# Properties

 Label 2-624-13.9-c3-0-39 Degree $2$ Conductor $624$ Sign $-0.872 - 0.488i$ Analytic cond. $36.8171$ Root an. cond. $6.06771$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.5 − 2.59i)3-s + 7·5-s + (−5 − 8.66i)7-s + (−4.5 − 7.79i)9-s + (−11 + 19.0i)11-s + (−45.5 − 11.2i)13-s + (10.5 − 18.1i)15-s + (−18.5 − 32.0i)17-s + (15 + 25.9i)19-s − 30.0·21-s + (−81 + 140. i)23-s − 76·25-s − 27·27-s + (56.5 − 97.8i)29-s − 196·31-s + ⋯
 L(s)  = 1 + (0.288 − 0.499i)3-s + 0.626·5-s + (−0.269 − 0.467i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + (−0.970 − 0.240i)13-s + (0.180 − 0.313i)15-s + (−0.263 − 0.457i)17-s + (0.181 + 0.313i)19-s − 0.311·21-s + (−0.734 + 1.27i)23-s − 0.607·25-s − 0.192·27-s + (0.361 − 0.626i)29-s − 1.13·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$624$$    =    $$2^{4} \cdot 3 \cdot 13$$ Sign: $-0.872 - 0.488i$ Analytic conductor: $$36.8171$$ Root analytic conductor: $$6.06771$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{624} (529, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 624,\ (\ :3/2),\ -0.872 - 0.488i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.06927072486$$ $$L(\frac12)$$ $$\approx$$ $$0.06927072486$$ $$L(\frac{5}{2})$$ 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 + (-1.5 + 2.59i)T$$
13 $$1 + (45.5 + 11.2i)T$$
good5 $$1 - 7T + 125T^{2}$$
7 $$1 + (5 + 8.66i)T + (-171.5 + 297. i)T^{2}$$
11 $$1 + (11 - 19.0i)T + (-665.5 - 1.15e3i)T^{2}$$
17 $$1 + (18.5 + 32.0i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-15 - 25.9i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (81 - 140. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (-56.5 + 97.8i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + 196T + 2.97e4T^{2}$$
37 $$1 + (6.5 - 11.2i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + (142.5 - 246. i)T + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (123 + 213. i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 - 462T + 1.03e5T^{2}$$
53 $$1 + 537T + 1.48e5T^{2}$$
59 $$1 + (-288 - 498. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-317.5 - 549. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-101 + 174. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + (543 + 940. i)T + (-1.78e5 + 3.09e5i)T^{2}$$
73 $$1 + 805T + 3.89e5T^{2}$$
79 $$1 + 884T + 4.93e5T^{2}$$
83 $$1 + 518T + 5.71e5T^{2}$$
89 $$1 + (97 - 168. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + (-601 - 1.04e3i)T + (-4.56e5 + 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$