Properties

Label 2-462-33.2-c1-0-21
Degree $2$
Conductor $462$
Sign $-0.0547 + 0.998i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (1.48 − 0.898i)3-s + (−0.809 − 0.587i)4-s + (3.37 − 1.09i)5-s + (−0.396 − 1.68i)6-s + (0.587 − 0.809i)7-s + (−0.809 + 0.587i)8-s + (1.38 − 2.66i)9-s − 3.54i·10-s + (−1.81 + 2.77i)11-s + (−1.72 − 0.143i)12-s + (−5.58 − 1.81i)13-s + (−0.587 − 0.809i)14-s + (4.01 − 4.65i)15-s + (0.309 + 0.951i)16-s + (2.31 + 7.12i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.854 − 0.518i)3-s + (−0.404 − 0.293i)4-s + (1.50 − 0.490i)5-s + (−0.161 − 0.688i)6-s + (0.222 − 0.305i)7-s + (−0.286 + 0.207i)8-s + (0.462 − 0.886i)9-s − 1.12i·10-s + (−0.546 + 0.837i)11-s + (−0.498 − 0.0414i)12-s + (−1.54 − 0.503i)13-s + (−0.157 − 0.216i)14-s + (1.03 − 1.20i)15-s + (0.0772 + 0.237i)16-s + (0.561 + 1.72i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0547 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0547 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.0547 + 0.998i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.0547 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63089 - 1.72269i\)
\(L(\frac12)\) \(\approx\) \(1.63089 - 1.72269i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (-1.48 + 0.898i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (1.81 - 2.77i)T \)
good5 \( 1 + (-3.37 + 1.09i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (5.58 + 1.81i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.31 - 7.12i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.87 - 3.95i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.38iT - 23T^{2} \)
29 \( 1 + (4.51 + 3.27i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.123 - 0.381i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.42 + 6.11i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.577 - 0.419i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 0.722iT - 43T^{2} \)
47 \( 1 + (-0.656 - 0.903i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.334 + 0.108i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.843 + 1.16i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.21 - 0.393i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.40T + 67T^{2} \)
71 \( 1 + (-3.46 + 1.12i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.72 - 2.38i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-11.2 - 3.65i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.11 + 9.57i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 4.97iT - 89T^{2} \)
97 \( 1 + (0.155 - 0.479i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.42146454022669015306764791552, −9.921099921752706937119926708659, −9.348469434688128608710026538244, −8.133204609097727716354417728492, −7.33065212714555601836366533163, −5.88868542921729196894708079103, −5.09101739626569858053168937526, −3.67503575256767997420267876302, −2.25213164400008195824469119676, −1.60039845886661667579119971485, 2.36101477055815314402421461533, 3.08728281236138897054672294903, 5.03222973523932779690301379078, 5.27465670029317893100869892323, 6.78648205991033481465769428383, 7.49505180955795117965257350498, 8.734942873872131831570382259264, 9.496968636966356315070505138137, 9.968353716945920375577684154277, 11.08913347883294459705741395063

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