# Properties

 Label 2-252-7.4-c11-0-1 Degree $2$ Conductor $252$ Sign $0.252 - 0.967i$ 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 + (−4.94e3 − 8.57e3i)5-s + (2.91e4 − 3.35e4i)7-s + (−2.50e4 + 4.33e4i)11-s + 1.46e6·13-s + (−3.16e6 + 5.48e6i)17-s + (−1.02e7 − 1.77e7i)19-s + (2.36e7 + 4.08e7i)23-s + (−2.45e7 + 4.25e7i)25-s + 9.10e7·29-s + (−4.83e7 + 8.37e7i)31-s + (−4.32e8 − 8.38e7i)35-s + (−7.14e7 − 1.23e8i)37-s − 8.44e8·41-s − 1.55e9·43-s + (4.22e8 + 7.32e8i)47-s + ⋯
 L(s)  = 1 + (−0.708 − 1.22i)5-s + (0.655 − 0.754i)7-s + (−0.0468 + 0.0810i)11-s + 1.09·13-s + (−0.541 + 0.937i)17-s + (−0.949 − 1.64i)19-s + (0.764 + 1.32i)23-s + (−0.503 + 0.872i)25-s + 0.824·29-s + (−0.303 + 0.525i)31-s + (−1.39 − 0.269i)35-s + (−0.169 − 0.293i)37-s − 1.13·41-s − 1.61·43-s + (0.268 + 0.465i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(12-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $0.252 - 0.967i$ 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: $$(2,\ 252,\ (\ :11/2),\ 0.252 - 0.967i)$$

## Particular Values

 $$L(6)$$ $$\approx$$ $$0.6611598767$$ $$L(\frac12)$$ $$\approx$$ $$0.6611598767$$ $$L(\frac{13}{2})$$ 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 + (-2.91e4 + 3.35e4i)T$$
good5 $$1 + (4.94e3 + 8.57e3i)T + (-2.44e7 + 4.22e7i)T^{2}$$
11 $$1 + (2.50e4 - 4.33e4i)T + (-1.42e11 - 2.47e11i)T^{2}$$
13 $$1 - 1.46e6T + 1.79e12T^{2}$$
17 $$1 + (3.16e6 - 5.48e6i)T + (-1.71e13 - 2.96e13i)T^{2}$$
19 $$1 + (1.02e7 + 1.77e7i)T + (-5.82e13 + 1.00e14i)T^{2}$$
23 $$1 + (-2.36e7 - 4.08e7i)T + (-4.76e14 + 8.25e14i)T^{2}$$
29 $$1 - 9.10e7T + 1.22e16T^{2}$$
31 $$1 + (4.83e7 - 8.37e7i)T + (-1.27e16 - 2.20e16i)T^{2}$$
37 $$1 + (7.14e7 + 1.23e8i)T + (-8.89e16 + 1.54e17i)T^{2}$$
41 $$1 + 8.44e8T + 5.50e17T^{2}$$
43 $$1 + 1.55e9T + 9.29e17T^{2}$$
47 $$1 + (-4.22e8 - 7.32e8i)T + (-1.23e18 + 2.14e18i)T^{2}$$
53 $$1 + (1.26e9 - 2.19e9i)T + (-4.63e18 - 8.02e18i)T^{2}$$
59 $$1 + (2.80e9 - 4.85e9i)T + (-1.50e19 - 2.61e19i)T^{2}$$
61 $$1 + (5.98e8 + 1.03e9i)T + (-2.17e19 + 3.76e19i)T^{2}$$
67 $$1 + (2.73e9 - 4.73e9i)T + (-6.10e19 - 1.05e20i)T^{2}$$
71 $$1 + 1.81e10T + 2.31e20T^{2}$$
73 $$1 + (-9.87e9 + 1.71e10i)T + (-1.56e20 - 2.71e20i)T^{2}$$
79 $$1 + (2.14e10 + 3.71e10i)T + (-3.73e20 + 6.47e20i)T^{2}$$
83 $$1 - 5.21e10T + 1.28e21T^{2}$$
89 $$1 + (-1.39e10 - 2.42e10i)T + (-1.38e21 + 2.40e21i)T^{2}$$
97 $$1 - 5.42e10T + 7.15e21T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$