Properties

Label 2-462-77.10-c1-0-7
Degree $2$
Conductor $462$
Sign $0.959 - 0.283i$
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  + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (1.92 − 1.11i)5-s − 0.999·6-s + (2.45 + 0.982i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.11 + 1.92i)10-s + (−0.0571 − 3.31i)11-s + (0.866 − 0.499i)12-s − 0.112·13-s + (−2.61 + 0.377i)14-s + 2.22·15-s + (−0.5 − 0.866i)16-s + (0.119 − 0.207i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (0.861 − 0.497i)5-s − 0.408·6-s + (0.928 + 0.371i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.351 + 0.609i)10-s + (−0.0172 − 0.999i)11-s + (0.249 − 0.144i)12-s − 0.0312·13-s + (−0.699 + 0.100i)14-s + 0.574·15-s + (−0.125 − 0.216i)16-s + (0.0290 − 0.0503i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.959 - 0.283i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.959 - 0.283i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53997 + 0.222653i\)
\(L(\frac12)\) \(\approx\) \(1.53997 + 0.222653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-2.45 - 0.982i)T \)
11 \( 1 + (0.0571 + 3.31i)T \)
good5 \( 1 + (-1.92 + 1.11i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.112T + 13T^{2} \)
17 \( 1 + (-0.119 + 0.207i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.218 + 0.377i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.401 + 0.695i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.50iT - 29T^{2} \)
31 \( 1 + (-0.306 - 0.177i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.67 - 6.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.14T + 41T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 + (11.2 - 6.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.28 + 2.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.27 - 1.89i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.524 - 0.909i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.48 + 4.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.58T + 71T^{2} \)
73 \( 1 + (-2.39 + 4.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.429 + 0.247i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.569T + 83T^{2} \)
89 \( 1 + (12.1 - 6.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−10.99959158848147629626013827672, −9.867595947045076637362718060817, −9.287885402053347894190973868756, −8.344285685704793480240052622531, −7.88709267197939988022081412000, −6.34217944289783198818799502332, −5.52182281531965290422701002615, −4.50499403991984032493135047740, −2.73247791987475130990420548460, −1.42868530370239496974236557087, 1.59513444495641555386660291311, 2.45587949938776929624967548817, 3.95010940282503246150501560039, 5.30797886091083104753865942464, 6.72443223092521714757548984622, 7.41718338079091164568263336021, 8.350008046615697249671184083006, 9.299153634486488591735479192193, 10.12961662188683478321725116254, 10.76080195033486513147786933444

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