (−0.214 + 0.437i)20-s + (1.35 + 0.314i)22-s + 0.0713·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.440 + 0.897i$
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: \((2,\ 324,\ (\ :3/2),\ -0.440 + 0.897i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2798735361\)
\(L(\frac12)\) \(\approx\) \(0.2798735361\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.75 + 0.639i)T \)
3 \( 1 \)
good5 \( 1 - 5.44iT - 125T^{2} \)
7 \( 1 - 24.1iT - 343T^{2} \)
11 \( 1 + 50.7T + 1.33e3T^{2} \)
13 \( 1 + 50.1T + 2.19e3T^{2} \)
17 \( 1 + 51.7iT - 4.91e3T^{2} \)
19 \( 1 + 27.9iT - 6.85e3T^{2} \)
23 \( 1 - 7.87T + 1.21e4T^{2} \)
29 \( 1 + 245. iT - 2.43e4T^{2} \)
31 \( 1 - 59.3iT - 2.97e4T^{2} \)
37 \( 1 - 295.T + 5.06e4T^{2} \)
41 \( 1 - 169. iT - 6.89e4T^{2} \)
43 \( 1 + 329. iT - 7.95e4T^{2} \)
47 \( 1 - 95.9T + 1.03e5T^{2} \)
53 \( 1 + 300. iT - 1.48e5T^{2} \)
59 \( 1 + 226.T + 2.05e5T^{2} \)
61 \( 1 + 347.T + 2.26e5T^{2} \)
67 \( 1 + 1.04e3iT - 3.00e5T^{2} \)
71 \( 1 - 243.T + 3.57e5T^{2} \)
73 \( 1 + 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + 612. iT - 4.93e5T^{2} \)
83 \( 1 + 566.T + 5.71e5T^{2} \)
89 \( 1 - 212. iT - 7.04e5T^{2} \)
97 \( 1 + 468.T + 9.12e5T^{2} \)
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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

−10.77298937943182970991288715903, −9.878876065335550171400288389271, −9.123555977902056816215201259259, −8.066335674502540352978450057986, −7.30830597961316916009917493478, −6.12486272476913372438807138854, −4.99036353309236218700643625218, −2.84093805948003619980126803909, −2.37197570907333748002185924896, −0.14599700573241743932743387521, 1.19535987191887017351015722753, 2.80196905400205087752220534258, 4.54813098298114498997711666778, 5.67822822858369579530004971614, 7.07508037962909443965770179637, 7.66751298312190579130350318761, 8.564101931644139993144990987378, 9.731223582948818698344621848259, 10.44905184970628585991390952967, 11.03418514311321608081315073359

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