# Properties

 Label 2-624-1.1-c3-0-8 Degree $2$ Conductor $624$ Sign $1$ Analytic cond. $36.8171$ Root an. cond. $6.06771$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 3·3-s − 20·5-s + 32·7-s + 9·9-s − 50·11-s − 13·13-s − 60·15-s − 30·17-s + 120·19-s + 96·21-s + 20·23-s + 275·25-s + 27·27-s + 82·29-s + 44·31-s − 150·33-s − 640·35-s − 306·37-s − 39·39-s + 108·41-s + 356·43-s − 180·45-s + 178·47-s + 681·49-s − 90·51-s + 198·53-s + 1.00e3·55-s + ⋯
 L(s)  = 1 + 0.577·3-s − 1.78·5-s + 1.72·7-s + 1/3·9-s − 1.37·11-s − 0.277·13-s − 1.03·15-s − 0.428·17-s + 1.44·19-s + 0.997·21-s + 0.181·23-s + 11/5·25-s + 0.192·27-s + 0.525·29-s + 0.254·31-s − 0.791·33-s − 3.09·35-s − 1.35·37-s − 0.160·39-s + 0.411·41-s + 1.26·43-s − 0.596·45-s + 0.552·47-s + 1.98·49-s − 0.247·51-s + 0.513·53-s + 2.45·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$624$$    =    $$2^{4} \cdot 3 \cdot 13$$ Sign: $1$ Analytic conductor: $$36.8171$$ Root analytic conductor: $$6.06771$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 624,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.929615039$$ $$L(\frac12)$$ $$\approx$$ $$1.929615039$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - p T$$
13 $$1 + p T$$
good5 $$1 + 4 p T + p^{3} T^{2}$$
7 $$1 - 32 T + p^{3} T^{2}$$
11 $$1 + 50 T + p^{3} T^{2}$$
17 $$1 + 30 T + p^{3} T^{2}$$
19 $$1 - 120 T + p^{3} T^{2}$$
23 $$1 - 20 T + p^{3} T^{2}$$
29 $$1 - 82 T + p^{3} T^{2}$$
31 $$1 - 44 T + p^{3} T^{2}$$
37 $$1 + 306 T + p^{3} T^{2}$$
41 $$1 - 108 T + p^{3} T^{2}$$
43 $$1 - 356 T + p^{3} T^{2}$$
47 $$1 - 178 T + p^{3} T^{2}$$
53 $$1 - 198 T + p^{3} T^{2}$$
59 $$1 + 94 T + p^{3} T^{2}$$
61 $$1 + 62 T + p^{3} T^{2}$$
67 $$1 - 140 T + p^{3} T^{2}$$
71 $$1 - 778 T + p^{3} T^{2}$$
73 $$1 - 62 T + p^{3} T^{2}$$
79 $$1 - 1096 T + p^{3} T^{2}$$
83 $$1 - 462 T + p^{3} T^{2}$$
89 $$1 - 1224 T + p^{3} T^{2}$$
97 $$1 - 614 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$