# Properties

 Label 2-4002-1.1-c1-0-16 Degree $2$ Conductor $4002$ Sign $-1$ Analytic cond. $31.9561$ Root an. cond. $5.65297$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s − 3.83·5-s + 6-s − 4.91·7-s − 8-s + 9-s + 3.83·10-s − 1.87·11-s − 12-s − 3.25·13-s + 4.91·14-s + 3.83·15-s + 16-s + 5.70·17-s − 18-s + 0.533·19-s − 3.83·20-s + 4.91·21-s + 1.87·22-s + 23-s + 24-s + 9.66·25-s + 3.25·26-s − 27-s − 4.91·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.71·5-s + 0.408·6-s − 1.85·7-s − 0.353·8-s + 0.333·9-s + 1.21·10-s − 0.565·11-s − 0.288·12-s − 0.902·13-s + 1.31·14-s + 0.988·15-s + 0.250·16-s + 1.38·17-s − 0.235·18-s + 0.122·19-s − 0.856·20-s + 1.07·21-s + 0.399·22-s + 0.208·23-s + 0.204·24-s + 1.93·25-s + 0.638·26-s − 0.192·27-s − 0.928·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4002$$    =    $$2 \cdot 3 \cdot 23 \cdot 29$$ Sign: $-1$ Analytic conductor: $$31.9561$$ Root analytic conductor: $$5.65297$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 4002,\ (\ :1/2),\ -1)$$

## 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 + T$$
3 $$1 + T$$
23 $$1 - T$$
29 $$1 - T$$
good5 $$1 + 3.83T + 5T^{2}$$
7 $$1 + 4.91T + 7T^{2}$$
11 $$1 + 1.87T + 11T^{2}$$
13 $$1 + 3.25T + 13T^{2}$$
17 $$1 - 5.70T + 17T^{2}$$
19 $$1 - 0.533T + 19T^{2}$$
31 $$1 + 6.65T + 31T^{2}$$
37 $$1 - 0.389T + 37T^{2}$$
41 $$1 - 2.74T + 41T^{2}$$
43 $$1 - 12.3T + 43T^{2}$$
47 $$1 + 3.23T + 47T^{2}$$
53 $$1 + 3.80T + 53T^{2}$$
59 $$1 - 7.62T + 59T^{2}$$
61 $$1 - 5.94T + 61T^{2}$$
67 $$1 + 6.17T + 67T^{2}$$
71 $$1 - 2.94T + 71T^{2}$$
73 $$1 + 9.05T + 73T^{2}$$
79 $$1 + 5.70T + 79T^{2}$$
83 $$1 - 3.17T + 83T^{2}$$
89 $$1 - 11.3T + 89T^{2}$$
97 $$1 - 3.97T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$