Properties

 Label 2-252-7.4-c11-0-27 Degree $2$ Conductor $252$ Sign $0.234 + 0.972i$ Analytic cond. $193.622$ Root an. cond. $13.9148$ Motivic weight $11$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−84.6 − 146. i)5-s + (−9.85e3 + 4.33e4i)7-s + (1.43e5 − 2.49e5i)11-s + 2.48e6·13-s + (3.58e6 − 6.20e6i)17-s + (−1.63e6 − 2.83e6i)19-s + (−1.56e7 − 2.71e7i)23-s + (2.43e7 − 4.22e7i)25-s − 5.63e7·29-s + (−8.04e7 + 1.39e8i)31-s + (7.18e6 − 2.22e6i)35-s + (1.48e8 + 2.57e8i)37-s − 1.39e9·41-s − 4.76e8·43-s + (1.19e9 + 2.06e9i)47-s + ⋯
 L(s)  = 1 + (−0.0121 − 0.0209i)5-s + (−0.221 + 0.975i)7-s + (0.269 − 0.466i)11-s + 1.85·13-s + (0.611 − 1.05i)17-s + (−0.151 − 0.262i)19-s + (−0.507 − 0.878i)23-s + (0.499 − 0.865i)25-s − 0.509·29-s + (−0.504 + 0.874i)31-s + (0.0231 − 0.00716i)35-s + (0.352 + 0.611i)37-s − 1.88·41-s − 0.494·43-s + (0.758 + 1.31i)47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(12-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $0.234 + 0.972i$ Analytic conductor: $$193.622$$ Root analytic conductor: $$13.9148$$ Motivic weight: $$11$$ Rational: no Arithmetic: yes Character: $\chi_{252} (109, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :11/2),\ 0.234 + 0.972i)$$

Particular Values

 $$L(6)$$ $$\approx$$ $$1.934258009$$ $$L(\frac12)$$ $$\approx$$ $$1.934258009$$ $$L(\frac{13}{2})$$ 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 + (9.85e3 - 4.33e4i)T$$
good5 $$1 + (84.6 + 146. i)T + (-2.44e7 + 4.22e7i)T^{2}$$
11 $$1 + (-1.43e5 + 2.49e5i)T + (-1.42e11 - 2.47e11i)T^{2}$$
13 $$1 - 2.48e6T + 1.79e12T^{2}$$
17 $$1 + (-3.58e6 + 6.20e6i)T + (-1.71e13 - 2.96e13i)T^{2}$$
19 $$1 + (1.63e6 + 2.83e6i)T + (-5.82e13 + 1.00e14i)T^{2}$$
23 $$1 + (1.56e7 + 2.71e7i)T + (-4.76e14 + 8.25e14i)T^{2}$$
29 $$1 + 5.63e7T + 1.22e16T^{2}$$
31 $$1 + (8.04e7 - 1.39e8i)T + (-1.27e16 - 2.20e16i)T^{2}$$
37 $$1 + (-1.48e8 - 2.57e8i)T + (-8.89e16 + 1.54e17i)T^{2}$$
41 $$1 + 1.39e9T + 5.50e17T^{2}$$
43 $$1 + 4.76e8T + 9.29e17T^{2}$$
47 $$1 + (-1.19e9 - 2.06e9i)T + (-1.23e18 + 2.14e18i)T^{2}$$
53 $$1 + (-2.34e9 + 4.05e9i)T + (-4.63e18 - 8.02e18i)T^{2}$$
59 $$1 + (-8.21e8 + 1.42e9i)T + (-1.50e19 - 2.61e19i)T^{2}$$
61 $$1 + (1.55e9 + 2.69e9i)T + (-2.17e19 + 3.76e19i)T^{2}$$
67 $$1 + (5.33e9 - 9.24e9i)T + (-6.10e19 - 1.05e20i)T^{2}$$
71 $$1 - 2.59e10T + 2.31e20T^{2}$$
73 $$1 + (2.45e8 - 4.24e8i)T + (-1.56e20 - 2.71e20i)T^{2}$$
79 $$1 + (4.61e9 + 7.98e9i)T + (-3.73e20 + 6.47e20i)T^{2}$$
83 $$1 - 3.48e10T + 1.28e21T^{2}$$
89 $$1 + (1.81e10 + 3.15e10i)T + (-1.38e21 + 2.40e21i)T^{2}$$
97 $$1 + 1.51e11T + 7.15e21T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$