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Properties

Label 2-462-11.3-c1-0-9
Degree $2$
Conductor $462$
Sign $0.254 + 0.966i$
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  + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (−2.26 − 1.64i)5-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 2.80·10-s + (3.31 + 0.108i)11-s + 12-s + (5.56 − 4.03i)13-s + (−0.309 − 0.951i)14-s + (0.866 − 2.66i)15-s + (−0.809 − 0.587i)16-s + (−1.68 − 1.22i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 − 0.475i)4-s + (−1.01 − 0.736i)5-s + (0.330 + 0.239i)6-s + (0.116 − 0.359i)7-s + (−0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s − 0.886·10-s + (0.999 + 0.0326i)11-s + 0.288·12-s + (1.54 − 1.12i)13-s + (−0.0825 − 0.254i)14-s + (0.223 − 0.688i)15-s + (−0.202 − 0.146i)16-s + (−0.408 − 0.296i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.254 + 0.966i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.254 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39954 - 1.07848i\)
\(L(\frac12)\) \(\approx\) \(1.39954 - 1.07848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-3.31 - 0.108i)T \)
good5 \( 1 + (2.26 + 1.64i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-5.56 + 4.03i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.68 + 1.22i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.10 + 6.48i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.57T + 23T^{2} \)
29 \( 1 + (2.16 - 6.66i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.70 + 2.69i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.36 - 10.3i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.0475 + 0.146i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.20T + 43T^{2} \)
47 \( 1 + (-1.91 - 5.89i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.49 + 2.53i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.89 - 8.92i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.96 - 4.33i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.66T + 67T^{2} \)
71 \( 1 + (-4.26 - 3.09i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.23 - 3.80i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.97 - 5.79i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.00 + 5.09i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.48T + 89T^{2} \)
97 \( 1 + (-4.74 + 3.44i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.13297822548943259178471517385, −10.17358407378461004393214400463, −8.890816390146523247146747288256, −8.470227316924513779808416158312, −7.15422199354851453741216321398, −5.95354204073492570854538631306, −4.68291846048525069387313727089, −4.07990367737332918872323447510, −3.10025752310759595141525478417, −0.993385182787748045894151527706, 1.94495867303911310367797456016, 3.69773011825359036172571449269, 4.04137956229547846298880715261, 5.98500493458990943331358836495, 6.49124162570017169128402642274, 7.47532091390443272409658667857, 8.341722225774052673170104346884, 9.081491420422471351006742869133, 10.68531399374811749211414486702, 11.55887742478011030269830412678

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