# Properties

 Label 2-29-29.16-c5-0-7 Degree $2$ Conductor $29$ Sign $0.974 + 0.224i$ Analytic cond. $4.65113$ Root an. cond. $2.15664$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.98 + 1.91i)2-s + (6.52 − 8.18i)3-s + (−7.76 − 9.73i)4-s + (64.0 + 30.8i)5-s + (41.6 − 20.0i)6-s + (85.0 − 106. i)7-s + (−43.7 − 191. i)8-s + (29.6 + 130. i)9-s + (196. + 245. i)10-s + (−58.7 + 257. i)11-s − 130.·12-s + (2.84 − 12.4i)13-s + (543. − 261. i)14-s + (670. − 323. i)15-s + (104. − 458. i)16-s − 2.14e3·17-s + ⋯
 L(s)  = 1 + (0.704 + 0.339i)2-s + (0.418 − 0.525i)3-s + (−0.242 − 0.304i)4-s + (1.14 + 0.552i)5-s + (0.472 − 0.227i)6-s + (0.656 − 0.822i)7-s + (−0.241 − 1.05i)8-s + (0.122 + 0.535i)9-s + (0.620 + 0.777i)10-s + (−0.146 + 0.641i)11-s − 0.261·12-s + (0.00467 − 0.0204i)13-s + (0.740 − 0.356i)14-s + (0.769 − 0.370i)15-s + (0.102 − 0.447i)16-s − 1.80·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$29$$ Sign: $0.974 + 0.224i$ Analytic conductor: $$4.65113$$ Root analytic conductor: $$2.15664$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{29} (16, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 29,\ (\ :5/2),\ 0.974 + 0.224i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.52469 - 0.286535i$$ $$L(\frac12)$$ $$\approx$$ $$2.52469 - 0.286535i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad29 $$1 + (3.25e3 + 3.15e3i)T$$
good2 $$1 + (-3.98 - 1.91i)T + (19.9 + 25.0i)T^{2}$$
3 $$1 + (-6.52 + 8.18i)T + (-54.0 - 236. i)T^{2}$$
5 $$1 + (-64.0 - 30.8i)T + (1.94e3 + 2.44e3i)T^{2}$$
7 $$1 + (-85.0 + 106. i)T + (-3.73e3 - 1.63e4i)T^{2}$$
11 $$1 + (58.7 - 257. i)T + (-1.45e5 - 6.98e4i)T^{2}$$
13 $$1 + (-2.84 + 12.4i)T + (-3.34e5 - 1.61e5i)T^{2}$$
17 $$1 + 2.14e3T + 1.41e6T^{2}$$
19 $$1 + (-721. - 904. i)T + (-5.50e5 + 2.41e6i)T^{2}$$
23 $$1 + (2.67e3 - 1.28e3i)T + (4.01e6 - 5.03e6i)T^{2}$$
31 $$1 + (-2.73e3 - 1.31e3i)T + (1.78e7 + 2.23e7i)T^{2}$$
37 $$1 + (-755. - 3.30e3i)T + (-6.24e7 + 3.00e7i)T^{2}$$
41 $$1 - 1.70e4T + 1.15e8T^{2}$$
43 $$1 + (-1.22e4 + 5.88e3i)T + (9.16e7 - 1.14e8i)T^{2}$$
47 $$1 + (2.58e3 - 1.13e4i)T + (-2.06e8 - 9.95e7i)T^{2}$$
53 $$1 + (2.92e4 + 1.40e4i)T + (2.60e8 + 3.26e8i)T^{2}$$
59 $$1 + 1.24e4T + 7.14e8T^{2}$$
61 $$1 + (-1.48e4 + 1.86e4i)T + (-1.87e8 - 8.23e8i)T^{2}$$
67 $$1 + (1.06e4 + 4.66e4i)T + (-1.21e9 + 5.85e8i)T^{2}$$
71 $$1 + (-6.51e3 + 2.85e4i)T + (-1.62e9 - 7.82e8i)T^{2}$$
73 $$1 + (9.95e3 - 4.79e3i)T + (1.29e9 - 1.62e9i)T^{2}$$
79 $$1 + (1.06e4 + 4.68e4i)T + (-2.77e9 + 1.33e9i)T^{2}$$
83 $$1 + (-1.29e4 - 1.62e4i)T + (-8.76e8 + 3.84e9i)T^{2}$$
89 $$1 + (2.15e4 + 1.03e4i)T + (3.48e9 + 4.36e9i)T^{2}$$
97 $$1 + (4.86e4 + 6.09e4i)T + (-1.91e9 + 8.37e9i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$