# Properties

 Label 2-29645-1.1-c1-0-13 Degree $2$ Conductor $29645$ Sign $1$ Analytic cond. $236.716$ Root an. cond. $15.3855$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $2$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 4-s − 5-s + 3·8-s − 3·9-s + 10-s − 4·13-s − 16-s + 3·18-s + 4·19-s + 20-s − 6·23-s + 25-s + 4·26-s − 6·31-s − 5·32-s + 3·36-s + 2·37-s − 4·38-s − 3·40-s − 2·41-s − 12·43-s + 3·45-s + 6·46-s − 8·47-s − 50-s + 4·52-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s − 9-s + 0.316·10-s − 1.10·13-s − 1/4·16-s + 0.707·18-s + 0.917·19-s + 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.784·26-s − 1.07·31-s − 0.883·32-s + 1/2·36-s + 0.328·37-s − 0.648·38-s − 0.474·40-s − 0.312·41-s − 1.82·43-s + 0.447·45-s + 0.884·46-s − 1.16·47-s − 0.141·50-s + 0.554·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$29645$$    =    $$5 \cdot 7^{2} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$236.716$$ Root analytic conductor: $$15.3855$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{29645} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(2,\ 29645,\ (\ :1/2),\ 1)$$

## 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$$F_p(T)$
bad5 $$1 + T$$
7 $$1$$
11 $$1$$
good2 $$1 + T + p T^{2}$$
3 $$1 + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 + 6 T + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 + 6 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 + 2 T + p T^{2}$$
43 $$1 + 12 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 + 14 T + p T^{2}$$
59 $$1 + 6 T + p T^{2}$$
61 $$1 - 14 T + p T^{2}$$
67 $$1 + 10 T + p T^{2}$$
71 $$1 - 12 T + p T^{2}$$
73 $$1 - 4 T + p T^{2}$$
79 $$1 + 4 T + p T^{2}$$
83 $$1 + 4 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 + 14 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$