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Properties

Label 2-462-11.4-c1-0-4
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: $\chi_{462} (169, \cdot )$
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

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

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