# Properties

 Degree 2 Conductor $3 \cdot 17 \cdot 157$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.206·2-s − 3-s − 1.95·4-s − 2.96·5-s + 0.206·6-s − 3.29·7-s + 0.817·8-s + 9-s + 0.612·10-s − 2.02·11-s + 1.95·12-s − 4.52·13-s + 0.681·14-s + 2.96·15-s + 3.74·16-s − 17-s − 0.206·18-s + 2.84·19-s + 5.80·20-s + 3.29·21-s + 0.418·22-s − 6.15·23-s − 0.817·24-s + 3.78·25-s + 0.934·26-s − 27-s + 6.45·28-s + ⋯
 L(s)  = 1 − 0.146·2-s − 0.577·3-s − 0.978·4-s − 1.32·5-s + 0.0843·6-s − 1.24·7-s + 0.289·8-s + 0.333·9-s + 0.193·10-s − 0.610·11-s + 0.565·12-s − 1.25·13-s + 0.182·14-s + 0.765·15-s + 0.936·16-s − 0.242·17-s − 0.0486·18-s + 0.652·19-s + 1.29·20-s + 0.719·21-s + 0.0891·22-s − 1.28·23-s − 0.166·24-s + 0.756·25-s + 0.183·26-s − 0.192·27-s + 1.22·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8007$$    =    $$3 \cdot 17 \cdot 157$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{8007} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 8007,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;17,\;157\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + T$$
17 $$1 + T$$
157 $$1 + T$$
good2 $$1 + 0.206T + 2T^{2}$$
5 $$1 + 2.96T + 5T^{2}$$
7 $$1 + 3.29T + 7T^{2}$$
11 $$1 + 2.02T + 11T^{2}$$
13 $$1 + 4.52T + 13T^{2}$$
19 $$1 - 2.84T + 19T^{2}$$
23 $$1 + 6.15T + 23T^{2}$$
29 $$1 - 3.35T + 29T^{2}$$
31 $$1 + 3.97T + 31T^{2}$$
37 $$1 + 6.02T + 37T^{2}$$
41 $$1 + 5.50T + 41T^{2}$$
43 $$1 + 0.805T + 43T^{2}$$
47 $$1 - 2.71T + 47T^{2}$$
53 $$1 - 7.81T + 53T^{2}$$
59 $$1 - 0.000491T + 59T^{2}$$
61 $$1 - 10.2T + 61T^{2}$$
67 $$1 - 6.98T + 67T^{2}$$
71 $$1 - 12.7T + 71T^{2}$$
73 $$1 + 0.595T + 73T^{2}$$
79 $$1 - 10.0T + 79T^{2}$$
83 $$1 + 10.2T + 83T^{2}$$
89 $$1 - 6.31T + 89T^{2}$$
97 $$1 - 6.62T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}