# Properties

 Label 2-252-7.5-c8-0-15 Degree $2$ Conductor $252$ Sign $-0.302 + 0.953i$ Analytic cond. $102.659$ Root an. cond. $10.1320$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.01e3 − 584. i)5-s + (−1.82e3 − 1.55e3i)7-s + (1.30e4 + 2.26e4i)11-s − 2.85e4i·13-s + (1.80e4 − 1.04e4i)17-s + (4.29e4 + 2.48e4i)19-s + (−1.70e5 + 2.96e5i)23-s + (4.87e5 + 8.43e5i)25-s + 1.12e6·29-s + (1.47e6 − 8.51e5i)31-s + (9.43e5 + 2.64e6i)35-s + (3.20e5 − 5.54e5i)37-s + 1.34e6i·41-s − 1.80e6·43-s + (−7.88e6 − 4.55e6i)47-s + ⋯
 L(s)  = 1 + (−1.61 − 0.934i)5-s + (−0.762 − 0.647i)7-s + (0.893 + 1.54i)11-s − 1.00i·13-s + (0.215 − 0.124i)17-s + (0.329 + 0.190i)19-s + (−0.610 + 1.05i)23-s + (1.24 + 2.16i)25-s + 1.59·29-s + (1.59 − 0.922i)31-s + (0.628 + 1.76i)35-s + (0.170 − 0.295i)37-s + 0.476i·41-s − 0.526·43-s + (−1.61 − 0.933i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $-0.302 + 0.953i$ Analytic conductor: $$102.659$$ Root analytic conductor: $$10.1320$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{252} (145, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :4),\ -0.302 + 0.953i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.000679539$$ $$L(\frac12)$$ $$\approx$$ $$1.000679539$$ $$L(5)$$ 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 + (1.82e3 + 1.55e3i)T$$
good5 $$1 + (1.01e3 + 584. i)T + (1.95e5 + 3.38e5i)T^{2}$$
11 $$1 + (-1.30e4 - 2.26e4i)T + (-1.07e8 + 1.85e8i)T^{2}$$
13 $$1 + 2.85e4iT - 8.15e8T^{2}$$
17 $$1 + (-1.80e4 + 1.04e4i)T + (3.48e9 - 6.04e9i)T^{2}$$
19 $$1 + (-4.29e4 - 2.48e4i)T + (8.49e9 + 1.47e10i)T^{2}$$
23 $$1 + (1.70e5 - 2.96e5i)T + (-3.91e10 - 6.78e10i)T^{2}$$
29 $$1 - 1.12e6T + 5.00e11T^{2}$$
31 $$1 + (-1.47e6 + 8.51e5i)T + (4.26e11 - 7.38e11i)T^{2}$$
37 $$1 + (-3.20e5 + 5.54e5i)T + (-1.75e12 - 3.04e12i)T^{2}$$
41 $$1 - 1.34e6iT - 7.98e12T^{2}$$
43 $$1 + 1.80e6T + 1.16e13T^{2}$$
47 $$1 + (7.88e6 + 4.55e6i)T + (1.19e13 + 2.06e13i)T^{2}$$
53 $$1 + (-1.82e6 - 3.16e6i)T + (-3.11e13 + 5.39e13i)T^{2}$$
59 $$1 + (9.97e6 - 5.76e6i)T + (7.34e13 - 1.27e14i)T^{2}$$
61 $$1 + (8.77e6 + 5.06e6i)T + (9.58e13 + 1.66e14i)T^{2}$$
67 $$1 + (8.86e6 + 1.53e7i)T + (-2.03e14 + 3.51e14i)T^{2}$$
71 $$1 - 3.79e7T + 6.45e14T^{2}$$
73 $$1 + (-2.94e7 + 1.70e7i)T + (4.03e14 - 6.98e14i)T^{2}$$
79 $$1 + (9.66e6 - 1.67e7i)T + (-7.58e14 - 1.31e15i)T^{2}$$
83 $$1 - 9.60e5iT - 2.25e15T^{2}$$
89 $$1 + (5.62e7 + 3.24e7i)T + (1.96e15 + 3.40e15i)T^{2}$$
97 $$1 - 2.66e7iT - 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$