# Properties

 Degree 2 Conductor 29 Sign $0.957 - 0.289i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.45 + 1.45i)2-s + (3.81 − 3.81i)3-s − 0.234i·4-s + 3.14i·5-s + 11.0i·6-s + 0.342·7-s + (−5.47 − 5.47i)8-s − 20.0i·9-s + (−4.57 − 4.57i)10-s + (−14.0 + 14.0i)11-s + (−0.895 − 0.895i)12-s − 11.5i·13-s + (−0.498 + 0.498i)14-s + (11.9 + 11.9i)15-s + 16.8·16-s + (−6.37 + 6.37i)17-s + ⋯
 L(s)  = 1 + (−0.727 + 0.727i)2-s + (1.27 − 1.27i)3-s − 0.0587i·4-s + 0.629i·5-s + 1.84i·6-s + 0.0489·7-s + (−0.684 − 0.684i)8-s − 2.22i·9-s + (−0.457 − 0.457i)10-s + (−1.27 + 1.27i)11-s + (−0.0746 − 0.0746i)12-s − 0.885i·13-s + (−0.0355 + 0.0355i)14-s + (0.799 + 0.799i)15-s + 1.05·16-s + (−0.374 + 0.374i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$29$$ $$\varepsilon$$ = $0.957 - 0.289i$ motivic weight = $$2$$ character : $\chi_{29} (17, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 29,\ (\ :1),\ 0.957 - 0.289i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.909737 + 0.134355i$$ $$L(\frac12)$$ $$\approx$$ $$0.909737 + 0.134355i$$ $$L(2)$$ 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 + (-25.6 + 13.4i)T$$
good2 $$1 + (1.45 - 1.45i)T - 4iT^{2}$$
3 $$1 + (-3.81 + 3.81i)T - 9iT^{2}$$
5 $$1 - 3.14iT - 25T^{2}$$
7 $$1 - 0.342T + 49T^{2}$$
11 $$1 + (14.0 - 14.0i)T - 121iT^{2}$$
13 $$1 + 11.5iT - 169T^{2}$$
17 $$1 + (6.37 - 6.37i)T - 289iT^{2}$$
19 $$1 + (6.11 - 6.11i)T - 361iT^{2}$$
23 $$1 - 15.8T + 529T^{2}$$
31 $$1 + (-6.18 + 6.18i)T - 961iT^{2}$$
37 $$1 + (-24.2 - 24.2i)T + 1.36e3iT^{2}$$
41 $$1 + (34.9 + 34.9i)T + 1.68e3iT^{2}$$
43 $$1 + (15.9 - 15.9i)T - 1.84e3iT^{2}$$
47 $$1 + (-7.57 - 7.57i)T + 2.20e3iT^{2}$$
53 $$1 - 62.1T + 2.80e3T^{2}$$
59 $$1 + 21.4T + 3.48e3T^{2}$$
61 $$1 + (-13.1 + 13.1i)T - 3.72e3iT^{2}$$
67 $$1 - 17.8iT - 4.48e3T^{2}$$
71 $$1 + 53.0iT - 5.04e3T^{2}$$
73 $$1 + (-5.93 - 5.93i)T + 5.32e3iT^{2}$$
79 $$1 + (81.6 - 81.6i)T - 6.24e3iT^{2}$$
83 $$1 + 77.1T + 6.88e3T^{2}$$
89 $$1 + (-27.4 + 27.4i)T - 7.92e3iT^{2}$$
97 $$1 + (-48.7 - 48.7i)T + 9.40e3iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}