# Properties

 Degree $2$ Conductor $5610$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s + 5-s + 6-s + 3·7-s − 8-s + 9-s − 10-s + 11-s − 12-s − 13-s − 3·14-s − 15-s + 16-s − 17-s − 18-s − 5·19-s + 20-s − 3·21-s − 22-s + 5·23-s + 24-s + 25-s + 26-s − 27-s + 3·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.654·21-s − 0.213·22-s + 1.04·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.566·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5610$$    =    $$2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{5610} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 5610,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + T$$
5 $$1 - T$$
11 $$1 - T$$
17 $$1 + T$$
good7 $$1 - 3 T + p T^{2}$$
13 $$1 + T + p T^{2}$$
19 $$1 + 5 T + p T^{2}$$
23 $$1 - 5 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 - T + p T^{2}$$
37 $$1 + 3 T + p T^{2}$$
41 $$1 - 2 T + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 + 6 T + p T^{2}$$
53 $$1 + 10 T + p T^{2}$$
59 $$1 + 12 T + p T^{2}$$
61 $$1 - 11 T + p T^{2}$$
67 $$1 + 7 T + p T^{2}$$
71 $$1 + 10 T + p T^{2}$$
73 $$1 + p T^{2}$$
79 $$1 + 2 T + p T^{2}$$
83 $$1 - 7 T + p T^{2}$$
89 $$1 + 10 T + p T^{2}$$
97 $$1 + 7 T + p 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

−17.74013586619440, −17.28911356968163, −16.88432840932579, −16.32816772742226, −15.42776542166839, −14.84392954781967, −14.52451316045866, −13.54651530493959, −12.92979098706956, −12.25184732800681, −11.51651354022915, −11.02582084141891, −10.61929626410382, −9.780982404796473, −9.177365629270733, −8.472524341814639, −7.880581496281212, −7.032121244958564, −6.508557745470195, −5.667974872399758, −4.948094213591206, −4.296627976645541, −3.077017588293166, −1.933174153077318, −1.412520403962990, 0, 1.412520403962990, 1.933174153077318, 3.077017588293166, 4.296627976645541, 4.948094213591206, 5.667974872399758, 6.508557745470195, 7.032121244958564, 7.880581496281212, 8.472524341814639, 9.177365629270733, 9.780982404796473, 10.61929626410382, 11.02582084141891, 11.51651354022915, 12.25184732800681, 12.92979098706956, 13.54651530493959, 14.52451316045866, 14.84392954781967, 15.42776542166839, 16.32816772742226, 16.88432840932579, 17.28911356968163, 17.74013586619440