Properties

Label 2-462-77.4-c1-0-1
Degree $2$
Conductor $462$
Sign $-0.260 - 0.965i$
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  + (−0.104 + 0.994i)2-s + (−0.669 − 0.743i)3-s + (−0.978 − 0.207i)4-s + (−3.99 − 1.77i)5-s + (0.809 − 0.587i)6-s + (2.64 + 0.160i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (2.18 − 3.78i)10-s + (−1.81 + 2.77i)11-s + (0.5 + 0.866i)12-s + (3.79 + 2.75i)13-s + (−0.435 + 2.60i)14-s + (1.35 + 4.15i)15-s + (0.913 + 0.406i)16-s + (0.0781 + 0.743i)17-s + ⋯
L(s)  = 1  + (−0.0739 + 0.703i)2-s + (−0.386 − 0.429i)3-s + (−0.489 − 0.103i)4-s + (−1.78 − 0.795i)5-s + (0.330 − 0.239i)6-s + (0.998 + 0.0604i)7-s + (0.109 − 0.336i)8-s + (−0.0348 + 0.331i)9-s + (0.691 − 1.19i)10-s + (−0.546 + 0.837i)11-s + (0.144 + 0.249i)12-s + (1.05 + 0.764i)13-s + (−0.116 + 0.697i)14-s + (0.348 + 1.07i)15-s + (0.228 + 0.101i)16-s + (0.0189 + 0.180i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.260 - 0.965i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.260 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394127 + 0.514831i\)
\(L(\frac12)\) \(\approx\) \(0.394127 + 0.514831i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.104 - 0.994i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + (-2.64 - 0.160i)T \)
11 \( 1 + (1.81 - 2.77i)T \)
good5 \( 1 + (3.99 + 1.77i)T + (3.34 + 3.71i)T^{2} \)
13 \( 1 + (-3.79 - 2.75i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.0781 - 0.743i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (3.87 - 0.822i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (-2.65 - 4.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.839 - 2.58i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.91 - 3.07i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (2.02 - 2.25i)T + (-3.86 - 36.7i)T^{2} \)
41 \( 1 + (-1.28 + 3.95i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.23T + 43T^{2} \)
47 \( 1 + (-11.4 + 2.44i)T + (42.9 - 19.1i)T^{2} \)
53 \( 1 + (10.0 - 4.46i)T + (35.4 - 39.3i)T^{2} \)
59 \( 1 + (-7.20 - 1.53i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (7.12 + 3.17i)T + (40.8 + 45.3i)T^{2} \)
67 \( 1 + (0.607 - 1.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.79 - 4.21i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.29 + 0.487i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (0.753 - 7.17i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (9.08 - 6.60i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.31 - 2.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.68 + 5.58i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.30025435553499681900684505913, −10.77773743361590442134437980361, −8.975595062769925939781874402689, −8.463708927227075559045809495912, −7.56305328560426652590229078110, −7.07997620703637117834251797742, −5.54540214742280846882968727849, −4.64465183220521341816265390334, −3.89164567851505376964596561200, −1.40252228455209967109534341512, 0.48616926162205912017490174676, 2.85448786785677873396402410272, 3.84733458714186754062582686092, 4.62185191660314934940493773422, 5.95715957412891888142966129865, 7.38502980074203708725377309334, 8.198331338070337590738805534635, 8.791033243623069439456271087837, 10.59821425843982539814010950746, 10.86125570329807877139146019170

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