Properties

 Degree $2$ Conductor $4$ Sign $-0.0106 + 0.999i$ Motivic weight $32$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (4.60e4 + 4.65e4i)2-s + 8.00e7i·3-s + (−4.59e7 + 4.29e9i)4-s − 3.25e10·5-s + (−3.73e12 + 3.69e12i)6-s − 4.74e13i·7-s + (−2.02e14 + 1.95e14i)8-s − 4.55e15·9-s + (−1.49e15 − 1.51e15i)10-s − 1.35e16i·11-s + (−3.43e17 − 3.67e15i)12-s + 9.45e16·13-s + (2.21e18 − 2.18e18i)14-s − 2.60e18i·15-s + (−1.84e19 − 3.94e17i)16-s + 6.35e18·17-s + ⋯
 L(s)  = 1 + (0.703 + 0.710i)2-s + 1.85i·3-s + (−0.0106 + 0.999i)4-s − 0.213·5-s + (−1.32 + 1.30i)6-s − 1.42i·7-s + (−0.718 + 0.695i)8-s − 2.45·9-s + (−0.149 − 0.151i)10-s − 0.294i·11-s + (−1.85 − 0.0198i)12-s + 0.142·13-s + (1.01 − 1.00i)14-s − 0.396i·15-s + (−0.999 − 0.0213i)16-s + 0.130·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(33-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$4$$    =    $$2^{2}$$ Sign: $-0.0106 + 0.999i$ Motivic weight: $$32$$ Character: $\chi_{4} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4,\ (\ :16),\ -0.0106 + 0.999i)$$

Particular Values

 $$L(\frac{33}{2})$$ $$\approx$$ $$1.126043129$$ $$L(\frac12)$$ $$\approx$$ $$1.126043129$$ $$L(17)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-4.60e4 - 4.65e4i)T$$
good3 $$1 - 8.00e7iT - 1.85e15T^{2}$$
5 $$1 + 3.25e10T + 2.32e22T^{2}$$
7 $$1 + 4.74e13iT - 1.10e27T^{2}$$
11 $$1 + 1.35e16iT - 2.11e33T^{2}$$
13 $$1 - 9.45e16T + 4.42e35T^{2}$$
17 $$1 - 6.35e18T + 2.36e39T^{2}$$
19 $$1 - 4.93e20iT - 8.31e40T^{2}$$
23 $$1 - 4.92e21iT - 3.76e43T^{2}$$
29 $$1 + 2.04e23T + 6.26e46T^{2}$$
31 $$1 + 8.15e23iT - 5.29e47T^{2}$$
37 $$1 - 8.64e24T + 1.52e50T^{2}$$
41 $$1 + 9.37e25T + 4.06e51T^{2}$$
43 $$1 - 9.13e25iT - 1.86e52T^{2}$$
47 $$1 - 4.69e26iT - 3.21e53T^{2}$$
53 $$1 - 3.40e27T + 1.50e55T^{2}$$
59 $$1 + 2.87e27iT - 4.64e56T^{2}$$
61 $$1 - 7.72e27T + 1.35e57T^{2}$$
67 $$1 - 8.28e28iT - 2.71e58T^{2}$$
71 $$1 + 2.37e29iT - 1.73e59T^{2}$$
73 $$1 + 4.61e27T + 4.22e59T^{2}$$
79 $$1 - 3.06e30iT - 5.29e60T^{2}$$
83 $$1 + 3.67e29iT - 2.57e61T^{2}$$
89 $$1 + 2.12e31T + 2.40e62T^{2}$$
97 $$1 - 2.45e30T + 3.77e63T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$