Properties

 Label 2-3800-5.4-c1-0-69 Degree $2$ Conductor $3800$ Sign $-0.894 + 0.447i$ Analytic cond. $30.3431$ Root an. cond. $5.50846$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 2.34i·3-s − 1.19i·7-s − 2.48·9-s + 4.97·11-s − 6.63i·13-s − 1.48i·17-s − 19-s − 2.80·21-s − 0.510i·23-s − 1.19i·27-s + 7.88·29-s − 2.97·31-s − 11.6i·33-s + 7.14i·37-s − 15.5·39-s + ⋯
 L(s)  = 1 − 1.35i·3-s − 0.452i·7-s − 0.829·9-s + 1.50·11-s − 1.84i·13-s − 0.361i·17-s − 0.229·19-s − 0.611·21-s − 0.106i·23-s − 0.230i·27-s + 1.46·29-s − 0.534·31-s − 2.03i·33-s + 1.17i·37-s − 2.48·39-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$3800$$    =    $$2^{3} \cdot 5^{2} \cdot 19$$ Sign: $-0.894 + 0.447i$ Analytic conductor: $$30.3431$$ Root analytic conductor: $$5.50846$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3800} (3649, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3800,\ (\ :1/2),\ -0.894 + 0.447i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$2.026733684$$ $$L(\frac12)$$ $$\approx$$ $$2.026733684$$ $$L(\frac{3}{2})$$ 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$$
19 $$1 + T$$
good3 $$1 + 2.34iT - 3T^{2}$$
7 $$1 + 1.19iT - 7T^{2}$$
11 $$1 - 4.97T + 11T^{2}$$
13 $$1 + 6.63iT - 13T^{2}$$
17 $$1 + 1.48iT - 17T^{2}$$
23 $$1 + 0.510iT - 23T^{2}$$
29 $$1 - 7.88T + 29T^{2}$$
31 $$1 + 2.97T + 31T^{2}$$
37 $$1 - 7.14iT - 37T^{2}$$
41 $$1 - 1.66T + 41T^{2}$$
43 $$1 + 6.39iT - 43T^{2}$$
47 $$1 - 9.95iT - 47T^{2}$$
53 $$1 + 11.4iT - 53T^{2}$$
59 $$1 - 11.8T + 59T^{2}$$
61 $$1 - 3.66T + 61T^{2}$$
67 $$1 + 7.61iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 13.8iT - 73T^{2}$$
79 $$1 + 12.6T + 79T^{2}$$
83 $$1 + 8.68iT - 83T^{2}$$
89 $$1 - 4.87T + 89T^{2}$$
97 $$1 - 6.81iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$