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Properties

Label 2-462-21.5-c1-0-0
Degree $2$
Conductor $462$
Sign $-0.999 + 0.0394i$
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  + (−0.866 − 0.5i)2-s + (−0.530 + 1.64i)3-s + (0.499 + 0.866i)4-s + (−0.417 + 0.723i)5-s + (1.28 − 1.16i)6-s + (−1.42 + 2.22i)7-s − 0.999i·8-s + (−2.43 − 1.75i)9-s + (0.723 − 0.417i)10-s + (0.866 − 0.5i)11-s + (−1.69 + 0.364i)12-s + 3.86i·13-s + (2.34 − 1.21i)14-s + (−0.971 − 1.07i)15-s + (−0.5 + 0.866i)16-s + (−1.06 − 1.84i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.306 + 0.951i)3-s + (0.249 + 0.433i)4-s + (−0.186 + 0.323i)5-s + (0.524 − 0.474i)6-s + (−0.539 + 0.842i)7-s − 0.353i·8-s + (−0.812 − 0.583i)9-s + (0.228 − 0.132i)10-s + (0.261 − 0.150i)11-s + (−0.488 + 0.105i)12-s + 1.07i·13-s + (0.627 − 0.325i)14-s + (−0.250 − 0.277i)15-s + (−0.125 + 0.216i)16-s + (−0.258 − 0.447i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.999 + 0.0394i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.999 + 0.0394i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00646359 - 0.327284i\)
\(L(\frac12)\) \(\approx\) \(0.00646359 - 0.327284i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.530 - 1.64i)T \)
7 \( 1 + (1.42 - 2.22i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (0.417 - 0.723i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 3.86iT - 13T^{2} \)
17 \( 1 + (1.06 + 1.84i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.35 + 1.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.08 + 3.51i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.25iT - 29T^{2} \)
31 \( 1 + (4.85 - 2.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.76 + 3.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.835T + 41T^{2} \)
43 \( 1 + 9.85T + 43T^{2} \)
47 \( 1 + (5.84 - 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.26 - 4.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.60 - 6.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.31 + 1.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.35 - 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.66iT - 71T^{2} \)
73 \( 1 + (-10.7 + 6.21i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.81 + 4.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.41T + 83T^{2} \)
89 \( 1 + (-0.355 + 0.616i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.84iT - 97T^{2} \)
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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

−11.43950157356076649958447344095, −10.59042975497601568443180017686, −9.576733817135871214766729487437, −9.152867768780142686742361056550, −8.221434469863960684967961221109, −6.76419421673771820625916586260, −6.02287768627388550912606868518, −4.61871904820553306198168997414, −3.55173233677854250781434937813, −2.35390646510984031941839632665, 0.24575454677872382642398416254, 1.73713505650390112629153251776, 3.51449189568595587795721601399, 5.10635668805356140129659724180, 6.23249491718760213773008786511, 6.90916957897611273582593516915, 7.924054643934769034955361847630, 8.428848939438300494580605376365, 9.764598512818716795367233814448, 10.53436868936869637263930796054

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