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Properties

Label 2-462-33.2-c1-0-20
Degree $2$
Conductor $462$
Sign $-0.975 + 0.220i$
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 + (−0.740 + 1.56i)3-s + (−0.809 − 0.587i)4-s + (−0.826 + 0.268i)5-s + (1.26 + 1.18i)6-s + (−0.587 + 0.809i)7-s + (−0.809 + 0.587i)8-s + (−1.90 − 2.31i)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 + (0.191 − 1.49i)15-s + (0.309 + 0.951i)16-s + (−0.798 − 2.45i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.427 + 0.903i)3-s + (−0.404 − 0.293i)4-s + (−0.369 + 0.120i)5-s + (0.514 + 0.485i)6-s + (−0.222 + 0.305i)7-s + (−0.286 + 0.207i)8-s + (−0.634 − 0.773i)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 + (0.0495 − 0.385i)15-s + (0.0772 + 0.237i)16-s + (−0.193 − 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.975 + 0.220i$
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: \((2,\ 462,\ (\ :1/2),\ -0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0267993 - 0.240305i\)
\(L(\frac12)\) \(\approx\) \(0.0267993 - 0.240305i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.740 - 1.56i)T \)
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

−10.81436173625207673475634288468, −9.920137432128925444870874085893, −9.094930314608896565499013820155, −8.182203264054377364844773551772, −6.72143650480849475304302809974, −5.53600548893182272708053986099, −4.80236178201240199842754644083, −3.61129530345929695265321779444, −2.66148139242374104976575242989, −0.13780980491559365456860006111, 2.04475707790278975587853246167, 3.79796607235742106878527119390, 5.03900669926049093526592046313, 5.90581506893077681487577492881, 7.04430194556718756177226737094, 7.54419477359118116547434794531, 8.378211566668284703425912212793, 9.634275627596398445283058714944, 10.61577328004781593154651746263, 11.78779256573099487072862538042

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