Properties

 Label 2-252-7.5-c8-0-0 Degree $2$ Conductor $252$ Sign $-0.995 - 0.0931i$ 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 + (203. + 117. i)5-s + (1.13e3 − 2.11e3i)7-s + (7.47e3 + 1.29e4i)11-s + 3.96e4i·13-s + (−8.46e4 + 4.88e4i)17-s + (−1.77e5 − 1.02e5i)19-s + (1.85e5 − 3.20e5i)23-s + (−1.67e5 − 2.90e5i)25-s − 1.17e5·29-s + (4.37e5 − 2.52e5i)31-s + (4.78e5 − 2.98e5i)35-s + (−9.86e5 + 1.70e6i)37-s + 3.25e6i·41-s − 4.13e6·43-s + (4.79e6 + 2.76e6i)47-s + ⋯
 L(s)  = 1 + (0.325 + 0.187i)5-s + (0.470 − 0.882i)7-s + (0.510 + 0.884i)11-s + 1.38i·13-s + (−1.01 + 0.585i)17-s + (−1.36 − 0.787i)19-s + (0.662 − 1.14i)23-s + (−0.429 − 0.743i)25-s − 0.165·29-s + (0.474 − 0.273i)31-s + (0.318 − 0.198i)35-s + (−0.526 + 0.911i)37-s + 1.15i·41-s − 1.20·43-s + (0.982 + 0.567i)47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0931i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.995 - 0.0931i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $-0.995 - 0.0931i$ 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: $$(2,\ 252,\ (\ :4),\ -0.995 - 0.0931i)$$

Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.2364175435$$ $$L(\frac12)$$ $$\approx$$ $$0.2364175435$$ $$L(5)$$ 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 + (-1.13e3 + 2.11e3i)T$$
good5 $$1 + (-203. - 117. i)T + (1.95e5 + 3.38e5i)T^{2}$$
11 $$1 + (-7.47e3 - 1.29e4i)T + (-1.07e8 + 1.85e8i)T^{2}$$
13 $$1 - 3.96e4iT - 8.15e8T^{2}$$
17 $$1 + (8.46e4 - 4.88e4i)T + (3.48e9 - 6.04e9i)T^{2}$$
19 $$1 + (1.77e5 + 1.02e5i)T + (8.49e9 + 1.47e10i)T^{2}$$
23 $$1 + (-1.85e5 + 3.20e5i)T + (-3.91e10 - 6.78e10i)T^{2}$$
29 $$1 + 1.17e5T + 5.00e11T^{2}$$
31 $$1 + (-4.37e5 + 2.52e5i)T + (4.26e11 - 7.38e11i)T^{2}$$
37 $$1 + (9.86e5 - 1.70e6i)T + (-1.75e12 - 3.04e12i)T^{2}$$
41 $$1 - 3.25e6iT - 7.98e12T^{2}$$
43 $$1 + 4.13e6T + 1.16e13T^{2}$$
47 $$1 + (-4.79e6 - 2.76e6i)T + (1.19e13 + 2.06e13i)T^{2}$$
53 $$1 + (-3.39e6 - 5.87e6i)T + (-3.11e13 + 5.39e13i)T^{2}$$
59 $$1 + (-1.91e7 + 1.10e7i)T + (7.34e13 - 1.27e14i)T^{2}$$
61 $$1 + (3.97e6 + 2.29e6i)T + (9.58e13 + 1.66e14i)T^{2}$$
67 $$1 + (1.86e7 + 3.22e7i)T + (-2.03e14 + 3.51e14i)T^{2}$$
71 $$1 + 4.19e7T + 6.45e14T^{2}$$
73 $$1 + (-1.08e7 + 6.27e6i)T + (4.03e14 - 6.98e14i)T^{2}$$
79 $$1 + (2.01e7 - 3.48e7i)T + (-7.58e14 - 1.31e15i)T^{2}$$
83 $$1 - 5.31e6iT - 2.25e15T^{2}$$
89 $$1 + (6.45e7 + 3.72e7i)T + (1.96e15 + 3.40e15i)T^{2}$$
97 $$1 - 3.76e7iT - 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$