# Properties

 Degree $2$ Conductor $400$ Sign $1$ Motivic weight $5$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 25.5·3-s + 131.·7-s + 408.·9-s − 290.·11-s − 68.3·13-s + 310.·17-s + 2.13e3·19-s + 3.34e3·21-s + 873.·23-s + 4.22e3·27-s − 2.58e3·29-s + 9.08e3·31-s − 7.40e3·33-s − 3.99e3·37-s − 1.74e3·39-s + 1.69e4·41-s − 1.80e4·43-s + 2.48e4·47-s + 366.·49-s + 7.92e3·51-s + 7.65e3·53-s + 5.44e4·57-s + 9.23e3·59-s + 3.32e3·61-s + 5.35e4·63-s − 3.23e4·67-s + 2.22e4·69-s + ⋯
 L(s)  = 1 + 1.63·3-s + 1.01·7-s + 1.68·9-s − 0.722·11-s − 0.112·13-s + 0.260·17-s + 1.35·19-s + 1.65·21-s + 0.344·23-s + 1.11·27-s − 0.569·29-s + 1.69·31-s − 1.18·33-s − 0.479·37-s − 0.183·39-s + 1.57·41-s − 1.48·43-s + 1.64·47-s + 0.0218·49-s + 0.426·51-s + 0.374·53-s + 2.21·57-s + 0.345·59-s + 0.114·61-s + 1.69·63-s − 0.880·67-s + 0.563·69-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$400$$    =    $$2^{4} \cdot 5^{2}$$ Sign: $1$ Motivic weight: $$5$$ Character: $\chi_{400} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 400,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$4.675817421$$ $$L(\frac12)$$ $$\approx$$ $$4.675817421$$ $$L(\frac{7}{2})$$ 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$$
good3 $$1 - 25.5T + 243T^{2}$$
7 $$1 - 131.T + 1.68e4T^{2}$$
11 $$1 + 290.T + 1.61e5T^{2}$$
13 $$1 + 68.3T + 3.71e5T^{2}$$
17 $$1 - 310.T + 1.41e6T^{2}$$
19 $$1 - 2.13e3T + 2.47e6T^{2}$$
23 $$1 - 873.T + 6.43e6T^{2}$$
29 $$1 + 2.58e3T + 2.05e7T^{2}$$
31 $$1 - 9.08e3T + 2.86e7T^{2}$$
37 $$1 + 3.99e3T + 6.93e7T^{2}$$
41 $$1 - 1.69e4T + 1.15e8T^{2}$$
43 $$1 + 1.80e4T + 1.47e8T^{2}$$
47 $$1 - 2.48e4T + 2.29e8T^{2}$$
53 $$1 - 7.65e3T + 4.18e8T^{2}$$
59 $$1 - 9.23e3T + 7.14e8T^{2}$$
61 $$1 - 3.32e3T + 8.44e8T^{2}$$
67 $$1 + 3.23e4T + 1.35e9T^{2}$$
71 $$1 - 3.58e4T + 1.80e9T^{2}$$
73 $$1 - 2.65e4T + 2.07e9T^{2}$$
79 $$1 + 7.17e4T + 3.07e9T^{2}$$
83 $$1 - 3.96e4T + 3.93e9T^{2}$$
89 $$1 + 1.17e5T + 5.58e9T^{2}$$
97 $$1 + 2.18e4T + 8.58e9T^{2}$$
show less
$$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

−10.21761518961975166188871211990, −9.427370150805717267392975804051, −8.475789529566974499214828139504, −7.86731576539804107883479821291, −7.17305070309182479320245476207, −5.46096694243240174390212588375, −4.41076396743122626779521326921, −3.20797351358336802516694669401, −2.33922417839937057309173227973, −1.15923660263639035114368770645, 1.15923660263639035114368770645, 2.33922417839937057309173227973, 3.20797351358336802516694669401, 4.41076396743122626779521326921, 5.46096694243240174390212588375, 7.17305070309182479320245476207, 7.86731576539804107883479821291, 8.475789529566974499214828139504, 9.427370150805717267392975804051, 10.21761518961975166188871211990