# Properties

 Label 2-930-1.1-c1-0-10 Degree $2$ Conductor $930$ Sign $1$ Analytic cond. $7.42608$ Root an. cond. $2.72508$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s + 5-s − 6-s + 4·7-s − 8-s + 9-s − 10-s + 2·11-s + 12-s + 2·13-s − 4·14-s + 15-s + 16-s − 18-s + 20-s + 4·21-s − 2·22-s − 6·23-s − 24-s + 25-s − 2·26-s + 27-s + 4·28-s − 30-s + 31-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.223·20-s + 0.872·21-s − 0.426·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s − 0.182·30-s + 0.179·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$930$$    =    $$2 \cdot 3 \cdot 5 \cdot 31$$ Sign: $1$ Analytic conductor: $$7.42608$$ Root analytic conductor: $$2.72508$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 930,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.870762571$$ $$L(\frac12)$$ $$\approx$$ $$1.870762571$$ $$L(\frac{3}{2})$$ 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$$
31 $$1 - T$$
good7 $$1 - 4 T + p T^{2}$$
11 $$1 - 2 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + 6 T + p T^{2}$$
29 $$1 + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 + 10 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 - 4 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + 16 T + p T^{2}$$
73 $$1 - 4 T + p T^{2}$$
79 $$1 - 4 T + p T^{2}$$
83 $$1 - 8 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 - 14 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

−10.04032737559915698815557889566, −9.055213210068326570869700775965, −8.440939755758636367305738722374, −7.83432923668169888093880069027, −6.85665676541103796632916894690, −5.84878626235490167704110394946, −4.74300402264492794222699242229, −3.63014594234784318412782833797, −2.15830058196252375078020040957, −1.38666625536959637751487418742, 1.38666625536959637751487418742, 2.15830058196252375078020040957, 3.63014594234784318412782833797, 4.74300402264492794222699242229, 5.84878626235490167704110394946, 6.85665676541103796632916894690, 7.83432923668169888093880069027, 8.440939755758636367305738722374, 9.055213210068326570869700775965, 10.04032737559915698815557889566