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Properties

Label 2-462-7.2-c1-0-10
Degree $2$
Conductor $462$
Sign $-0.605 + 0.795i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.5 − 2.59i)5-s − 0.999·6-s + (2.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 − 2.59i)10-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + 2·13-s + (0.500 − 2.59i)14-s − 3·15-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.670 − 1.16i)5-s − 0.408·6-s + (0.944 − 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.474 − 0.821i)10-s + (0.150 + 0.261i)11-s + (−0.144 + 0.249i)12-s + 0.554·13-s + (0.133 − 0.694i)14-s − 0.774·15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.785432 - 1.58440i\)
\(L(\frac12)\) \(\approx\) \(0.785432 - 1.58440i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-8 - 13.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−10.93654216649603626104166968522, −9.927907516137854598349173685749, −8.967276400106426869910190980417, −8.185130982465278066784869731568, −6.95901501898229785890438689766, −5.65081568096164035135449734555, −5.04441095709232249172197879272, −3.97342559205762001613961305927, −2.05750700543528053380001804863, −1.14550545010524135596556915406, 2.24133601191494122991144307934, 3.64220797129304656938683558994, 4.77496859845964703996739890820, 5.92248409012339033304400184717, 6.40988959433214787146001074595, 7.61737701820199576337490709246, 8.620128179545905334550509034037, 9.541855471454360321108817114676, 10.77998444165836686916566992690, 11.04597271497655279173690827563

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