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Properties

Label 2-462-33.2-c1-0-2
Degree $2$
Conductor $462$
Sign $-0.711 - 0.702i$
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  + (−0.309 + 0.951i)2-s + (−1.71 + 0.220i)3-s + (−0.809 − 0.587i)4-s + (0.826 − 0.268i)5-s + (0.320 − 1.70i)6-s + (−0.587 + 0.809i)7-s + (0.809 − 0.587i)8-s + (2.90 − 0.758i)9-s + 0.868i·10-s + (2.10 + 2.56i)11-s + (1.51 + 0.831i)12-s + (−2.27 − 0.737i)13-s + (−0.587 − 0.809i)14-s + (−1.35 + 0.643i)15-s + (0.309 + 0.951i)16-s + (0.798 + 2.45i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.991 + 0.127i)3-s + (−0.404 − 0.293i)4-s + (0.369 − 0.120i)5-s + (0.130 − 0.694i)6-s + (−0.222 + 0.305i)7-s + (0.286 − 0.207i)8-s + (0.967 − 0.252i)9-s + 0.274i·10-s + (0.633 + 0.773i)11-s + (0.438 + 0.239i)12-s + (−0.629 − 0.204i)13-s + (−0.157 − 0.216i)14-s + (−0.351 + 0.166i)15-s + (0.0772 + 0.237i)16-s + (0.193 + 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.711 - 0.702i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270823 + 0.659975i\)
\(L(\frac12)\) \(\approx\) \(0.270823 + 0.659975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (1.71 - 0.220i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (-2.10 - 2.56i)T \)
good5 \( 1 + (-0.826 + 0.268i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (2.27 + 0.737i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.798 - 2.45i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.826 + 1.13i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 7.05iT - 23T^{2} \)
29 \( 1 + (-2.21 - 1.60i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.56 - 7.90i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.66 + 5.56i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.38 + 1.00i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.09iT - 43T^{2} \)
47 \( 1 + (4.32 + 5.94i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-11.0 - 3.58i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.55 - 6.27i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-10.7 + 3.48i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 3.25T + 67T^{2} \)
71 \( 1 + (1.00 - 0.326i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.20 - 11.2i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-14.0 - 4.56i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.40 - 7.41i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 4.36iT - 89T^{2} \)
97 \( 1 + (-3.57 + 10.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.38132413398093508751817430195, −10.25532389689158972517999448333, −9.678604718352852774551862141582, −8.814604498381447257515063388462, −7.40408500823910197974814081526, −6.78655058989648209566638437363, −5.69912286127358102878124752149, −5.08805383546362032724712046719, −3.82459474417802233424275371737, −1.60987312794000636728454492040, 0.55647035005319194195357700050, 2.20437978230226759809803458798, 3.80551011349409121936355858728, 4.87562194582561329305496305186, 6.05486887076627243605511737507, 6.85180008131401093290185881343, 8.016763495019889678039145482096, 9.214125930602577560881966094834, 10.09199436955428881022906985818, 10.65114090349630975466530430462

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