# Properties

 Label 2-624-13.9-c3-0-24 Degree $2$ Conductor $624$ Sign $0.477 + 0.878i$ Analytic cond. $36.8171$ Root an. cond. $6.06771$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (1.5 − 2.59i)3-s − 9·5-s + (1 + 1.73i)7-s + (−4.5 − 7.79i)9-s + (15 − 25.9i)11-s + (32.5 + 33.7i)13-s + (−13.5 + 23.3i)15-s + (55.5 + 96.1i)17-s + (−23 − 39.8i)19-s + 6·21-s + (−3 + 5.19i)23-s − 44·25-s − 27·27-s + (52.5 − 90.9i)29-s + 100·31-s + ⋯
 L(s)  = 1 + (0.288 − 0.499i)3-s − 0.804·5-s + (0.0539 + 0.0935i)7-s + (−0.166 − 0.288i)9-s + (0.411 − 0.712i)11-s + (0.693 + 0.720i)13-s + (−0.232 + 0.402i)15-s + (0.791 + 1.37i)17-s + (−0.277 − 0.481i)19-s + 0.0623·21-s + (−0.0271 + 0.0471i)23-s − 0.351·25-s − 0.192·27-s + (0.336 − 0.582i)29-s + 0.579·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$624$$    =    $$2^{4} \cdot 3 \cdot 13$$ Sign: $0.477 + 0.878i$ 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: $$(2,\ 624,\ (\ :3/2),\ 0.477 + 0.878i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.855679336$$ $$L(\frac12)$$ $$\approx$$ $$1.855679336$$ $$L(\frac{5}{2})$$ 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 + 9T + 125T^{2}$$
7 $$1 + (-1 - 1.73i)T + (-171.5 + 297. i)T^{2}$$
11 $$1 + (-15 + 25.9i)T + (-665.5 - 1.15e3i)T^{2}$$
17 $$1 + (-55.5 - 96.1i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (23 + 39.8i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (3 - 5.19i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (-52.5 + 90.9i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 - 100T + 2.97e4T^{2}$$
37 $$1 + (8.5 - 14.7i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + (-115.5 + 200. i)T + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (257 + 445. i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 - 162T + 1.03e5T^{2}$$
53 $$1 - 639T + 1.48e5T^{2}$$
59 $$1 + (-300 - 519. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (116.5 + 201. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-463 + 801. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + (465 + 805. i)T + (-1.78e5 + 3.09e5i)T^{2}$$
73 $$1 + 253T + 3.89e5T^{2}$$
79 $$1 - 1.32e3T + 4.93e5T^{2}$$
83 $$1 + 810T + 5.71e5T^{2}$$
89 $$1 + (249 - 431. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + (679 + 1.17e3i)T + (-4.56e5 + 7.90e5i)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

−10.12228070641966794824395147730, −8.791863087322136956645716701519, −8.440931726225022840282961220656, −7.47619997275104978454817976580, −6.51612800706993630936987751885, −5.68448828610927526421410735388, −4.11845221358513004580139850647, −3.49818213348786110324906019496, −1.98306304860140211172678548734, −0.66056249898241450156180098984, 1.01552289057209752215427353112, 2.77018826177645645544112788036, 3.76592902164759883703600280283, 4.62523374874596951295113004997, 5.69286067065807715021971876738, 6.96947746243986636099100937777, 7.82734748335094717092900164805, 8.521507810361313081651397849990, 9.608747201184535224727786924457, 10.21759280088295748589653924631