Properties

Label 2-462-33.2-c1-0-1
Degree $2$
Conductor $462$
Sign $-0.714 - 0.699i$
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 + (−0.777 − 1.54i)3-s + (−0.809 − 0.587i)4-s + (−1.20 + 0.391i)5-s + (1.71 − 0.261i)6-s + (0.587 − 0.809i)7-s + (0.809 − 0.587i)8-s + (−1.79 + 2.40i)9-s − 1.26i·10-s + (−3.11 + 1.14i)11-s + (−0.280 + 1.70i)12-s + (−0.142 − 0.0463i)13-s + (0.587 + 0.809i)14-s + (1.54 + 1.55i)15-s + (0.309 + 0.951i)16-s + (0.379 + 1.16i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.448 − 0.893i)3-s + (−0.404 − 0.293i)4-s + (−0.538 + 0.174i)5-s + (0.699 − 0.106i)6-s + (0.222 − 0.305i)7-s + (0.286 − 0.207i)8-s + (−0.596 + 0.802i)9-s − 0.400i·10-s + (−0.938 + 0.345i)11-s + (−0.0809 + 0.493i)12-s + (−0.0395 − 0.0128i)13-s + (0.157 + 0.216i)14-s + (0.397 + 0.402i)15-s + (0.0772 + 0.237i)16-s + (0.0920 + 0.283i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.714 - 0.699i$
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: \((2,\ 462,\ (\ :1/2),\ -0.714 - 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149182 + 0.365677i\)
\(L(\frac12)\) \(\approx\) \(0.149182 + 0.365677i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.777 + 1.54i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (3.11 - 1.14i)T \)
good5 \( 1 + (1.20 - 0.391i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.142 + 0.0463i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.379 - 1.16i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.43 - 3.34i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 7.98iT - 23T^{2} \)
29 \( 1 + (4.28 + 3.11i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.38 - 4.26i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.04 + 4.39i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.89 - 5.73i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.38iT - 43T^{2} \)
47 \( 1 + (-1.64 - 2.26i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-5.52 - 1.79i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.64 + 11.8i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.40 - 0.783i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + (-5.97 + 1.94i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.32 + 8.70i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (10.3 + 3.36i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.01 - 9.28i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 1.07iT - 89T^{2} \)
97 \( 1 + (3.06 - 9.42i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.42174231189993923310619058319, −10.55756491396193158702037851600, −9.565316455728005580882204576445, −8.183521845881117541029587740112, −7.65227090242289741213453201588, −7.04834647710546791024687446374, −5.79656772732297222597566413427, −5.09118817958321721991621824568, −3.54274116862023213643241292221, −1.66638871396655521123657591238, 0.27556078931913919979748449398, 2.60117187952079934105969533410, 3.77356996901941953877877043659, 4.82841604256426514521406578430, 5.60720890349726383477687417920, 7.12361143774558206928745988319, 8.393395220359694957061533223298, 8.936354088221671309977954416853, 10.10959761283455236297898589667, 10.63826029908086226821309571635

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