# Properties

 Label 2-392-7.4-c5-0-41 Degree $2$ Conductor $392$ Sign $0.605 + 0.795i$ Analytic cond. $62.8704$ Root an. cond. $7.92908$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−10 + 17.3i)3-s + (37 + 64.0i)5-s + (−78.5 − 135. i)9-s + (−62 + 107. i)11-s + 478·13-s − 1.48e3·15-s + (599 − 1.03e3i)17-s + (−1.52e3 − 2.63e3i)19-s + (−92 − 159. i)23-s + (−1.17e3 + 2.03e3i)25-s − 1.71e3·27-s − 3.28e3·29-s + (2.86e3 − 4.96e3i)31-s + (−1.24e3 − 2.14e3i)33-s + (−5.16e3 − 8.94e3i)37-s + ⋯
 L(s)  = 1 + (−0.641 + 1.11i)3-s + (0.661 + 1.14i)5-s + (−0.323 − 0.559i)9-s + (−0.154 + 0.267i)11-s + 0.784·13-s − 1.69·15-s + (0.502 − 0.870i)17-s + (−0.967 − 1.67i)19-s + (−0.0362 − 0.0628i)23-s + (−0.376 + 0.651i)25-s − 0.454·27-s − 0.724·29-s + (0.535 − 0.927i)31-s + (−0.198 − 0.343i)33-s + (−0.620 − 1.07i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$392$$    =    $$2^{3} \cdot 7^{2}$$ Sign: $0.605 + 0.795i$ Analytic conductor: $$62.8704$$ Root analytic conductor: $$7.92908$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{392} (361, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 392,\ (\ :5/2),\ 0.605 + 0.795i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.6363756309$$ $$L(\frac12)$$ $$\approx$$ $$0.6363756309$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1$$
good3 $$1 + (10 - 17.3i)T + (-121.5 - 210. i)T^{2}$$
5 $$1 + (-37 - 64.0i)T + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (62 - 107. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 - 478T + 3.71e5T^{2}$$
17 $$1 + (-599 + 1.03e3i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (1.52e3 + 2.63e3i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (92 + 159. i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + 3.28e3T + 2.05e7T^{2}$$
31 $$1 + (-2.86e3 + 4.96e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (5.16e3 + 8.94e3i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 + 8.88e3T + 1.15e8T^{2}$$
43 $$1 + 9.18e3T + 1.47e8T^{2}$$
47 $$1 + (1.18e4 + 2.04e4i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (5.84e3 - 1.01e4i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (8.43e3 - 1.46e4i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (-9.24e3 - 1.60e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (-7.76e3 + 1.34e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 3.19e4T + 1.80e9T^{2}$$
73 $$1 + (-2.44e3 + 4.23e3i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (2.22e4 + 3.85e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 - 6.73e4T + 3.93e9T^{2}$$
89 $$1 + (3.59e4 + 6.23e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 - 4.88e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$