# Properties

 Label 2-700-1.1-c5-0-2 Degree $2$ Conductor $700$ Sign $1$ Analytic cond. $112.268$ Root an. cond. $10.5956$ Motivic weight $5$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2·3-s − 49·7-s − 239·9-s − 720·11-s − 572·13-s − 1.25e3·17-s − 94·19-s − 98·21-s − 96·23-s − 964·27-s − 4.37e3·29-s − 6.24e3·31-s − 1.44e3·33-s + 1.07e4·37-s − 1.14e3·39-s + 1.20e4·41-s + 9.16e3·43-s + 2.58e4·47-s + 2.40e3·49-s − 2.50e3·51-s − 1.01e3·53-s − 188·57-s + 1.24e3·59-s + 7.59e3·61-s + 1.17e4·63-s − 4.11e4·67-s − 192·69-s + ⋯
 L(s)  = 1 + 0.128·3-s − 0.377·7-s − 0.983·9-s − 1.79·11-s − 0.938·13-s − 1.05·17-s − 0.0597·19-s − 0.0484·21-s − 0.0378·23-s − 0.254·27-s − 0.965·29-s − 1.16·31-s − 0.230·33-s + 1.29·37-s − 0.120·39-s + 1.11·41-s + 0.755·43-s + 1.70·47-s + 1/7·49-s − 0.135·51-s − 0.0495·53-s − 0.00766·57-s + 0.0464·59-s + 0.261·61-s + 0.371·63-s − 1.11·67-s − 0.00485·69-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$700$$    =    $$2^{2} \cdot 5^{2} \cdot 7$$ Sign: $1$ Analytic conductor: $$112.268$$ Root analytic conductor: $$10.5956$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 700,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.5835743757$$ $$L(\frac12)$$ $$\approx$$ $$0.5835743757$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
7 $$1 + p^{2} T$$
good3 $$1 - 2 T + p^{5} T^{2}$$
11 $$1 + 720 T + p^{5} T^{2}$$
13 $$1 + 44 p T + p^{5} T^{2}$$
17 $$1 + 1254 T + p^{5} T^{2}$$
19 $$1 + 94 T + p^{5} T^{2}$$
23 $$1 + 96 T + p^{5} T^{2}$$
29 $$1 + 4374 T + p^{5} T^{2}$$
31 $$1 + 6244 T + p^{5} T^{2}$$
37 $$1 - 10798 T + p^{5} T^{2}$$
41 $$1 - 12006 T + p^{5} T^{2}$$
43 $$1 - 9160 T + p^{5} T^{2}$$
47 $$1 - 25836 T + p^{5} T^{2}$$
53 $$1 + 1014 T + p^{5} T^{2}$$
59 $$1 - 1242 T + p^{5} T^{2}$$
61 $$1 - 7592 T + p^{5} T^{2}$$
67 $$1 + 41132 T + p^{5} T^{2}$$
71 $$1 + 37632 T + p^{5} T^{2}$$
73 $$1 - 13438 T + p^{5} T^{2}$$
79 $$1 - 6248 T + p^{5} T^{2}$$
83 $$1 - 25254 T + p^{5} T^{2}$$
89 $$1 + 45126 T + p^{5} T^{2}$$
97 $$1 + 107222 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$