Properties

Label 2-462-33.8-c1-0-6
Degree $2$
Conductor $462$
Sign $0.344 - 0.938i$
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 + (1.61 − 0.622i)3-s + (0.309 − 0.951i)4-s + (−1.77 + 2.44i)5-s + (−0.941 + 1.45i)6-s + (−0.951 − 0.309i)7-s + (0.309 + 0.951i)8-s + (2.22 − 2.01i)9-s − 3.02i·10-s + (2.94 + 1.51i)11-s + (−0.0930 − 1.72i)12-s + (1.46 + 2.01i)13-s + (0.951 − 0.309i)14-s + (−1.34 + 5.05i)15-s + (−0.809 − 0.587i)16-s + (2.63 + 1.91i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.933 − 0.359i)3-s + (0.154 − 0.475i)4-s + (−0.794 + 1.09i)5-s + (−0.384 + 0.593i)6-s + (−0.359 − 0.116i)7-s + (0.109 + 0.336i)8-s + (0.741 − 0.671i)9-s − 0.955i·10-s + (0.889 + 0.457i)11-s + (−0.0268 − 0.499i)12-s + (0.406 + 0.559i)13-s + (0.254 − 0.0825i)14-s + (−0.347 + 1.30i)15-s + (−0.202 − 0.146i)16-s + (0.640 + 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.344 - 0.938i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.344 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04408 + 0.728639i\)
\(L(\frac12)\) \(\approx\) \(1.04408 + 0.728639i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 0.622i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-2.94 - 1.51i)T \)
good5 \( 1 + (1.77 - 2.44i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.46 - 2.01i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.63 - 1.91i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.06 - 0.671i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.69iT - 23T^{2} \)
29 \( 1 + (2.83 - 8.71i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.14 + 3.73i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.36 + 4.21i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.00 + 9.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.49iT - 43T^{2} \)
47 \( 1 + (-9.50 + 3.08i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.92 + 8.15i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.10 + 1.33i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.82 - 2.51i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + (6.23 - 8.57i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.37 + 1.74i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.33 + 1.83i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.58 + 5.51i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 11.7iT - 89T^{2} \)
97 \( 1 + (-10.8 + 7.87i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.13614903376590170150119351665, −10.11344523330064521233162465369, −9.312164949449737581066146917673, −8.459921550842876879044145537593, −7.36933944574107959175898999582, −7.06576101094258605889482919321, −6.04620461967057152314612127539, −4.03717036282022206316087304116, −3.28005776784754513979335721991, −1.68257812746593136614925838897, 0.953720701274878392070113158461, 2.75427289512853275067686047280, 3.86809433294082325453181863354, 4.68163818626694219423475133536, 6.35366079354053361308216482842, 7.74182176808444088953642316055, 8.395426312866232479632699641845, 8.952059581768980149947841570388, 9.797391336994483977826563377229, 10.71541103839368952405112033021

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