# Properties

 Label 2-4830-1.1-c1-0-13 Degree $2$ Conductor $4830$ Sign $1$ Analytic cond. $38.5677$ Root an. cond. $6.21029$ 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 + 7-s − 8-s + 9-s + 10-s + 12-s − 4·13-s − 14-s − 15-s + 16-s − 18-s + 2·19-s − 20-s + 21-s + 23-s − 24-s + 25-s + 4·26-s + 27-s + 28-s + 6·29-s + 30-s + 2·31-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 1.10·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.218·21-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.182·30-s + 0.359·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4830$$    =    $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 23$$ Sign: $1$ Analytic conductor: $$38.5677$$ Root analytic conductor: $$6.21029$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{4830} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 4830,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.510780206$$ $$L(\frac12)$$ $$\approx$$ $$1.510780206$$ $$L(\frac{3}{2})$$ 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$$
7 $$1 - T$$
23 $$1 - T$$
good11 $$1 + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 - 2 T + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 - 2 T + p T^{2}$$
37 $$1 + 10 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 - 12 T + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 14 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 - 6 T + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 - 8 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

−8.289978710462499911103408641219, −7.62967070335659537431995715532, −7.14395266017522711060430990120, −6.37028369552767253034796372345, −5.22507322117727091285237140059, −4.60107108263936642886562055588, −3.55624941343280070308943386477, −2.75265441640306274816311731263, −1.91174831431498191295734688911, −0.72894115116713518555393979924, 0.72894115116713518555393979924, 1.91174831431498191295734688911, 2.75265441640306274816311731263, 3.55624941343280070308943386477, 4.60107108263936642886562055588, 5.22507322117727091285237140059, 6.37028369552767253034796372345, 7.14395266017522711060430990120, 7.62967070335659537431995715532, 8.289978710462499911103408641219