# Properties

 Degree $2$ Conductor $29$ Sign $-0.229 + 0.973i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.07 − 1.07i)2-s + (−2.14 − 2.14i)3-s − 1.67i·4-s − 0.488i·5-s + 4.62i·6-s + 8.09·7-s + (−6.11 + 6.11i)8-s + 0.180i·9-s + (−0.527 + 0.527i)10-s + (11.3 + 11.3i)11-s + (−3.57 + 3.57i)12-s − 10.0i·13-s + (−8.73 − 8.73i)14-s + (−1.04 + 1.04i)15-s + 6.53·16-s + (−9.69 − 9.69i)17-s + ⋯
 L(s)  = 1 + (−0.539 − 0.539i)2-s + (−0.714 − 0.714i)3-s − 0.417i·4-s − 0.0977i·5-s + 0.770i·6-s + 1.15·7-s + (−0.764 + 0.764i)8-s + 0.0201i·9-s + (−0.0527 + 0.0527i)10-s + (1.02 + 1.02i)11-s + (−0.298 + 0.298i)12-s − 0.772i·13-s + (−0.623 − 0.623i)14-s + (−0.0698 + 0.0698i)15-s + 0.408·16-s + (−0.570 − 0.570i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$29$$ Sign: $-0.229 + 0.973i$ Motivic weight: $$2$$ Character: $\chi_{29} (12, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 29,\ (\ :1),\ -0.229 + 0.973i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.416099 - 0.525572i$$ $$L(\frac12)$$ $$\approx$$ $$0.416099 - 0.525572i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad29 $$1 + (11.8 - 26.4i)T$$
good2 $$1 + (1.07 + 1.07i)T + 4iT^{2}$$
3 $$1 + (2.14 + 2.14i)T + 9iT^{2}$$
5 $$1 + 0.488iT - 25T^{2}$$
7 $$1 - 8.09T + 49T^{2}$$
11 $$1 + (-11.3 - 11.3i)T + 121iT^{2}$$
13 $$1 + 10.0iT - 169T^{2}$$
17 $$1 + (9.69 + 9.69i)T + 289iT^{2}$$
19 $$1 + (-8.58 - 8.58i)T + 361iT^{2}$$
23 $$1 + 14.4T + 529T^{2}$$
31 $$1 + (-31.2 - 31.2i)T + 961iT^{2}$$
37 $$1 + (31.2 - 31.2i)T - 1.36e3iT^{2}$$
41 $$1 + (-19.6 + 19.6i)T - 1.68e3iT^{2}$$
43 $$1 + (50.2 + 50.2i)T + 1.84e3iT^{2}$$
47 $$1 + (-42.2 + 42.2i)T - 2.20e3iT^{2}$$
53 $$1 - 16.5T + 2.80e3T^{2}$$
59 $$1 + 14.5T + 3.48e3T^{2}$$
61 $$1 + (45.3 + 45.3i)T + 3.72e3iT^{2}$$
67 $$1 - 133. iT - 4.48e3T^{2}$$
71 $$1 - 33.9iT - 5.04e3T^{2}$$
73 $$1 + (22.1 - 22.1i)T - 5.32e3iT^{2}$$
79 $$1 + (-9.72 - 9.72i)T + 6.24e3iT^{2}$$
83 $$1 - 64.2T + 6.88e3T^{2}$$
89 $$1 + (119. + 119. i)T + 7.92e3iT^{2}$$
97 $$1 + (33.5 - 33.5i)T - 9.40e3iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$