# Properties

 Degree 2 Conductor 29 Sign $0.974 - 0.225i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.415 + 0.0467i)2-s + (2.68 + 1.68i)3-s + (−3.72 − 0.851i)4-s + (0.738 − 0.589i)5-s + (1.03 + 0.825i)6-s + (−0.577 − 2.53i)7-s + (−3.08 − 1.07i)8-s + (0.449 + 0.932i)9-s + (0.334 − 0.209i)10-s + (−8.65 + 3.02i)11-s + (−8.56 − 8.56i)12-s + (−4.51 + 9.37i)13-s + (−0.121 − 1.07i)14-s + (2.97 − 0.335i)15-s + (12.5 + 6.04i)16-s + (21.4 − 21.4i)17-s + ⋯
 L(s)  = 1 + (0.207 + 0.0233i)2-s + (0.894 + 0.561i)3-s + (−0.932 − 0.212i)4-s + (0.147 − 0.117i)5-s + (0.172 + 0.137i)6-s + (−0.0824 − 0.361i)7-s + (−0.385 − 0.134i)8-s + (0.0499 + 0.103i)9-s + (0.0334 − 0.0209i)10-s + (−0.786 + 0.275i)11-s + (−0.714 − 0.714i)12-s + (−0.347 + 0.720i)13-s + (−0.00866 − 0.0769i)14-s + (0.198 − 0.0223i)15-s + (0.784 + 0.377i)16-s + (1.26 − 1.26i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$29$$ $$\varepsilon$$ = $0.974 - 0.225i$ motivic weight = $$2$$ character : $\chi_{29} (2, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 29,\ (\ :1),\ 0.974 - 0.225i)$ $L(\frac{3}{2})$ $\approx$ $1.11334 + 0.127015i$ $L(\frac12)$ $\approx$ $1.11334 + 0.127015i$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 29$, $$F_p$$ is a polynomial of degree 2. If $p = 29$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad29 $$1 + (24.3 + 15.7i)T$$
good2 $$1 + (-0.415 - 0.0467i)T + (3.89 + 0.890i)T^{2}$$
3 $$1 + (-2.68 - 1.68i)T + (3.90 + 8.10i)T^{2}$$
5 $$1 + (-0.738 + 0.589i)T + (5.56 - 24.3i)T^{2}$$
7 $$1 + (0.577 + 2.53i)T + (-44.1 + 21.2i)T^{2}$$
11 $$1 + (8.65 - 3.02i)T + (94.6 - 75.4i)T^{2}$$
13 $$1 + (4.51 - 9.37i)T + (-105. - 132. i)T^{2}$$
17 $$1 + (-21.4 + 21.4i)T - 289iT^{2}$$
19 $$1 + (-14.2 - 22.6i)T + (-156. + 325. i)T^{2}$$
23 $$1 + (2.27 - 2.85i)T + (-117. - 515. i)T^{2}$$
31 $$1 + (-21.7 - 2.44i)T + (936. + 213. i)T^{2}$$
37 $$1 + (46.0 + 16.1i)T + (1.07e3 + 853. i)T^{2}$$
41 $$1 + (25.3 + 25.3i)T + 1.68e3iT^{2}$$
43 $$1 + (-2.78 - 24.6i)T + (-1.80e3 + 411. i)T^{2}$$
47 $$1 + (-7.49 - 21.4i)T + (-1.72e3 + 1.37e3i)T^{2}$$
53 $$1 + (18.6 + 23.3i)T + (-625. + 2.73e3i)T^{2}$$
59 $$1 - 18.2T + 3.48e3T^{2}$$
61 $$1 + (78.8 + 49.5i)T + (1.61e3 + 3.35e3i)T^{2}$$
67 $$1 + (-24.5 - 50.9i)T + (-2.79e3 + 3.50e3i)T^{2}$$
71 $$1 + (-56.1 + 116. i)T + (-3.14e3 - 3.94e3i)T^{2}$$
73 $$1 + (-47.2 + 5.32i)T + (5.19e3 - 1.18e3i)T^{2}$$
79 $$1 + (13.7 - 39.2i)T + (-4.87e3 - 3.89e3i)T^{2}$$
83 $$1 + (29.4 - 129. i)T + (-6.20e3 - 2.98e3i)T^{2}$$
89 $$1 + (-32.3 - 3.64i)T + (7.72e3 + 1.76e3i)T^{2}$$
97 $$1 + (36.1 - 22.6i)T + (4.08e3 - 8.47e3i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}