# Properties

 Label 2-28e2-112.37-c1-0-66 Degree $2$ Conductor $784$ Sign $-0.615 + 0.788i$ Analytic cond. $6.26027$ Root an. cond. $2.50205$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.36 − 0.366i)2-s + (−1.36 − 0.366i)3-s + (1.73 − i)4-s + (−1.36 + 0.366i)5-s − 2·6-s + (1.99 − 2i)8-s + (−0.866 − 0.5i)9-s + (−1.73 + i)10-s + (0.366 − 1.36i)11-s + (−2.73 + 0.732i)12-s + (1 − i)13-s + 2·15-s + (1.99 − 3.46i)16-s + (−1 − 1.73i)17-s + (−1.36 − 0.366i)18-s + (−1.09 − 4.09i)19-s + ⋯
 L(s)  = 1 + (0.965 − 0.258i)2-s + (−0.788 − 0.211i)3-s + (0.866 − 0.5i)4-s + (−0.610 + 0.163i)5-s − 0.816·6-s + (0.707 − 0.707i)8-s + (−0.288 − 0.166i)9-s + (−0.547 + 0.316i)10-s + (0.110 − 0.411i)11-s + (−0.788 + 0.211i)12-s + (0.277 − 0.277i)13-s + 0.516·15-s + (0.499 − 0.866i)16-s + (−0.242 − 0.420i)17-s + (−0.321 − 0.0862i)18-s + (−0.251 − 0.940i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$784$$    =    $$2^{4} \cdot 7^{2}$$ Sign: $-0.615 + 0.788i$ Analytic conductor: $$6.26027$$ Root analytic conductor: $$2.50205$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{784} (373, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 784,\ (\ :1/2),\ -0.615 + 0.788i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.637806 - 1.30698i$$ $$L(\frac12)$$ $$\approx$$ $$0.637806 - 1.30698i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.36 + 0.366i)T$$
7 $$1$$
good3 $$1 + (1.36 + 0.366i)T + (2.59 + 1.5i)T^{2}$$
5 $$1 + (1.36 - 0.366i)T + (4.33 - 2.5i)T^{2}$$
11 $$1 + (-0.366 + 1.36i)T + (-9.52 - 5.5i)T^{2}$$
13 $$1 + (-1 + i)T - 13iT^{2}$$
17 $$1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (1.09 + 4.09i)T + (-16.4 + 9.5i)T^{2}$$
23 $$1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (-3 + 3i)T - 29iT^{2}$$
31 $$1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (4.09 - 1.09i)T + (32.0 - 18.5i)T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 + (-5 - 5i)T + 43iT^{2}$$
47 $$1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (1.83 - 6.83i)T + (-45.8 - 26.5i)T^{2}$$
59 $$1 + (-1.09 + 4.09i)T + (-51.0 - 29.5i)T^{2}$$
61 $$1 + (-3.29 - 12.2i)T + (-52.8 + 30.5i)T^{2}$$
67 $$1 + (-6.83 - 1.83i)T + (58.0 + 33.5i)T^{2}$$
71 $$1 + 10iT - 71T^{2}$$
73 $$1 + (-3.46 + 2i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-1 + i)T - 83iT^{2}$$
89 $$1 + (-3.46 - 2i)T + (44.5 + 77.0i)T^{2}$$
97 $$1 - 2T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$