# Properties

 Label 2-1875-1.1-c3-0-132 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 − 2.33·2-s + 3·3-s − 2.52·4-s − 7.01·6-s − 32.9·7-s + 24.6·8-s + 9·9-s + 49.5·11-s − 7.57·12-s − 51.9·13-s + 77.0·14-s − 37.4·16-s − 59.1·17-s − 21.0·18-s + 146.·19-s − 98.7·21-s − 115.·22-s − 48.6·23-s + 73.8·24-s + 121.·26-s + 27·27-s + 83.2·28-s − 32.7·29-s − 81.7·31-s − 109.·32-s + 148.·33-s + 138.·34-s + ⋯
 L(s)  = 1 − 0.827·2-s + 0.577·3-s − 0.315·4-s − 0.477·6-s − 1.77·7-s + 1.08·8-s + 0.333·9-s + 1.35·11-s − 0.182·12-s − 1.10·13-s + 1.47·14-s − 0.584·16-s − 0.843·17-s − 0.275·18-s + 1.76·19-s − 1.02·21-s − 1.12·22-s − 0.440·23-s + 0.628·24-s + 0.917·26-s + 0.192·27-s + 0.561·28-s − 0.209·29-s − 0.473·31-s − 0.604·32-s + 0.783·33-s + 0.697·34-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 + 2.33T + 8T^{2}$$
7 $$1 + 32.9T + 343T^{2}$$
11 $$1 - 49.5T + 1.33e3T^{2}$$
13 $$1 + 51.9T + 2.19e3T^{2}$$
17 $$1 + 59.1T + 4.91e3T^{2}$$
19 $$1 - 146.T + 6.85e3T^{2}$$
23 $$1 + 48.6T + 1.21e4T^{2}$$
29 $$1 + 32.7T + 2.43e4T^{2}$$
31 $$1 + 81.7T + 2.97e4T^{2}$$
37 $$1 - 278.T + 5.06e4T^{2}$$
41 $$1 + 296.T + 6.89e4T^{2}$$
43 $$1 + 64.8T + 7.95e4T^{2}$$
47 $$1 - 154.T + 1.03e5T^{2}$$
53 $$1 - 157.T + 1.48e5T^{2}$$
59 $$1 - 409.T + 2.05e5T^{2}$$
61 $$1 + 164.T + 2.26e5T^{2}$$
67 $$1 + 149.T + 3.00e5T^{2}$$
71 $$1 - 440.T + 3.57e5T^{2}$$
73 $$1 - 943.T + 3.89e5T^{2}$$
79 $$1 - 392.T + 4.93e5T^{2}$$
83 $$1 + 1.14e3T + 5.71e5T^{2}$$
89 $$1 - 1.33e3T + 7.04e5T^{2}$$
97 $$1 + 108.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$