# Properties

 Label 2-1815-1.1-c3-0-178 Degree $2$ Conductor $1815$ Sign $-1$ Analytic cond. $107.088$ Root an. cond. $10.3483$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.70·2-s + 3·3-s + 5.70·4-s + 5·5-s − 11.1·6-s + 9.10·7-s + 8.50·8-s + 9·9-s − 18.5·10-s + 17.1·12-s − 20.2·13-s − 33.7·14-s + 15·15-s − 77.1·16-s + 61.8·17-s − 33.3·18-s + 8.68·19-s + 28.5·20-s + 27.3·21-s − 127.·23-s + 25.5·24-s + 25·25-s + 75.1·26-s + 27·27-s + 51.9·28-s + 85.9·29-s − 55.5·30-s + ⋯
 L(s)  = 1 − 1.30·2-s + 0.577·3-s + 0.712·4-s + 0.447·5-s − 0.755·6-s + 0.491·7-s + 0.375·8-s + 0.333·9-s − 0.585·10-s + 0.411·12-s − 0.433·13-s − 0.643·14-s + 0.258·15-s − 1.20·16-s + 0.881·17-s − 0.436·18-s + 0.104·19-s + 0.318·20-s + 0.283·21-s − 1.15·23-s + 0.217·24-s + 0.200·25-s + 0.566·26-s + 0.192·27-s + 0.350·28-s + 0.550·29-s − 0.337·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1815$$    =    $$3 \cdot 5 \cdot 11^{2}$$ Sign: $-1$ Analytic conductor: $$107.088$$ Root analytic conductor: $$10.3483$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1815} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1815,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 3T$$
5 $$1 - 5T$$
11 $$1$$
good2 $$1 + 3.70T + 8T^{2}$$
7 $$1 - 9.10T + 343T^{2}$$
13 $$1 + 20.2T + 2.19e3T^{2}$$
17 $$1 - 61.8T + 4.91e3T^{2}$$
19 $$1 - 8.68T + 6.85e3T^{2}$$
23 $$1 + 127.T + 1.21e4T^{2}$$
29 $$1 - 85.9T + 2.43e4T^{2}$$
31 $$1 + 158.T + 2.97e4T^{2}$$
37 $$1 - 138.T + 5.06e4T^{2}$$
41 $$1 + 28.2T + 6.89e4T^{2}$$
43 $$1 + 265.T + 7.95e4T^{2}$$
47 $$1 + 515.T + 1.03e5T^{2}$$
53 $$1 + 466.T + 1.48e5T^{2}$$
59 $$1 - 373.T + 2.05e5T^{2}$$
61 $$1 + 84.8T + 2.26e5T^{2}$$
67 $$1 + 88.7T + 3.00e5T^{2}$$
71 $$1 + 536.T + 3.57e5T^{2}$$
73 $$1 + 322.T + 3.89e5T^{2}$$
79 $$1 + 265.T + 4.93e5T^{2}$$
83 $$1 - 520.T + 5.71e5T^{2}$$
89 $$1 + 464.T + 7.04e5T^{2}$$
97 $$1 + 204.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$