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Properties

Label 2-462-77.62-c1-0-1
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  + (−0.587 + 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (0.946 + 1.30i)5-s + (0.809 − 0.587i)6-s + (−1.50 + 2.17i)7-s + (0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s − 1.60·10-s + (−2.29 − 2.39i)11-s + i·12-s + (1.19 + 0.865i)13-s + (−0.877 − 2.49i)14-s + (−0.497 − 1.53i)15-s + (−0.809 + 0.587i)16-s + (0.309 − 0.224i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.549 − 0.178i)3-s + (−0.154 − 0.475i)4-s + (0.423 + 0.582i)5-s + (0.330 − 0.239i)6-s + (−0.568 + 0.822i)7-s + (0.336 + 0.109i)8-s + (0.269 + 0.195i)9-s − 0.509·10-s + (−0.693 − 0.720i)11-s + 0.288i·12-s + (0.330 + 0.240i)13-s + (−0.234 − 0.667i)14-s + (−0.128 − 0.395i)15-s + (−0.202 + 0.146i)16-s + (0.0750 − 0.0545i)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: $\chi_{462} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0540087 + 0.484250i\)
\(L(\frac12)\) \(\approx\) \(0.0540087 + 0.484250i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + (1.50 - 2.17i)T \)
11 \( 1 + (2.29 + 2.39i)T \)
good5 \( 1 + (-0.946 - 1.30i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.19 - 0.865i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.309 + 0.224i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.11 - 6.49i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.63T + 23T^{2} \)
29 \( 1 + (8.63 - 2.80i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.73 - 6.51i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.778 - 2.39i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.174 - 0.537i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.15iT - 43T^{2} \)
47 \( 1 + (3.78 + 1.22i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.56 + 1.13i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.74 - 1.54i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.93 + 2.13i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 + (-1.45 + 1.05i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.29 - 13.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.135 - 0.186i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-11.4 + 8.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 7.19iT - 89T^{2} \)
97 \( 1 + (-9.25 + 12.7i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.26634660240171364928060676152, −10.45951551543704242413581098540, −9.750606172914245471273801713324, −8.678237869734950054234607822083, −7.83515137870920110477493365672, −6.62563999521472831826711314234, −5.99785039530407411453226194162, −5.31337275395451916358698647236, −3.52868796553444231473869758436, −1.98333374445396605201966741553, 0.35083097440621369256514863911, 2.06771352697317021600629372984, 3.70658980209615543324843856203, 4.72992221756183850676277496827, 5.82928617157443270688041833018, 7.05336184513527349952173408961, 7.906411876499159935206455258732, 9.285836170442902415649504192590, 9.687933687503959590440535301169, 10.70544813651809476324143114423

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