Properties

Degree 2
Conductor $ 2 \cdot 4001 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.40·3-s + 4-s + 2.18·5-s − 3.40·6-s − 1.24·7-s + 8-s + 8.60·9-s + 2.18·10-s + 3.12·11-s − 3.40·12-s − 1.99·13-s − 1.24·14-s − 7.44·15-s + 16-s − 2.59·17-s + 8.60·18-s + 6.38·19-s + 2.18·20-s + 4.22·21-s + 3.12·22-s − 2.47·23-s − 3.40·24-s − 0.229·25-s − 1.99·26-s − 19.0·27-s − 1.24·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.96·3-s + 0.5·4-s + 0.976·5-s − 1.39·6-s − 0.468·7-s + 0.353·8-s + 2.86·9-s + 0.690·10-s + 0.943·11-s − 0.983·12-s − 0.553·13-s − 0.331·14-s − 1.92·15-s + 0.250·16-s − 0.628·17-s + 2.02·18-s + 1.46·19-s + 0.488·20-s + 0.922·21-s + 0.667·22-s − 0.515·23-s − 0.695·24-s − 0.0458·25-s − 0.391·26-s − 3.67·27-s − 0.234·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8002\)    =    \(2 \cdot 4001\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8002} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8002,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;4001\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;4001\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
4001 \( 1+O(T) \)
good3 \( 1 + 3.40T + 3T^{2} \)
5 \( 1 - 2.18T + 5T^{2} \)
7 \( 1 + 1.24T + 7T^{2} \)
11 \( 1 - 3.12T + 11T^{2} \)
13 \( 1 + 1.99T + 13T^{2} \)
17 \( 1 + 2.59T + 17T^{2} \)
19 \( 1 - 6.38T + 19T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + 7.09T + 29T^{2} \)
31 \( 1 - 1.66T + 31T^{2} \)
37 \( 1 + 7.59T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 7.73T + 43T^{2} \)
47 \( 1 - 2.02T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + 5.00T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 - 5.74T + 67T^{2} \)
71 \( 1 - 3.73T + 71T^{2} \)
73 \( 1 - 0.446T + 73T^{2} \)
79 \( 1 - 5.38T + 79T^{2} \)
83 \( 1 + 4.50T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.07799805143329015280225701692, −6.49753463868835343491189956727, −6.02232349428877101603676069384, −5.49254781764549007663278776119, −4.89795888606228962889502789686, −4.18977381110710892415089173679, −3.32856466727307393540612839385, −1.95732354167965823809033440983, −1.32268585516474470274220687038, 0, 1.32268585516474470274220687038, 1.95732354167965823809033440983, 3.32856466727307393540612839385, 4.18977381110710892415089173679, 4.89795888606228962889502789686, 5.49254781764549007663278776119, 6.02232349428877101603676069384, 6.49753463868835343491189956727, 7.07799805143329015280225701692

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