# Properties

 Label 2-1260-105.104-c1-0-15 Degree $2$ Conductor $1260$ Sign $-0.975 - 0.220i$ Analytic cond. $10.0611$ Root an. cond. $3.17193$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.08 − 1.95i)5-s + (−0.595 − 2.57i)7-s − 3.74i·11-s − 3.36·13-s − 0.841i·17-s + 5.59i·19-s − 2.35·23-s + (−2.64 + 4.24i)25-s + 1.41i·29-s + 8.66i·31-s + (−4.39 + 3.96i)35-s − 5.15i·37-s − 5.74·41-s + 3.32i·43-s − 6.43i·47-s + ⋯
 L(s)  = 1 + (−0.485 − 0.874i)5-s + (−0.224 − 0.974i)7-s − 1.12i·11-s − 0.931·13-s − 0.204i·17-s + 1.28i·19-s − 0.490·23-s + (−0.529 + 0.848i)25-s + 0.262i·29-s + 1.55i·31-s + (−0.742 + 0.669i)35-s − 0.847i·37-s − 0.896·41-s + 0.507i·43-s − 0.938i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1260$$    =    $$2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Sign: $-0.975 - 0.220i$ Analytic conductor: $$10.0611$$ Root analytic conductor: $$3.17193$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1260} (629, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1260,\ (\ :1/2),\ -0.975 - 0.220i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4531763094$$ $$L(\frac12)$$ $$\approx$$ $$0.4531763094$$ $$L(\frac{3}{2})$$ 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 + (1.08 + 1.95i)T$$
7 $$1 + (0.595 + 2.57i)T$$
good11 $$1 + 3.74iT - 11T^{2}$$
13 $$1 + 3.36T + 13T^{2}$$
17 $$1 + 0.841iT - 17T^{2}$$
19 $$1 - 5.59iT - 19T^{2}$$
23 $$1 + 2.35T + 23T^{2}$$
29 $$1 - 1.41iT - 29T^{2}$$
31 $$1 - 8.66iT - 31T^{2}$$
37 $$1 + 5.15iT - 37T^{2}$$
41 $$1 + 5.74T + 41T^{2}$$
43 $$1 - 3.32iT - 43T^{2}$$
47 $$1 + 6.43iT - 47T^{2}$$
53 $$1 - 9.64T + 53T^{2}$$
59 $$1 + 12.2T + 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 + 1.82iT - 67T^{2}$$
71 $$1 + 3.74iT - 71T^{2}$$
73 $$1 - 0.979T + 73T^{2}$$
79 $$1 + 6.58T + 79T^{2}$$
83 $$1 + 12.5iT - 83T^{2}$$
89 $$1 - 2.16T + 89T^{2}$$
97 $$1 + 12.0T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$