# 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 + 1.18·2-s − 3-s − 0.588·4-s − 3.84·5-s − 1.18·6-s − 5.02·7-s − 3.07·8-s + 9-s − 4.56·10-s + 1.17·11-s + 0.588·12-s + 4.56·13-s − 5.96·14-s + 3.84·15-s − 2.47·16-s − 17-s + 1.18·18-s − 5.09·19-s + 2.26·20-s + 5.02·21-s + 1.39·22-s + 0.118·23-s + 3.07·24-s + 9.77·25-s + 5.41·26-s − 27-s + 2.95·28-s + ⋯
 L(s)  = 1 + 0.840·2-s − 0.577·3-s − 0.294·4-s − 1.71·5-s − 0.485·6-s − 1.89·7-s − 1.08·8-s + 0.333·9-s − 1.44·10-s + 0.354·11-s + 0.169·12-s + 1.26·13-s − 1.59·14-s + 0.992·15-s − 0.619·16-s − 0.242·17-s + 0.280·18-s − 1.16·19-s + 0.505·20-s + 1.09·21-s + 0.297·22-s + 0.0247·23-s + 0.627·24-s + 1.95·25-s + 1.06·26-s − 0.192·27-s + 0.558·28-s + ⋯

## Functional equation

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

## 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 - 1.18T + 2T^{2}$$
5 $$1 + 3.84T + 5T^{2}$$
7 $$1 + 5.02T + 7T^{2}$$
11 $$1 - 1.17T + 11T^{2}$$
13 $$1 - 4.56T + 13T^{2}$$
19 $$1 + 5.09T + 19T^{2}$$
23 $$1 - 0.118T + 23T^{2}$$
29 $$1 - 0.483T + 29T^{2}$$
31 $$1 - 2.83T + 31T^{2}$$
37 $$1 + 0.774T + 37T^{2}$$
41 $$1 - 1.91T + 41T^{2}$$
43 $$1 - 0.0806T + 43T^{2}$$
47 $$1 - 0.641T + 47T^{2}$$
53 $$1 - 1.13T + 53T^{2}$$
59 $$1 - 14.7T + 59T^{2}$$
61 $$1 - 4.40T + 61T^{2}$$
67 $$1 - 3.88T + 67T^{2}$$
71 $$1 + 2.48T + 71T^{2}$$
73 $$1 + 9.15T + 73T^{2}$$
79 $$1 - 0.193T + 79T^{2}$$
83 $$1 + 6.84T + 83T^{2}$$
89 $$1 + 4.55T + 89T^{2}$$
97 $$1 + 1.32T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}