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Properties

Label 2-462-21.17-c1-0-13
Degree $2$
Conductor $462$
Sign $0.921 - 0.388i$
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 + (1.60 − 0.656i)3-s + (0.499 − 0.866i)4-s + (1.92 + 3.33i)5-s + (−1.05 + 1.37i)6-s + (1.58 − 2.12i)7-s + 0.999i·8-s + (2.13 − 2.10i)9-s + (−3.33 − 1.92i)10-s + (0.866 + 0.5i)11-s + (0.232 − 1.71i)12-s − 3.23i·13-s + (−0.308 + 2.62i)14-s + (5.27 + 4.08i)15-s + (−0.5 − 0.866i)16-s + (−0.751 + 1.30i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.925 − 0.379i)3-s + (0.249 − 0.433i)4-s + (0.861 + 1.49i)5-s + (−0.432 + 0.559i)6-s + (0.597 − 0.801i)7-s + 0.353i·8-s + (0.712 − 0.701i)9-s + (−1.05 − 0.608i)10-s + (0.261 + 0.150i)11-s + (0.0671 − 0.495i)12-s − 0.897i·13-s + (−0.0823 + 0.702i)14-s + (1.36 + 1.05i)15-s + (−0.125 − 0.216i)16-s + (−0.182 + 0.315i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.921 - 0.388i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.921 - 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69934 + 0.343582i\)
\(L(\frac12)\) \(\approx\) \(1.69934 + 0.343582i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 + 0.656i)T \)
7 \( 1 + (-1.58 + 2.12i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (-1.92 - 3.33i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.23iT - 13T^{2} \)
17 \( 1 + (0.751 - 1.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.17 - 1.25i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.70 - 3.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.71iT - 29T^{2} \)
31 \( 1 + (-5.23 - 3.02i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0683 + 0.118i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.85T + 41T^{2} \)
43 \( 1 - 7.22T + 43T^{2} \)
47 \( 1 + (-5.40 - 9.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.98 + 5.18i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.68 + 2.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.45 - 4.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.19 - 8.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.857iT - 71T^{2} \)
73 \( 1 + (2.12 + 1.22i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.11 - 3.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.67T + 83T^{2} \)
89 \( 1 + (3.33 + 5.77i)T + (-44.5 + 77.0i)T^{2} \)
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

−10.64680591337319978864303368394, −10.18228746939713167193576732352, −9.434136615212933065144513593368, −8.076764662653890291696872889184, −7.63615540739723704928181203276, −6.64414595283493410949290590508, −5.94091418498501664109819775805, −4.04748123986609631774045478775, −2.74896266137406355516590205801, −1.65827291992550441399966302957, 1.59423554295649071864622374238, 2.41694431670313631251557840071, 4.21540389413451570666197352378, 5.01403707179754635440291339251, 6.32483786809262956591375770435, 7.86459608521059001206282353687, 8.751523474528538913592159646543, 8.984931466687400985535955900448, 9.763072409165666611586450910281, 10.76809660952915359146437565858

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