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Properties

Label 2-462-231.32-c1-0-11
Degree $2$
Conductor $462$
Sign $-0.785 - 0.619i$
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 + (−0.470 + 1.66i)3-s + (−0.499 + 0.866i)4-s + (1.00 − 0.580i)5-s + (−1.67 + 0.426i)6-s + (2.19 + 1.47i)7-s − 0.999·8-s + (−2.55 − 1.56i)9-s + (1.00 + 0.580i)10-s + (−0.864 + 3.20i)11-s + (−1.20 − 1.24i)12-s + 3.56i·13-s + (−0.173 + 2.64i)14-s + (0.494 + 1.94i)15-s + (−0.5 − 0.866i)16-s + (1.71 − 2.96i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.271 + 0.962i)3-s + (−0.249 + 0.433i)4-s + (0.449 − 0.259i)5-s + (−0.685 + 0.173i)6-s + (0.831 + 0.555i)7-s − 0.353·8-s + (−0.852 − 0.522i)9-s + (0.318 + 0.183i)10-s + (−0.260 + 0.965i)11-s + (−0.348 − 0.358i)12-s + 0.989i·13-s + (−0.0463 + 0.705i)14-s + (0.127 + 0.503i)15-s + (−0.125 − 0.216i)16-s + (0.415 − 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.785 - 0.619i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.785 - 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516222 + 1.48791i\)
\(L(\frac12)\) \(\approx\) \(0.516222 + 1.48791i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.470 - 1.66i)T \)
7 \( 1 + (-2.19 - 1.47i)T \)
11 \( 1 + (0.864 - 3.20i)T \)
good5 \( 1 + (-1.00 + 0.580i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 3.56iT - 13T^{2} \)
17 \( 1 + (-1.71 + 2.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.30 - 2.48i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.67 + 4.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.04T + 29T^{2} \)
31 \( 1 + (0.547 - 0.947i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.31 + 9.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.05T + 41T^{2} \)
43 \( 1 - 6.97iT - 43T^{2} \)
47 \( 1 + (-4.23 + 2.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.75 + 1.01i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.87 - 5.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.96 + 5.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.911 - 1.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + (-8.95 - 5.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.06 + 4.65i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.21T + 83T^{2} \)
89 \( 1 + (-2.99 + 1.72i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.20T + 97T^{2} \)
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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

−11.38185000514154794737412603457, −10.53643062853466896545220259431, −9.269639200438405575849953323209, −9.015122164411706902814101405040, −7.72790178331321420745368036445, −6.61799683357192912711229007151, −5.43185759535938220960617234568, −4.92237149840466717426025051639, −3.93949912448704290032336950046, −2.23624999473812101810817641237, 0.948877743644108299564204230254, 2.27514466697640113401891937218, 3.52369833018325757427602982618, 5.13547017234936708676309292778, 5.81471817347804954321065200604, 6.93197206234735619027452915671, 7.986705421880595912816455028240, 8.739326931470665602310524565693, 10.22626096048602028454203256688, 10.94311168823133098625051821912

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