Properties

 Degree 2 Conductor 29 Sign $0.357 + 0.934i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−1.12 − 1.40i)2-s + (−0.0990 − 0.433i)3-s + (−0.277 + 1.21i)4-s + (0.222 + 0.279i)5-s + (−0.500 + 0.626i)6-s + (0.900 + 3.94i)7-s + (−1.22 + 0.588i)8-s + (2.52 − 1.21i)9-s + (0.143 − 0.626i)10-s + (−2.62 − 1.26i)11-s + 0.554·12-s + (−4.67 − 2.25i)13-s + (4.54 − 5.70i)14-s + (0.0990 − 0.124i)15-s + (4.44 + 2.14i)16-s + 1.10·17-s + ⋯
 L(s)  = 1 + (−0.794 − 0.996i)2-s + (−0.0571 − 0.250i)3-s + (−0.138 + 0.607i)4-s + (0.0995 + 0.124i)5-s + (−0.204 + 0.255i)6-s + (0.340 + 1.49i)7-s + (−0.432 + 0.208i)8-s + (0.841 − 0.405i)9-s + (0.0452 − 0.198i)10-s + (−0.791 − 0.380i)11-s + 0.160·12-s + (−1.29 − 0.624i)13-s + (1.21 − 1.52i)14-s + (0.0255 − 0.0320i)15-s + (1.11 + 0.535i)16-s + 0.269·17-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$29$$ $$\varepsilon$$ = $0.357 + 0.934i$ motivic weight = $$1$$ character : $\chi_{29} (24, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 29,\ (\ :1/2),\ 0.357 + 0.934i)$ $L(1)$ $\approx$ $0.411375 - 0.283092i$ $L(\frac12)$ $\approx$ $0.411375 - 0.283092i$ $L(\frac{3}{2})$ 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 + (-4.38 - 3.12i)T$$
good2 $$1 + (1.12 + 1.40i)T + (-0.445 + 1.94i)T^{2}$$
3 $$1 + (0.0990 + 0.433i)T + (-2.70 + 1.30i)T^{2}$$
5 $$1 + (-0.222 - 0.279i)T + (-1.11 + 4.87i)T^{2}$$
7 $$1 + (-0.900 - 3.94i)T + (-6.30 + 3.03i)T^{2}$$
11 $$1 + (2.62 + 1.26i)T + (6.85 + 8.60i)T^{2}$$
13 $$1 + (4.67 + 2.25i)T + (8.10 + 10.1i)T^{2}$$
17 $$1 - 1.10T + 17T^{2}$$
19 $$1 + (0.455 - 1.99i)T + (-17.1 - 8.24i)T^{2}$$
23 $$1 + (2.57 - 3.23i)T + (-5.11 - 22.4i)T^{2}$$
31 $$1 + (3.96 + 4.97i)T + (-6.89 + 30.2i)T^{2}$$
37 $$1 + (-2.62 + 1.26i)T + (23.0 - 28.9i)T^{2}$$
41 $$1 - 0.396T + 41T^{2}$$
43 $$1 + (-3.57 + 4.48i)T + (-9.56 - 41.9i)T^{2}$$
47 $$1 + (-7.02 - 3.38i)T + (29.3 + 36.7i)T^{2}$$
53 $$1 + (2.71 + 3.40i)T + (-11.7 + 51.6i)T^{2}$$
59 $$1 + 9.10T + 59T^{2}$$
61 $$1 + (-1.34 - 5.89i)T + (-54.9 + 26.4i)T^{2}$$
67 $$1 + (0.337 - 0.162i)T + (41.7 - 52.3i)T^{2}$$
71 $$1 + (-10.2 - 4.94i)T + (44.2 + 55.5i)T^{2}$$
73 $$1 + (5.57 - 6.99i)T + (-16.2 - 71.1i)T^{2}$$
79 $$1 + (-0.535 + 0.257i)T + (49.2 - 61.7i)T^{2}$$
83 $$1 + (-2.09 + 9.19i)T + (-74.7 - 36.0i)T^{2}$$
89 $$1 + (-0.887 - 1.11i)T + (-19.8 + 86.7i)T^{2}$$
97 $$1 + (3.50 - 15.3i)T + (-87.3 - 42.0i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}