# Properties

 Label 2-462-1.1-c1-0-5 Degree $2$ Conductor $462$ Sign $-1$ Analytic cond. $3.68908$ Root an. cond. $1.92070$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s − 2·5-s + 6-s + 7-s − 8-s + 9-s + 2·10-s + 11-s − 12-s + 2·13-s − 14-s + 2·15-s + 16-s − 6·17-s − 18-s − 4·19-s − 2·20-s − 21-s − 22-s − 4·23-s + 24-s − 25-s − 2·26-s − 27-s + 28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$462$$    =    $$2 \cdot 3 \cdot 7 \cdot 11$$ Sign: $-1$ Analytic conductor: $$3.68908$$ Root analytic conductor: $$1.92070$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 462,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + T$$
7 $$1 - T$$
11 $$1 - T$$
good5 $$1 + 2 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 - 2 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 + 14 T + p T^{2}$$
59 $$1 - 12 T + p T^{2}$$
61 $$1 + 14 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 - 12 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 + 14 T + p 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

−10.87433876808485933510030987853, −9.736475583363369767193155110475, −8.635812633514396852158410641488, −8.036862420200993690310137538450, −6.91394332771191655575545668718, −6.17569848665337782312463950999, −4.72832473817225769292057457440, −3.72716170584453900058590617219, −1.87596574858564202101556695192, 0, 1.87596574858564202101556695192, 3.72716170584453900058590617219, 4.72832473817225769292057457440, 6.17569848665337782312463950999, 6.91394332771191655575545668718, 8.036862420200993690310137538450, 8.635812633514396852158410641488, 9.736475583363369767193155110475, 10.87433876808485933510030987853