# Properties

 Degree 2 Conductor 7 Sign $0.968 - 0.250i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + 1.73i)2-s + (−3.5 − 6.06i)3-s + (2.00 + 3.46i)4-s + (−3.5 + 6.06i)5-s + 14·6-s + (14 − 12.1i)7-s − 24·8-s + (−11 + 19.0i)9-s + (−7 − 12.1i)10-s + (2.5 + 4.33i)11-s + (13.9 − 24.2i)12-s − 14·13-s + (7 + 36.3i)14-s + 49·15-s + (8.00 − 13.8i)16-s + (10.5 + 18.1i)17-s + ⋯
 L(s)  = 1 + (−0.353 + 0.612i)2-s + (−0.673 − 1.16i)3-s + (0.250 + 0.433i)4-s + (−0.313 + 0.542i)5-s + 0.952·6-s + (0.755 − 0.654i)7-s − 1.06·8-s + (−0.407 + 0.705i)9-s + (−0.221 − 0.383i)10-s + (0.0685 + 0.118i)11-s + (0.336 − 0.583i)12-s − 0.298·13-s + (0.133 + 0.694i)14-s + 0.843·15-s + (0.125 − 0.216i)16-s + (0.149 + 0.259i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$7$$ $$\varepsilon$$ = $0.968 - 0.250i$ motivic weight = $$3$$ character : $\chi_{7} (2, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 7,\ (\ :3/2),\ 0.968 - 0.250i)$ $L(2)$ $\approx$ $0.592926 + 0.0755587i$ $L(\frac12)$ $\approx$ $0.592926 + 0.0755587i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 7$, $$F_p$$ is a polynomial of degree 2. If $p = 7$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 $$1 + (-14 + 12.1i)T$$
good2 $$1 + (1 - 1.73i)T + (-4 - 6.92i)T^{2}$$
3 $$1 + (3.5 + 6.06i)T + (-13.5 + 23.3i)T^{2}$$
5 $$1 + (3.5 - 6.06i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (-2.5 - 4.33i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + 14T + 2.19e3T^{2}$$
17 $$1 + (-10.5 - 18.1i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (24.5 - 42.4i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-79.5 + 137. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 58T + 2.43e4T^{2}$$
31 $$1 + (73.5 + 127. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (109.5 - 189. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 - 350T + 6.89e4T^{2}$$
43 $$1 + 124T + 7.95e4T^{2}$$
47 $$1 + (262.5 - 454. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (151.5 + 262. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-52.5 - 90.9i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-206.5 + 357. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (207.5 + 359. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 432T + 3.57e5T^{2}$$
73 $$1 + (-556.5 - 963. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-51.5 + 89.2i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 - 1.09e3T + 5.71e5T^{2}$$
89 $$1 + (-164.5 + 284. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + 882T + 9.12e5T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}