Properties

 Label 2-42e2-7.4-c3-0-6 Degree $2$ Conductor $1764$ Sign $-0.605 - 0.795i$ Analytic cond. $104.079$ Root an. cond. $10.2019$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−2 − 3.46i)5-s + (−10 + 17.3i)11-s + 4·13-s + (−12 + 20.7i)17-s + (22 + 38.1i)19-s + (36 + 62.3i)23-s + (54.5 − 94.3i)25-s + 38·29-s + (92 − 159. i)31-s + (15 + 25.9i)37-s − 216·41-s − 164·43-s + (−260 − 450. i)47-s + (−73 + 126. i)53-s + 80·55-s + ⋯
 L(s)  = 1 + (−0.178 − 0.309i)5-s + (−0.274 + 0.474i)11-s + 0.0853·13-s + (−0.171 + 0.296i)17-s + (0.265 + 0.460i)19-s + (0.326 + 0.565i)23-s + (0.435 − 0.755i)25-s + 0.243·29-s + (0.533 − 0.923i)31-s + (0.0666 + 0.115i)37-s − 0.822·41-s − 0.581·43-s + (−0.806 − 1.39i)47-s + (−0.189 + 0.327i)53-s + 0.196·55-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-0.605 - 0.795i$ Analytic conductor: $$104.079$$ Root analytic conductor: $$10.2019$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1764} (361, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1764,\ (\ :3/2),\ -0.605 - 0.795i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$0.8014906983$$ $$L(\frac12)$$ $$\approx$$ $$0.8014906983$$ $$L(\frac{5}{2})$$ 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$$
good5 $$1 + (2 + 3.46i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (10 - 17.3i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 - 4T + 2.19e3T^{2}$$
17 $$1 + (12 - 20.7i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-22 - 38.1i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-36 - 62.3i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 - 38T + 2.43e4T^{2}$$
31 $$1 + (-92 + 159. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (-15 - 25.9i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 216T + 6.89e4T^{2}$$
43 $$1 + 164T + 7.95e4T^{2}$$
47 $$1 + (260 + 450. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (73 - 126. i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (230 - 398. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-314 - 543. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (278 - 481. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 592T + 3.57e5T^{2}$$
73 $$1 + (-512 + 886. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (-52 - 90.0i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 324T + 5.71e5T^{2}$$
89 $$1 + (448 + 775. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 920T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$