Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.22·3-s + 4-s − 3.24·5-s − 3.22·6-s + 4.15·7-s + 8-s + 7.38·9-s − 3.24·10-s + 4.01·11-s − 3.22·12-s − 1.31·13-s + 4.15·14-s + 10.4·15-s + 16-s − 5.05·17-s + 7.38·18-s − 3.54·19-s − 3.24·20-s − 13.3·21-s + 4.01·22-s + 0.910·23-s − 3.22·24-s + 5.55·25-s − 1.31·26-s − 14.1·27-s + 4.15·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.86·3-s + 0.5·4-s − 1.45·5-s − 1.31·6-s + 1.57·7-s + 0.353·8-s + 2.46·9-s − 1.02·10-s + 1.20·11-s − 0.930·12-s − 0.365·13-s + 1.11·14-s + 2.70·15-s + 0.250·16-s − 1.22·17-s + 1.74·18-s − 0.812·19-s − 0.726·20-s − 2.92·21-s + 0.855·22-s + 0.189·23-s − 0.657·24-s + 1.11·25-s − 0.258·26-s − 2.71·27-s + 0.785·28-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
4001 \( 1+O(T) \)
good3 \( 1 + 3.22T + 3T^{2} \)
5 \( 1 + 3.24T + 5T^{2} \)
7 \( 1 - 4.15T + 7T^{2} \)
11 \( 1 - 4.01T + 11T^{2} \)
13 \( 1 + 1.31T + 13T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 + 3.54T + 19T^{2} \)
23 \( 1 - 0.910T + 23T^{2} \)
29 \( 1 - 1.83T + 29T^{2} \)
31 \( 1 - 4.10T + 31T^{2} \)
37 \( 1 - 0.633T + 37T^{2} \)
41 \( 1 + 1.02T + 41T^{2} \)
43 \( 1 + 6.99T + 43T^{2} \)
47 \( 1 + 9.55T + 47T^{2} \)
53 \( 1 + 8.16T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 + 8.71T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 0.173T + 73T^{2} \)
79 \( 1 - 0.271T + 79T^{2} \)
83 \( 1 - 0.170T + 83T^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
97 \( 1 + 3.23T + 97T^{2} \)
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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

−7.21689213340900163358178750167, −6.63029777157785274362475658322, −6.19883632867825441834573468711, −5.07577269800032439109603128234, −4.60959155923216467080761337013, −4.40525534913470808453491446242, −3.56599611969575692180454803615, −1.97558916227538715395771362009, −1.14969200072867739345850125049, 0, 1.14969200072867739345850125049, 1.97558916227538715395771362009, 3.56599611969575692180454803615, 4.40525534913470808453491446242, 4.60959155923216467080761337013, 5.07577269800032439109603128234, 6.19883632867825441834573468711, 6.63029777157785274362475658322, 7.21689213340900163358178750167

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