# Properties

 Degree 2 Conductor $2^{4} \cdot 251$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.32·3-s − 0.316·5-s − 0.268·7-s + 8.06·9-s + 0.366·11-s − 3.69·13-s + 1.05·15-s + 5.91·17-s − 7.61·19-s + 0.891·21-s + 6.61·23-s − 4.89·25-s − 16.8·27-s + 5.48·29-s + 4.15·31-s − 1.21·33-s + 0.0849·35-s + 8.30·37-s + 12.2·39-s + 3.92·41-s − 6.01·43-s − 2.55·45-s − 1.79·47-s − 6.92·49-s − 19.6·51-s − 10.5·53-s − 0.116·55-s + ⋯
 L(s)  = 1 − 1.92·3-s − 0.141·5-s − 0.101·7-s + 2.68·9-s + 0.110·11-s − 1.02·13-s + 0.272·15-s + 1.43·17-s − 1.74·19-s + 0.194·21-s + 1.37·23-s − 0.979·25-s − 3.23·27-s + 1.01·29-s + 0.746·31-s − 0.212·33-s + 0.0143·35-s + 1.36·37-s + 1.96·39-s + 0.612·41-s − 0.917·43-s − 0.380·45-s − 0.261·47-s − 0.989·49-s − 2.75·51-s − 1.44·53-s − 0.0156·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4016$$    =    $$2^{4} \cdot 251$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{4016} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 4016,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.6850768891$ $L(\frac12)$ $\approx$ $0.6850768891$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;251\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
251 $$1 - T$$
good3 $$1 + 3.32T + 3T^{2}$$
5 $$1 + 0.316T + 5T^{2}$$
7 $$1 + 0.268T + 7T^{2}$$
11 $$1 - 0.366T + 11T^{2}$$
13 $$1 + 3.69T + 13T^{2}$$
17 $$1 - 5.91T + 17T^{2}$$
19 $$1 + 7.61T + 19T^{2}$$
23 $$1 - 6.61T + 23T^{2}$$
29 $$1 - 5.48T + 29T^{2}$$
31 $$1 - 4.15T + 31T^{2}$$
37 $$1 - 8.30T + 37T^{2}$$
41 $$1 - 3.92T + 41T^{2}$$
43 $$1 + 6.01T + 43T^{2}$$
47 $$1 + 1.79T + 47T^{2}$$
53 $$1 + 10.5T + 53T^{2}$$
59 $$1 - 3.99T + 59T^{2}$$
61 $$1 - 1.88T + 61T^{2}$$
67 $$1 + 1.28T + 67T^{2}$$
71 $$1 + 8.15T + 71T^{2}$$
73 $$1 + 16.0T + 73T^{2}$$
79 $$1 + 7.93T + 79T^{2}$$
83 $$1 - 11.7T + 83T^{2}$$
89 $$1 + 1.39T + 89T^{2}$$
97 $$1 + 9.53T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}