# Properties

 Label 2-29-29.2-c8-0-13 Degree $2$ Conductor $29$ Sign $0.635 + 0.772i$ Analytic cond. $11.8139$ Root an. cond. $3.43714$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (14.6 + 1.65i)2-s + (1.69 + 1.06i)3-s + (−36.4 − 8.32i)4-s + (784. − 625. i)5-s + (23.1 + 18.4i)6-s + (313. + 1.37e3i)7-s + (−4.09e3 − 1.43e3i)8-s + (−2.84e3 − 5.90e3i)9-s + (1.25e4 − 7.88e3i)10-s + (2.25e4 − 7.90e3i)11-s + (−52.9 − 52.9i)12-s + (7.33e3 − 1.52e4i)13-s + (2.33e3 + 2.07e4i)14-s + (1.99e3 − 224. i)15-s + (−4.91e4 − 2.36e4i)16-s + (5.00e4 − 5.00e4i)17-s + ⋯
 L(s)  = 1 + (0.918 + 0.103i)2-s + (0.0209 + 0.0131i)3-s + (−0.142 − 0.0325i)4-s + (1.25 − 1.00i)5-s + (0.0178 + 0.0142i)6-s + (0.130 + 0.572i)7-s + (−0.999 − 0.349i)8-s + (−0.433 − 0.900i)9-s + (1.25 − 0.788i)10-s + (1.54 − 0.539i)11-s + (−0.00255 − 0.00255i)12-s + (0.256 − 0.533i)13-s + (0.0607 + 0.538i)14-s + (0.0394 − 0.00444i)15-s + (−0.750 − 0.361i)16-s + (0.599 − 0.599i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$29$$ Sign: $0.635 + 0.772i$ Analytic conductor: $$11.8139$$ Root analytic conductor: $$3.43714$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{29} (2, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 29,\ (\ :4),\ 0.635 + 0.772i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$2.72709 - 1.28770i$$ $$L(\frac12)$$ $$\approx$$ $$2.72709 - 1.28770i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad29 $$1 + (8.10e4 - 7.02e5i)T$$
good2 $$1 + (-14.6 - 1.65i)T + (249. + 56.9i)T^{2}$$
3 $$1 + (-1.69 - 1.06i)T + (2.84e3 + 5.91e3i)T^{2}$$
5 $$1 + (-784. + 625. i)T + (8.69e4 - 3.80e5i)T^{2}$$
7 $$1 + (-313. - 1.37e3i)T + (-5.19e6 + 2.50e6i)T^{2}$$
11 $$1 + (-2.25e4 + 7.90e3i)T + (1.67e8 - 1.33e8i)T^{2}$$
13 $$1 + (-7.33e3 + 1.52e4i)T + (-5.08e8 - 6.37e8i)T^{2}$$
17 $$1 + (-5.00e4 + 5.00e4i)T - 6.97e9iT^{2}$$
19 $$1 + (-6.28e4 - 9.99e4i)T + (-7.36e9 + 1.53e10i)T^{2}$$
23 $$1 + (1.16e5 - 1.46e5i)T + (-1.74e10 - 7.63e10i)T^{2}$$
31 $$1 + (1.40e6 + 1.58e5i)T + (8.31e11 + 1.89e11i)T^{2}$$
37 $$1 + (1.81e6 + 6.36e5i)T + (2.74e12 + 2.18e12i)T^{2}$$
41 $$1 + (-4.23e5 - 4.23e5i)T + 7.98e12iT^{2}$$
43 $$1 + (-5.96e5 - 5.29e6i)T + (-1.13e13 + 2.60e12i)T^{2}$$
47 $$1 + (-1.80e6 - 5.15e6i)T + (-1.86e13 + 1.48e13i)T^{2}$$
53 $$1 + (-2.74e6 - 3.44e6i)T + (-1.38e13 + 6.06e13i)T^{2}$$
59 $$1 - 2.07e7T + 1.46e14T^{2}$$
61 $$1 + (1.56e7 + 9.81e6i)T + (8.31e13 + 1.72e14i)T^{2}$$
67 $$1 + (-4.64e5 - 9.65e5i)T + (-2.53e14 + 3.17e14i)T^{2}$$
71 $$1 + (-6.07e6 + 1.26e7i)T + (-4.02e14 - 5.04e14i)T^{2}$$
73 $$1 + (1.06e6 - 1.19e5i)T + (7.86e14 - 1.79e14i)T^{2}$$
79 $$1 + (1.13e7 - 3.24e7i)T + (-1.18e15 - 9.45e14i)T^{2}$$
83 $$1 + (-9.48e6 + 4.15e7i)T + (-2.02e15 - 9.77e14i)T^{2}$$
89 $$1 + (-9.20e7 - 1.03e7i)T + (3.83e15 + 8.75e14i)T^{2}$$
97 $$1 + (-9.91e7 + 6.22e7i)T + (3.40e15 - 7.06e15i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$