Properties

 Label 2-48-48.11-c1-0-5 Degree $2$ Conductor $48$ Sign $0.208 + 0.978i$ Analytic cond. $0.383281$ Root an. cond. $0.619097$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−0.416 − 1.35i)2-s + (0.966 − 1.43i)3-s + (−1.65 + 1.12i)4-s + (−1.57 + 1.57i)5-s + (−2.34 − 0.708i)6-s + 2.24·7-s + (2.20 + 1.76i)8-s + (−1.13 − 2.77i)9-s + (2.77 + 1.47i)10-s + (1.13 + 1.13i)11-s + (0.0176 + 3.46i)12-s + (−3.24 + 3.24i)13-s + (−0.935 − 3.04i)14-s + (0.739 + 3.77i)15-s + (1.47 − 3.71i)16-s + 1.66i·17-s + ⋯
 L(s)  = 1 + (−0.294 − 0.955i)2-s + (0.558 − 0.829i)3-s + (−0.826 + 0.562i)4-s + (−0.702 + 0.702i)5-s + (−0.957 − 0.289i)6-s + 0.850·7-s + (0.780 + 0.624i)8-s + (−0.377 − 0.926i)9-s + (0.878 + 0.465i)10-s + (0.341 + 0.341i)11-s + (0.00510 + 0.999i)12-s + (−0.901 + 0.901i)13-s + (−0.250 − 0.812i)14-s + (0.191 + 0.975i)15-s + (0.367 − 0.929i)16-s + 0.403i·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$48$$    =    $$2^{4} \cdot 3$$ Sign: $0.208 + 0.978i$ Analytic conductor: $$0.383281$$ Root analytic conductor: $$0.619097$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{48} (11, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 48,\ (\ :1/2),\ 0.208 + 0.978i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.582356 - 0.471527i$$ $$L(\frac12)$$ $$\approx$$ $$0.582356 - 0.471527i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.416 + 1.35i)T$$
3 $$1 + (-0.966 + 1.43i)T$$
good5 $$1 + (1.57 - 1.57i)T - 5iT^{2}$$
7 $$1 - 2.24T + 7T^{2}$$
11 $$1 + (-1.13 - 1.13i)T + 11iT^{2}$$
13 $$1 + (3.24 - 3.24i)T - 13iT^{2}$$
17 $$1 - 1.66iT - 17T^{2}$$
19 $$1 + (3.77 + 3.77i)T + 19iT^{2}$$
23 $$1 + 2.26iT - 23T^{2}$$
29 $$1 + (3.23 + 3.23i)T + 29iT^{2}$$
31 $$1 + 1.30iT - 31T^{2}$$
37 $$1 + (-2.30 - 2.30i)T + 37iT^{2}$$
41 $$1 - 10.2T + 41T^{2}$$
43 $$1 + (-3.77 + 3.77i)T - 43iT^{2}$$
47 $$1 + 3.74T + 47T^{2}$$
53 $$1 + (-0.972 + 0.972i)T - 53iT^{2}$$
59 $$1 + (-3.88 - 3.88i)T + 59iT^{2}$$
61 $$1 + (-4.19 + 4.19i)T - 61iT^{2}$$
67 $$1 + (-8.02 - 8.02i)T + 67iT^{2}$$
71 $$1 + 11.0iT - 71T^{2}$$
73 $$1 - 6.38iT - 73T^{2}$$
79 $$1 + 2.69iT - 79T^{2}$$
83 $$1 + (-2.61 + 2.61i)T - 83iT^{2}$$
89 $$1 + 7.35T + 89T^{2}$$
97 $$1 + 5.67T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$