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 − 3.04·3-s + 4-s − 0.106·5-s − 3.04·6-s − 0.613·7-s + 8-s + 6.29·9-s − 0.106·10-s + 4.91·11-s − 3.04·12-s − 1.74·13-s − 0.613·14-s + 0.325·15-s + 16-s + 0.753·17-s + 6.29·18-s + 0.918·19-s − 0.106·20-s + 1.87·21-s + 4.91·22-s − 3.35·23-s − 3.04·24-s − 4.98·25-s − 1.74·26-s − 10.0·27-s − 0.613·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.76·3-s + 0.5·4-s − 0.0477·5-s − 1.24·6-s − 0.232·7-s + 0.353·8-s + 2.09·9-s − 0.0337·10-s + 1.48·11-s − 0.880·12-s − 0.483·13-s − 0.164·14-s + 0.0839·15-s + 0.250·16-s + 0.182·17-s + 1.48·18-s + 0.210·19-s − 0.0238·20-s + 0.408·21-s + 1.04·22-s − 0.700·23-s − 0.622·24-s − 0.997·25-s − 0.341·26-s − 1.93·27-s − 0.116·28-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 \( 1 + 3.04T + 3T^{2} \)
5 \( 1 + 0.106T + 5T^{2} \)
7 \( 1 + 0.613T + 7T^{2} \)
11 \( 1 - 4.91T + 11T^{2} \)
13 \( 1 + 1.74T + 13T^{2} \)
17 \( 1 - 0.753T + 17T^{2} \)
19 \( 1 - 0.918T + 19T^{2} \)
23 \( 1 + 3.35T + 23T^{2} \)
29 \( 1 - 3.52T + 29T^{2} \)
31 \( 1 + 6.13T + 31T^{2} \)
37 \( 1 + 1.78T + 37T^{2} \)
41 \( 1 + 9.94T + 41T^{2} \)
43 \( 1 - 5.22T + 43T^{2} \)
47 \( 1 + 8.14T + 47T^{2} \)
53 \( 1 - 6.17T + 53T^{2} \)
59 \( 1 - 8.55T + 59T^{2} \)
61 \( 1 + 3.22T + 61T^{2} \)
67 \( 1 + 4.18T + 67T^{2} \)
71 \( 1 + 0.630T + 71T^{2} \)
73 \( 1 + 0.360T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 4.38T + 83T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 + 2.30T + 97T^{2} \)
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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

−7.13731209469954236993970575713, −6.45997919747577840995502103024, −6.20015051143019931783091136223, −5.34377135376734460860416406484, −4.87349467302301645341708443399, −4.00591555012798511077906712325, −3.52432728347690741441070956572, −2.03907651895477965033810940874, −1.20640200799872019834787562197, 0, 1.20640200799872019834787562197, 2.03907651895477965033810940874, 3.52432728347690741441070956572, 4.00591555012798511077906712325, 4.87349467302301645341708443399, 5.34377135376734460860416406484, 6.20015051143019931783091136223, 6.45997919747577840995502103024, 7.13731209469954236993970575713

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