# Properties

 Degree 2 Conductor 13 Sign $0.973 + 0.230i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.219 − 0.379i)2-s + (1.84 − 3.19i)3-s + (3.90 + 6.76i)4-s − 17.8·5-s + (−0.807 − 1.39i)6-s + (−2.71 − 4.70i)7-s + 6.93·8-s + (6.71 + 11.6i)9-s + (−3.90 + 6.76i)10-s + (11.2 − 19.4i)11-s + 28.7·12-s + (21.9 − 41.4i)13-s − 2.38·14-s + (−32.8 + 56.8i)15-s + (−29.7 + 51.4i)16-s + (−33.9 − 58.8i)17-s + ⋯
 L(s)  = 1 + (0.0775 − 0.134i)2-s + (0.354 − 0.614i)3-s + (0.487 + 0.845i)4-s − 1.59·5-s + (−0.0549 − 0.0951i)6-s + (−0.146 − 0.254i)7-s + 0.306·8-s + (0.248 + 0.430i)9-s + (−0.123 + 0.213i)10-s + (0.307 − 0.532i)11-s + 0.692·12-s + (0.468 − 0.883i)13-s − 0.0455·14-s + (−0.564 + 0.978i)15-s + (−0.464 + 0.804i)16-s + (−0.484 − 0.839i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$13$$ $$\varepsilon$$ = $0.973 + 0.230i$ motivic weight = $$3$$ character : $\chi_{13} (9, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 13,\ (\ :3/2),\ 0.973 + 0.230i)$ $L(2)$ $\approx$ $0.982027 - 0.114691i$ $L(\frac12)$ $\approx$ $0.982027 - 0.114691i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 13$, $$F_p$$ is a polynomial of degree 2. If $p = 13$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad13 $$1 + (-21.9 + 41.4i)T$$
good2 $$1 + (-0.219 + 0.379i)T + (-4 - 6.92i)T^{2}$$
3 $$1 + (-1.84 + 3.19i)T + (-13.5 - 23.3i)T^{2}$$
5 $$1 + 17.8T + 125T^{2}$$
7 $$1 + (2.71 + 4.70i)T + (-171.5 + 297. i)T^{2}$$
11 $$1 + (-11.2 + 19.4i)T + (-665.5 - 1.15e3i)T^{2}$$
17 $$1 + (33.9 + 58.8i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-40.4 - 69.9i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (70.2 - 121. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (-53.3 + 92.3i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + 276.T + 2.97e4T^{2}$$
37 $$1 + (-2.14 + 3.71i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + (113. - 197. i)T + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (13.7 + 23.8i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 - 318.T + 1.03e5T^{2}$$
53 $$1 + 67.6T + 1.48e5T^{2}$$
59 $$1 + (-145. - 252. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (331. + 574. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-212. + 368. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + (-76.4 - 132. i)T + (-1.78e5 + 3.09e5i)T^{2}$$
73 $$1 - 117.T + 3.89e5T^{2}$$
79 $$1 - 202.T + 4.93e5T^{2}$$
83 $$1 - 336.T + 5.71e5T^{2}$$
89 $$1 + (359. - 621. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + (379. + 657. i)T + (-4.56e5 + 7.90e5i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}