Properties

Label 2-4001-1.1-c1-0-105
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.932·2-s − 1.34·3-s − 1.13·4-s − 0.776·5-s + 1.25·6-s − 4.61·7-s + 2.91·8-s − 1.19·9-s + 0.724·10-s − 4.05·11-s + 1.51·12-s + 4.65·13-s + 4.30·14-s + 1.04·15-s − 0.460·16-s − 7.05·17-s + 1.11·18-s + 3.82·19-s + 0.877·20-s + 6.19·21-s + 3.78·22-s + 3.00·23-s − 3.91·24-s − 4.39·25-s − 4.33·26-s + 5.63·27-s + 5.21·28-s + ⋯
L(s)  = 1  − 0.659·2-s − 0.775·3-s − 0.565·4-s − 0.347·5-s + 0.511·6-s − 1.74·7-s + 1.03·8-s − 0.399·9-s + 0.228·10-s − 1.22·11-s + 0.438·12-s + 1.29·13-s + 1.14·14-s + 0.269·15-s − 0.115·16-s − 1.71·17-s + 0.263·18-s + 0.876·19-s + 0.196·20-s + 1.35·21-s + 0.807·22-s + 0.626·23-s − 0.799·24-s − 0.879·25-s − 0.850·26-s + 1.08·27-s + 0.985·28-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: \(1\)
Selberg data: \((2,\ 4001,\ (\ :1/2),\ -1)\)

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)$
bad4001 \( 1+O(T) \)
good2 \( 1 + 0.932T + 2T^{2} \)
3 \( 1 + 1.34T + 3T^{2} \)
5 \( 1 + 0.776T + 5T^{2} \)
7 \( 1 + 4.61T + 7T^{2} \)
11 \( 1 + 4.05T + 11T^{2} \)
13 \( 1 - 4.65T + 13T^{2} \)
17 \( 1 + 7.05T + 17T^{2} \)
19 \( 1 - 3.82T + 19T^{2} \)
23 \( 1 - 3.00T + 23T^{2} \)
29 \( 1 - 2.12T + 29T^{2} \)
31 \( 1 + 3.23T + 31T^{2} \)
37 \( 1 - 2.48T + 37T^{2} \)
41 \( 1 - 5.43T + 41T^{2} \)
43 \( 1 + 6.26T + 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 - 9.74T + 53T^{2} \)
59 \( 1 - 2.38T + 59T^{2} \)
61 \( 1 - 3.43T + 61T^{2} \)
67 \( 1 + 3.42T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 2.12T + 73T^{2} \)
79 \( 1 - 0.0368T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 18.1T + 89T^{2} \)
97 \( 1 - 0.885T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.223063551968129877955215116295, −7.35884399278717688412707073731, −6.61714958714561685353380450461, −5.88488268296849098666763058776, −5.25891905079570533160214302943, −4.23712947294074065587780848859, −3.45593165533307940008464388671, −2.50319567851455884445771026672, −0.76789271653011250485522934832, 0, 0.76789271653011250485522934832, 2.50319567851455884445771026672, 3.45593165533307940008464388671, 4.23712947294074065587780848859, 5.25891905079570533160214302943, 5.88488268296849098666763058776, 6.61714958714561685353380450461, 7.35884399278717688412707073731, 8.223063551968129877955215116295

Graph of the $Z$-function along the critical line