# Properties

 Label 2-60e2-15.8-c1-0-10 Degree $2$ Conductor $3600$ Sign $0.391 - 0.920i$ Analytic cond. $28.7461$ Root an. cond. $5.36154$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.65i·11-s + (−3 − 3i)13-s − 4i·19-s + (2.82 − 2.82i)23-s + 1.41·29-s + 8·31-s + (−7 + 7i)37-s + 1.41i·41-s + (4 + 4i)43-s + (2.82 + 2.82i)47-s + 7i·49-s + (8.48 − 8.48i)53-s + 11.3·59-s − 12·61-s + (−8 + 8i)67-s + ⋯
 L(s)  = 1 + 1.70i·11-s + (−0.832 − 0.832i)13-s − 0.917i·19-s + (0.589 − 0.589i)23-s + 0.262·29-s + 1.43·31-s + (−1.15 + 1.15i)37-s + 0.220i·41-s + (0.609 + 0.609i)43-s + (0.412 + 0.412i)47-s + i·49-s + (1.16 − 1.16i)53-s + 1.47·59-s − 1.53·61-s + (−0.977 + 0.977i)67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3600$$    =    $$2^{4} \cdot 3^{2} \cdot 5^{2}$$ Sign: $0.391 - 0.920i$ Analytic conductor: $$28.7461$$ Root analytic conductor: $$5.36154$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3600} (593, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3600,\ (\ :1/2),\ 0.391 - 0.920i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.516488552$$ $$L(\frac12)$$ $$\approx$$ $$1.516488552$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
good7 $$1 - 7iT^{2}$$
11 $$1 - 5.65iT - 11T^{2}$$
13 $$1 + (3 + 3i)T + 13iT^{2}$$
17 $$1 + 17iT^{2}$$
19 $$1 + 4iT - 19T^{2}$$
23 $$1 + (-2.82 + 2.82i)T - 23iT^{2}$$
29 $$1 - 1.41T + 29T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 + (7 - 7i)T - 37iT^{2}$$
41 $$1 - 1.41iT - 41T^{2}$$
43 $$1 + (-4 - 4i)T + 43iT^{2}$$
47 $$1 + (-2.82 - 2.82i)T + 47iT^{2}$$
53 $$1 + (-8.48 + 8.48i)T - 53iT^{2}$$
59 $$1 - 11.3T + 59T^{2}$$
61 $$1 + 12T + 61T^{2}$$
67 $$1 + (8 - 8i)T - 67iT^{2}$$
71 $$1 - 5.65iT - 71T^{2}$$
73 $$1 + (-3 - 3i)T + 73iT^{2}$$
79 $$1 - 8iT - 79T^{2}$$
83 $$1 + (11.3 - 11.3i)T - 83iT^{2}$$
89 $$1 + 7.07T + 89T^{2}$$
97 $$1 + (5 - 5i)T - 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$