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Properties

Label 2-462-11.5-c1-0-0
Degree $2$
Conductor $462$
Sign $-0.957 - 0.288i$
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 + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 2.48i)5-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s − 2.61·10-s + (−2.54 + 2.12i)11-s − 0.999·12-s + (−1 − 3.07i)13-s + (−0.809 − 0.587i)14-s + (−2.11 + 1.53i)15-s + (0.309 − 0.951i)16-s + (−0.118 + 0.363i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.361 + 1.11i)5-s + (−0.126 + 0.388i)6-s + (−0.305 + 0.222i)7-s + (−0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s − 0.827·10-s + (−0.767 + 0.641i)11-s − 0.288·12-s + (−0.277 − 0.853i)13-s + (−0.216 − 0.157i)14-s + (−0.546 + 0.397i)15-s + (0.0772 − 0.237i)16-s + (−0.0286 + 0.0881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.957 - 0.288i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.957 - 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191263 + 1.29992i\)
\(L(\frac12)\) \(\approx\) \(0.191263 + 1.29992i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (2.54 - 2.12i)T \)
good5 \( 1 + (0.809 - 2.48i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1 + 3.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.118 - 0.363i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.927 - 0.673i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.61T + 23T^{2} \)
29 \( 1 + (6.85 - 4.97i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.572 - 1.76i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.5 + 1.08i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.35 - 3.88i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.23T + 43T^{2} \)
47 \( 1 + (-6.85 - 4.97i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.14 - 3.52i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.85 + 4.25i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (3.70 - 11.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.38 - 1.00i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.145 + 0.449i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.61 + 14.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 1.32T + 89T^{2} \)
97 \( 1 + (2.61 + 8.05i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.26296719217059978779952254791, −10.48726718057784905939704975207, −9.657131932331408062324444785817, −8.639701706820334611076111953686, −7.50585039142048161943660190477, −7.15911413790482620125378650986, −5.82818174467965514728716438335, −4.82804651143583961419060707935, −3.47357608642884438622978628211, −2.69013473891280655609065381768, 0.71989202668836843885049436713, 2.32409322166496963259124028668, 3.63586497400685841627895988775, 4.65392226947357496225787681835, 5.67734535281926251695015590647, 7.09180175340071885164077669385, 8.107070677163660253101869672168, 8.971411473790605264881928554957, 9.578298303053214884009649131967, 10.79125692049335639952082409257

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