Properties

Degree 2
Conductor $ 2 \cdot 4001 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3.14·3-s + 4-s − 2.17·5-s + 3.14·6-s + 4.77·7-s − 8-s + 6.90·9-s + 2.17·10-s + 0.464·11-s − 3.14·12-s − 3.54·13-s − 4.77·14-s + 6.84·15-s + 16-s + 2.17·17-s − 6.90·18-s − 7.81·19-s − 2.17·20-s − 15.0·21-s − 0.464·22-s − 2.67·23-s + 3.14·24-s − 0.276·25-s + 3.54·26-s − 12.2·27-s + 4.77·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.81·3-s + 0.5·4-s − 0.971·5-s + 1.28·6-s + 1.80·7-s − 0.353·8-s + 2.30·9-s + 0.687·10-s + 0.140·11-s − 0.908·12-s − 0.981·13-s − 1.27·14-s + 1.76·15-s + 0.250·16-s + 0.526·17-s − 1.62·18-s − 1.79·19-s − 0.485·20-s − 3.27·21-s − 0.0990·22-s − 0.558·23-s + 0.642·24-s − 0.0552·25-s + 0.694·26-s − 2.36·27-s + 0.901·28-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  =  0
Selberg data  =  $(2,\ 8002,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3501483418$
$L(\frac12)$  $\approx$  $0.3501483418$
$L(\frac{3}{2})$   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 \( 1 + 3.14T + 3T^{2} \)
5 \( 1 + 2.17T + 5T^{2} \)
7 \( 1 - 4.77T + 7T^{2} \)
11 \( 1 - 0.464T + 11T^{2} \)
13 \( 1 + 3.54T + 13T^{2} \)
17 \( 1 - 2.17T + 17T^{2} \)
19 \( 1 + 7.81T + 19T^{2} \)
23 \( 1 + 2.67T + 23T^{2} \)
29 \( 1 + 4.66T + 29T^{2} \)
31 \( 1 + 2.75T + 31T^{2} \)
37 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 - 0.752T + 41T^{2} \)
43 \( 1 - 0.446T + 43T^{2} \)
47 \( 1 - 6.33T + 47T^{2} \)
53 \( 1 + 5.24T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 + 6.29T + 61T^{2} \)
67 \( 1 + 7.37T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 + 2.14T + 79T^{2} \)
83 \( 1 + 1.26T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 5.61T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.69903475702647126925086440726, −7.30641864362545731124635931545, −6.53183327193618826168196003293, −5.68090824109715383381627270422, −5.14550039251633807784355677459, −4.38649255901985820421067083403, −3.94916395123625344301795292580, −2.18087023410271143343834259421, −1.48749134070606017195144576099, −0.37533032058123430276480051273, 0.37533032058123430276480051273, 1.48749134070606017195144576099, 2.18087023410271143343834259421, 3.94916395123625344301795292580, 4.38649255901985820421067083403, 5.14550039251633807784355677459, 5.68090824109715383381627270422, 6.53183327193618826168196003293, 7.30641864362545731124635931545, 7.69903475702647126925086440726

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