# Properties

 Label 2-252-7.2-c1-0-2 Degree $2$ Conductor $252$ Sign $0.386 + 0.922i$ Analytic cond. $2.01223$ Root an. cond. $1.41853$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.5 − 2.59i)5-s + (−2 − 1.73i)7-s + (−1.5 − 2.59i)11-s + 2·13-s + (1.5 + 2.59i)17-s + (0.5 − 0.866i)19-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s + 6·29-s + (3.5 + 6.06i)31-s + (−7.5 + 2.59i)35-s + (0.5 − 0.866i)37-s − 6·41-s − 4·43-s + (−4.5 + 7.79i)47-s + ⋯
 L(s)  = 1 + (0.670 − 1.16i)5-s + (−0.755 − 0.654i)7-s + (−0.452 − 0.783i)11-s + 0.554·13-s + (0.363 + 0.630i)17-s + (0.114 − 0.198i)19-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s + 1.11·29-s + (0.628 + 1.08i)31-s + (−1.26 + 0.439i)35-s + (0.0821 − 0.142i)37-s − 0.937·41-s − 0.609·43-s + (−0.656 + 1.13i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $0.386 + 0.922i$ Analytic conductor: $$2.01223$$ Root analytic conductor: $$1.41853$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{252} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :1/2),\ 0.386 + 0.922i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.04817 - 0.697228i$$ $$L(\frac12)$$ $$\approx$$ $$1.04817 - 0.697228i$$ $$L(\frac{3}{2})$$ 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$$
7 $$1 + (2 + 1.73i)T$$
good5 $$1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 - 2T + 13T^{2}$$
17 $$1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 - 6T + 29T^{2}$$
31 $$1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 + 4T + 43T^{2}$$
47 $$1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 12T + 83T^{2}$$
89 $$1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + 10T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$