Properties

 Degree $2$ Conductor $8004$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

Related objects

Dirichlet series

 L(s)  = 1 − 3-s + 0.900·5-s + 3.90·7-s + 9-s − 5.39·11-s − 2.18·13-s − 0.900·15-s − 2.64·17-s + 3.37·19-s − 3.90·21-s − 23-s − 4.18·25-s − 27-s + 29-s + 5.08·31-s + 5.39·33-s + 3.51·35-s − 3.79·37-s + 2.18·39-s + 8.82·41-s + 0.613·43-s + 0.900·45-s + 1.02·47-s + 8.24·49-s + 2.64·51-s + 5.94·53-s − 4.86·55-s + ⋯
 L(s)  = 1 − 0.577·3-s + 0.402·5-s + 1.47·7-s + 0.333·9-s − 1.62·11-s − 0.606·13-s − 0.232·15-s − 0.641·17-s + 0.773·19-s − 0.852·21-s − 0.208·23-s − 0.837·25-s − 0.192·27-s + 0.185·29-s + 0.913·31-s + 0.939·33-s + 0.594·35-s − 0.623·37-s + 0.349·39-s + 1.37·41-s + 0.0935·43-s + 0.134·45-s + 0.149·47-s + 1.17·49-s + 0.370·51-s + 0.816·53-s − 0.655·55-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

 Degree: $$2$$ Conductor: $$8004$$    =    $$2^{2} \cdot 3 \cdot 23 \cdot 29$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{8004} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 8004,\ (\ :1/2),\ -1)$$

Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + T$$
23 $$1 + T$$
29 $$1 - T$$
good5 $$1 - 0.900T + 5T^{2}$$
7 $$1 - 3.90T + 7T^{2}$$
11 $$1 + 5.39T + 11T^{2}$$
13 $$1 + 2.18T + 13T^{2}$$
17 $$1 + 2.64T + 17T^{2}$$
19 $$1 - 3.37T + 19T^{2}$$
31 $$1 - 5.08T + 31T^{2}$$
37 $$1 + 3.79T + 37T^{2}$$
41 $$1 - 8.82T + 41T^{2}$$
43 $$1 - 0.613T + 43T^{2}$$
47 $$1 - 1.02T + 47T^{2}$$
53 $$1 - 5.94T + 53T^{2}$$
59 $$1 + 10.0T + 59T^{2}$$
61 $$1 + 3.09T + 61T^{2}$$
67 $$1 + 12.6T + 67T^{2}$$
71 $$1 + 4.69T + 71T^{2}$$
73 $$1 - 9.03T + 73T^{2}$$
79 $$1 + 3.30T + 79T^{2}$$
83 $$1 + 1.01T + 83T^{2}$$
89 $$1 + 17.4T + 89T^{2}$$
97 $$1 - 4.91T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$