# Properties

 Degree 2 Conductor 29 Sign $0.913 + 0.407i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.65 − 0.578i)2-s + (−0.186 − 1.65i)3-s + (−0.727 + 0.579i)4-s + (−0.825 + 1.71i)5-s + (−1.26 − 2.62i)6-s + (−1.24 + 1.56i)7-s + (−4.59 + 7.31i)8-s + (6.07 − 1.38i)9-s + (−0.373 + 3.31i)10-s + (−6.63 − 10.5i)11-s + (1.09 + 1.09i)12-s + (−3.80 − 0.868i)13-s + (−1.15 + 3.30i)14-s + (2.98 + 1.04i)15-s + (−2.54 + 11.1i)16-s + (7.59 − 7.59i)17-s + ⋯
 L(s)  = 1 + (0.826 − 0.289i)2-s + (−0.0621 − 0.551i)3-s + (−0.181 + 0.144i)4-s + (−0.165 + 0.342i)5-s + (−0.210 − 0.437i)6-s + (−0.178 + 0.223i)7-s + (−0.574 + 0.914i)8-s + (0.674 − 0.154i)9-s + (−0.0373 + 0.331i)10-s + (−0.602 − 0.959i)11-s + (0.0912 + 0.0912i)12-s + (−0.292 − 0.0668i)13-s + (−0.0827 + 0.236i)14-s + (0.199 + 0.0697i)15-s + (−0.158 + 0.695i)16-s + (0.446 − 0.446i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$29$$ $$\varepsilon$$ = $0.913 + 0.407i$ motivic weight = $$2$$ character : $\chi_{29} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 29,\ (\ :1),\ 0.913 + 0.407i)$ $L(\frac{3}{2})$ $\approx$ $1.20315 - 0.256245i$ $L(\frac12)$ $\approx$ $1.20315 - 0.256245i$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 29$, $$F_p(T)$$ is a polynomial of degree 2. If $p = 29$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad29 $$1 + (-8.15 - 27.8i)T$$
good2 $$1 + (-1.65 + 0.578i)T + (3.12 - 2.49i)T^{2}$$
3 $$1 + (0.186 + 1.65i)T + (-8.77 + 2.00i)T^{2}$$
5 $$1 + (0.825 - 1.71i)T + (-15.5 - 19.5i)T^{2}$$
7 $$1 + (1.24 - 1.56i)T + (-10.9 - 47.7i)T^{2}$$
11 $$1 + (6.63 + 10.5i)T + (-52.4 + 109. i)T^{2}$$
13 $$1 + (3.80 + 0.868i)T + (152. + 73.3i)T^{2}$$
17 $$1 + (-7.59 + 7.59i)T - 289iT^{2}$$
19 $$1 + (0.137 + 0.0154i)T + (351. + 80.3i)T^{2}$$
23 $$1 + (-26.7 + 12.8i)T + (329. - 413. i)T^{2}$$
31 $$1 + (54.1 - 18.9i)T + (751. - 599. i)T^{2}$$
37 $$1 + (29.9 - 47.6i)T + (-593. - 1.23e3i)T^{2}$$
41 $$1 + (25.9 + 25.9i)T + 1.68e3iT^{2}$$
43 $$1 + (-5.16 + 14.7i)T + (-1.44e3 - 1.15e3i)T^{2}$$
47 $$1 + (-55.9 + 35.1i)T + (958. - 1.99e3i)T^{2}$$
53 $$1 + (-29.3 - 14.1i)T + (1.75e3 + 2.19e3i)T^{2}$$
59 $$1 - 0.396T + 3.48e3T^{2}$$
61 $$1 + (6.99 + 62.1i)T + (-3.62e3 + 828. i)T^{2}$$
67 $$1 + (21.7 - 4.97i)T + (4.04e3 - 1.94e3i)T^{2}$$
71 $$1 + (64.8 + 14.7i)T + (4.54e3 + 2.18e3i)T^{2}$$
73 $$1 + (-76.3 - 26.7i)T + (4.16e3 + 3.32e3i)T^{2}$$
79 $$1 + (-67.7 - 42.5i)T + (2.70e3 + 5.62e3i)T^{2}$$
83 $$1 + (68.4 + 85.8i)T + (-1.53e3 + 6.71e3i)T^{2}$$
89 $$1 + (77.7 - 27.2i)T + (6.19e3 - 4.93e3i)T^{2}$$
97 $$1 + (-19.5 + 173. i)T + (-9.17e3 - 2.09e3i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}