# Properties

 Label 2-1875-1.1-c3-0-176 Degree $2$ Conductor $1875$ Sign $-1$ Analytic cond. $110.628$ Root an. cond. $10.5180$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.955·2-s + 3·3-s − 7.08·4-s − 2.86·6-s + 12.4·7-s + 14.4·8-s + 9·9-s − 44.7·11-s − 21.2·12-s − 7.13·13-s − 11.9·14-s + 42.9·16-s + 26.5·17-s − 8.59·18-s − 46.3·19-s + 37.4·21-s + 42.7·22-s + 145.·23-s + 43.2·24-s + 6.81·26-s + 27·27-s − 88.4·28-s − 213.·29-s − 148.·31-s − 156.·32-s − 134.·33-s − 25.3·34-s + ⋯
 L(s)  = 1 − 0.337·2-s + 0.577·3-s − 0.885·4-s − 0.194·6-s + 0.674·7-s + 0.636·8-s + 0.333·9-s − 1.22·11-s − 0.511·12-s − 0.152·13-s − 0.227·14-s + 0.670·16-s + 0.378·17-s − 0.112·18-s − 0.560·19-s + 0.389·21-s + 0.414·22-s + 1.31·23-s + 0.367·24-s + 0.0514·26-s + 0.192·27-s − 0.597·28-s − 1.36·29-s − 0.857·31-s − 0.863·32-s − 0.707·33-s − 0.127·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1875$$    =    $$3 \cdot 5^{4}$$ Sign: $-1$ Analytic conductor: $$110.628$$ Root analytic conductor: $$10.5180$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1875} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1875,\ (\ :3/2),\ -1)$$

## 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$$
good2 $$1 + 0.955T + 8T^{2}$$
7 $$1 - 12.4T + 343T^{2}$$
11 $$1 + 44.7T + 1.33e3T^{2}$$
13 $$1 + 7.13T + 2.19e3T^{2}$$
17 $$1 - 26.5T + 4.91e3T^{2}$$
19 $$1 + 46.3T + 6.85e3T^{2}$$
23 $$1 - 145.T + 1.21e4T^{2}$$
29 $$1 + 213.T + 2.43e4T^{2}$$
31 $$1 + 148.T + 2.97e4T^{2}$$
37 $$1 - 402.T + 5.06e4T^{2}$$
41 $$1 - 233.T + 6.89e4T^{2}$$
43 $$1 - 87.5T + 7.95e4T^{2}$$
47 $$1 + 75.9T + 1.03e5T^{2}$$
53 $$1 - 167.T + 1.48e5T^{2}$$
59 $$1 + 595.T + 2.05e5T^{2}$$
61 $$1 + 865.T + 2.26e5T^{2}$$
67 $$1 + 406.T + 3.00e5T^{2}$$
71 $$1 - 526.T + 3.57e5T^{2}$$
73 $$1 - 1.00e3T + 3.89e5T^{2}$$
79 $$1 + 133.T + 4.93e5T^{2}$$
83 $$1 - 516.T + 5.71e5T^{2}$$
89 $$1 + 610.T + 7.04e5T^{2}$$
97 $$1 + 1.63e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$