# Properties

 Degree $2$ Conductor $252$ Sign $0.991 + 0.126i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.5 − 0.866i)7-s + 5·13-s + (0.5 − 0.866i)19-s + (2.5 + 4.33i)25-s + (−5.5 − 9.52i)31-s + (−5.5 + 9.52i)37-s − 13·43-s + (5.5 − 4.33i)49-s + (−7 + 12.1i)61-s + (−2.5 − 4.33i)67-s + (−8.5 − 14.7i)73-s + (−8.5 + 14.7i)79-s + (12.5 − 4.33i)91-s + 14·97-s + (6.5 − 11.2i)103-s + ⋯
 L(s)  = 1 + (0.944 − 0.327i)7-s + 1.38·13-s + (0.114 − 0.198i)19-s + (0.5 + 0.866i)25-s + (−0.987 − 1.71i)31-s + (−0.904 + 1.56i)37-s − 1.98·43-s + (0.785 − 0.618i)49-s + (−0.896 + 1.55i)61-s + (−0.305 − 0.529i)67-s + (−0.994 − 1.72i)73-s + (−0.956 + 1.65i)79-s + (1.31 − 0.453i)91-s + 1.42·97-s + (0.640 − 1.10i)103-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $0.991 + 0.126i$ Motivic weight: $$1$$ Character: $\chi_{252} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :1/2),\ 0.991 + 0.126i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.39091 - 0.0882674i$$ $$L(\frac12)$$ $$\approx$$ $$1.39091 - 0.0882674i$$ $$L(\frac{3}{2})$$ 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 + (-2.5 + 0.866i)T$$
good5 $$1 + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-5.5 + 9.52i)T^{2}$$
13 $$1 - 5T + 13T^{2}$$
17 $$1 + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-11.5 - 19.9i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (5.5 + 9.52i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + 41T^{2}$$
43 $$1 + 13T + 43T^{2}$$
47 $$1 + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (8.5 + 14.7i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (-44.5 - 77.0i)T^{2}$$
97 $$1 - 14T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$