Properties

 Label 2-75-15.14-c2-0-2 Degree $2$ Conductor $75$ Sign $-0.121 - 0.992i$ Analytic cond. $2.04360$ Root an. cond. $1.42954$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 3.31·2-s + (1.65 + 2.5i)3-s + 7·4-s + (−5.5 − 8.29i)6-s − 9.94·8-s + (−3.5 + 8.29i)9-s + 16.5i·11-s + (11.6 + 17.5i)12-s + 10i·13-s + 5.00·16-s − 3.31·17-s + (11.6 − 27.4i)18-s − 7·19-s − 55.0i·22-s + 19.8·23-s + (−16.5 − 24.8i)24-s + ⋯
 L(s)  = 1 − 1.65·2-s + (0.552 + 0.833i)3-s + 1.75·4-s + (−0.916 − 1.38i)6-s − 1.24·8-s + (−0.388 + 0.921i)9-s + 1.50i·11-s + (0.967 + 1.45i)12-s + 0.769i·13-s + 0.312·16-s − 0.195·17-s + (0.644 − 1.52i)18-s − 0.368·19-s − 2.50i·22-s + 0.865·23-s + (−0.687 − 1.03i)24-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$75$$    =    $$3 \cdot 5^{2}$$ Sign: $-0.121 - 0.992i$ Analytic conductor: $$2.04360$$ Root analytic conductor: $$1.42954$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{75} (74, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 75,\ (\ :1),\ -0.121 - 0.992i)$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.423227 + 0.478306i$$ $$L(\frac12)$$ $$\approx$$ $$0.423227 + 0.478306i$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.65 - 2.5i)T$$
5 $$1$$
good2 $$1 + 3.31T + 4T^{2}$$
7 $$1 - 49T^{2}$$
11 $$1 - 16.5iT - 121T^{2}$$
13 $$1 - 10iT - 169T^{2}$$
17 $$1 + 3.31T + 289T^{2}$$
19 $$1 + 7T + 361T^{2}$$
23 $$1 - 19.8T + 529T^{2}$$
29 $$1 + 33.1iT - 841T^{2}$$
31 $$1 - 42T + 961T^{2}$$
37 $$1 - 40iT - 1.36e3T^{2}$$
41 $$1 + 16.5iT - 1.68e3T^{2}$$
43 $$1 + 50iT - 1.84e3T^{2}$$
47 $$1 - 46.4T + 2.20e3T^{2}$$
53 $$1 + 46.4T + 2.80e3T^{2}$$
59 $$1 + 66.3iT - 3.48e3T^{2}$$
61 $$1 + 8T + 3.72e3T^{2}$$
67 $$1 + 45iT - 4.48e3T^{2}$$
71 $$1 - 33.1iT - 5.04e3T^{2}$$
73 $$1 + 35iT - 5.32e3T^{2}$$
79 $$1 + 12T + 6.24e3T^{2}$$
83 $$1 - 69.6T + 6.88e3T^{2}$$
89 $$1 - 149. iT - 7.92e3T^{2}$$
97 $$1 - 70iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$