Properties

 Label 2-4001-1.1-c1-0-172 Degree $2$ Conductor $4001$ Sign $1$ Analytic cond. $31.9481$ Root an. cond. $5.65226$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

Origins

Dirichlet series

 L(s)  = 1 − 2.55·2-s + 2.40·3-s + 4.52·4-s + 3.51·5-s − 6.14·6-s + 3.52·7-s − 6.45·8-s + 2.79·9-s − 8.97·10-s − 5.09·11-s + 10.8·12-s − 0.353·13-s − 9.00·14-s + 8.45·15-s + 7.43·16-s + 1.22·17-s − 7.13·18-s + 0.445·19-s + 15.8·20-s + 8.48·21-s + 13.0·22-s + 9.35·23-s − 15.5·24-s + 7.33·25-s + 0.902·26-s − 0.499·27-s + 15.9·28-s + ⋯
 L(s)  = 1 − 1.80·2-s + 1.38·3-s + 2.26·4-s + 1.57·5-s − 2.51·6-s + 1.33·7-s − 2.28·8-s + 0.930·9-s − 2.83·10-s − 1.53·11-s + 3.14·12-s − 0.0980·13-s − 2.40·14-s + 2.18·15-s + 1.85·16-s + 0.297·17-s − 1.68·18-s + 0.102·19-s + 3.55·20-s + 1.85·21-s + 2.77·22-s + 1.95·23-s − 3.17·24-s + 1.46·25-s + 0.177·26-s − 0.0961·27-s + 3.01·28-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$4001$$ Sign: $1$ Analytic conductor: $$31.9481$$ Root analytic conductor: $$5.65226$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 4001,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$2.155927920$$ $$L(\frac12)$$ $$\approx$$ $$2.155927920$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad4001 $$1+O(T)$$
good2 $$1 + 2.55T + 2T^{2}$$
3 $$1 - 2.40T + 3T^{2}$$
5 $$1 - 3.51T + 5T^{2}$$
7 $$1 - 3.52T + 7T^{2}$$
11 $$1 + 5.09T + 11T^{2}$$
13 $$1 + 0.353T + 13T^{2}$$
17 $$1 - 1.22T + 17T^{2}$$
19 $$1 - 0.445T + 19T^{2}$$
23 $$1 - 9.35T + 23T^{2}$$
29 $$1 + 6.41T + 29T^{2}$$
31 $$1 - 4.59T + 31T^{2}$$
37 $$1 + 0.0828T + 37T^{2}$$
41 $$1 - 4.27T + 41T^{2}$$
43 $$1 + 9.28T + 43T^{2}$$
47 $$1 - 2.53T + 47T^{2}$$
53 $$1 + 1.29T + 53T^{2}$$
59 $$1 + 7.77T + 59T^{2}$$
61 $$1 - 7.07T + 61T^{2}$$
67 $$1 - 15.5T + 67T^{2}$$
71 $$1 - 6.33T + 71T^{2}$$
73 $$1 - 12.7T + 73T^{2}$$
79 $$1 + 0.799T + 79T^{2}$$
83 $$1 - 2.66T + 83T^{2}$$
89 $$1 + 12.6T + 89T^{2}$$
97 $$1 - 7.86T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$