# Properties

 Label 2-624-13.10-c3-0-16 Degree $2$ Conductor $624$ Sign $0.711 + 0.702i$ 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 + 5.19i·5-s + (−9 − 5.19i)7-s + (−4.5 + 7.79i)9-s + (−45 + 25.9i)11-s + (−32.5 + 33.7i)13-s + (13.5 − 7.79i)15-s + (58.5 − 101. i)17-s + (21 + 12.1i)19-s + 31.1i·21-s + (9 + 15.5i)23-s + 98·25-s + 27·27-s + (49.5 + 85.7i)29-s − 193. i·31-s + ⋯
 L(s)  = 1 + (−0.288 − 0.499i)3-s + 0.464i·5-s + (−0.485 − 0.280i)7-s + (−0.166 + 0.288i)9-s + (−1.23 + 0.712i)11-s + (−0.693 + 0.720i)13-s + (0.232 − 0.134i)15-s + (0.834 − 1.44i)17-s + (0.253 + 0.146i)19-s + 0.323i·21-s + (0.0815 + 0.141i)23-s + 0.784·25-s + 0.192·27-s + (0.316 + 0.548i)29-s − 1.12i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$624$$    =    $$2^{4} \cdot 3 \cdot 13$$ Sign: $0.711 + 0.702i$ Analytic conductor: $$36.8171$$ Root analytic conductor: $$6.06771$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{624} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 624,\ (\ :3/2),\ 0.711 + 0.702i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.229565595$$ $$L(\frac12)$$ $$\approx$$ $$1.229565595$$ $$L(\frac{5}{2})$$ 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 + (32.5 - 33.7i)T$$
good5 $$1 - 5.19iT - 125T^{2}$$
7 $$1 + (9 + 5.19i)T + (171.5 + 297. i)T^{2}$$
11 $$1 + (45 - 25.9i)T + (665.5 - 1.15e3i)T^{2}$$
17 $$1 + (-58.5 + 101. i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-21 - 12.1i)T + (3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-9 - 15.5i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (-49.5 - 85.7i)T + (-1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + 193. iT - 2.97e4T^{2}$$
37 $$1 + (-97.5 + 56.2i)T + (2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + (31.5 - 18.1i)T + (3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (41 - 71.0i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 - 72.7iT - 1.03e5T^{2}$$
53 $$1 + 261T + 1.48e5T^{2}$$
59 $$1 + (-684 - 394. i)T + (1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-359.5 + 622. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-609 + 351. i)T + (1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + (-405 - 233. i)T + (1.78e5 + 3.09e5i)T^{2}$$
73 $$1 - 684. iT - 3.89e5T^{2}$$
79 $$1 - 440T + 4.93e5T^{2}$$
83 $$1 + 1.19e3iT - 5.71e5T^{2}$$
89 $$1 + (-1.31e3 + 758. i)T + (3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + (1.00e3 + 578. i)T + (4.56e5 + 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$