# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 23 \cdot 29$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 3-s + 0.138·5-s − 3.33·7-s + 9-s − 2.06·11-s − 0.702·13-s + 0.138·15-s + 4.32·17-s + 2.44·19-s − 3.33·21-s + 23-s − 4.98·25-s + 27-s + 29-s + 7.41·31-s − 2.06·33-s − 0.462·35-s − 2.90·37-s − 0.702·39-s − 4.05·41-s + 0.235·43-s + 0.138·45-s − 2.05·47-s + 4.13·49-s + 4.32·51-s − 3.22·53-s − 0.286·55-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.0619·5-s − 1.26·7-s + 0.333·9-s − 0.622·11-s − 0.194·13-s + 0.0357·15-s + 1.04·17-s + 0.560·19-s − 0.728·21-s + 0.208·23-s − 0.996·25-s + 0.192·27-s + 0.185·29-s + 1.33·31-s − 0.359·33-s − 0.0781·35-s − 0.477·37-s − 0.112·39-s − 0.633·41-s + 0.0358·43-s + 0.0206·45-s − 0.299·47-s + 0.590·49-s + 0.605·51-s − 0.442·53-s − 0.0386·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8004$$    =    $$2^{2} \cdot 3 \cdot 23 \cdot 29$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{8004} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 8004,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;23,\;29\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - T$$
23 $$1 - T$$
29 $$1 - T$$
good5 $$1 - 0.138T + 5T^{2}$$
7 $$1 + 3.33T + 7T^{2}$$
11 $$1 + 2.06T + 11T^{2}$$
13 $$1 + 0.702T + 13T^{2}$$
17 $$1 - 4.32T + 17T^{2}$$
19 $$1 - 2.44T + 19T^{2}$$
31 $$1 - 7.41T + 31T^{2}$$
37 $$1 + 2.90T + 37T^{2}$$
41 $$1 + 4.05T + 41T^{2}$$
43 $$1 - 0.235T + 43T^{2}$$
47 $$1 + 2.05T + 47T^{2}$$
53 $$1 + 3.22T + 53T^{2}$$
59 $$1 + 10.7T + 59T^{2}$$
61 $$1 + 2.78T + 61T^{2}$$
67 $$1 + 11.6T + 67T^{2}$$
71 $$1 - 7.47T + 71T^{2}$$
73 $$1 - 9.16T + 73T^{2}$$
79 $$1 - 11.0T + 79T^{2}$$
83 $$1 + 12.3T + 83T^{2}$$
89 $$1 - 3.05T + 89T^{2}$$
97 $$1 - 2.74T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}