# Properties

 Degree 2 Conductor $2 \cdot 7$ Sign $0.605 - 0.795i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + 1.73i)2-s + (2.5 + 4.33i)3-s + (−1.99 − 3.46i)4-s + (4.5 − 7.79i)5-s − 10·6-s + (−14 − 12.1i)7-s + 7.99·8-s + (0.999 − 1.73i)9-s + (9 + 15.5i)10-s + (28.5 + 49.3i)11-s + (10 − 17.3i)12-s − 70·13-s + (35 − 12.1i)14-s + 45.0·15-s + (−8 + 13.8i)16-s + (−25.5 − 44.1i)17-s + ⋯
 L(s)  = 1 + (−0.353 + 0.612i)2-s + (0.481 + 0.833i)3-s + (−0.249 − 0.433i)4-s + (0.402 − 0.697i)5-s − 0.680·6-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (0.0370 − 0.0641i)9-s + (0.284 + 0.492i)10-s + (0.781 + 1.35i)11-s + (0.240 − 0.416i)12-s − 1.49·13-s + (0.668 − 0.231i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$14$$    =    $$2 \cdot 7$$ $$\varepsilon$$ = $0.605 - 0.795i$ motivic weight = $$3$$ character : $\chi_{14} (9, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 14,\ (\ :3/2),\ 0.605 - 0.795i)$ $L(2)$ $\approx$ $0.822479 + 0.407725i$ $L(\frac12)$ $\approx$ $0.822479 + 0.407725i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (1 - 1.73i)T$$
7 $$1 + (14 + 12.1i)T$$
good3 $$1 + (-2.5 - 4.33i)T + (-13.5 + 23.3i)T^{2}$$
5 $$1 + (-4.5 + 7.79i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (-28.5 - 49.3i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + 70T + 2.19e3T^{2}$$
17 $$1 + (25.5 + 44.1i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (2.5 - 4.33i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (34.5 - 59.7i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 114T + 2.43e4T^{2}$$
31 $$1 + (11.5 + 19.9i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (-126.5 + 219. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + 42T + 6.89e4T^{2}$$
43 $$1 + 124T + 7.95e4T^{2}$$
47 $$1 + (100.5 - 174. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (-196.5 - 340. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (109.5 + 189. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-354.5 + 614. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (209.5 + 362. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 96T + 3.57e5T^{2}$$
73 $$1 + (-156.5 - 271. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (230.5 - 399. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + 588T + 5.71e5T^{2}$$
89 $$1 + (-508.5 + 880. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + 1.83e3T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}