Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 1667 $
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-s + 4-s − 2·5-s − 6-s − 2·7-s + 8-s + 9-s − 2·10-s + 3·11-s − 12-s + 4·13-s − 2·14-s + 2·15-s + 16-s + 18-s − 2·19-s − 2·20-s + 2·21-s + 3·22-s − 4·23-s − 24-s − 25-s + 4·26-s − 27-s − 2·28-s + 2·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.904·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.447·20-s + 0.436·21-s + 0.639·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10002\)    =    \(2 \cdot 3 \cdot 1667\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{10002} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 10002,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;1667\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;1667\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 + T \)
1667 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−16.67391534978190, −16.25795428384356, −15.78952141938611, −15.35943558381796, −14.61817049209255, −14.07501502586285, −13.36809151162345, −12.80783301676970, −12.32101825072007, −11.72733302239579, −11.22525300748871, −10.80478954682357, −9.906543858714935, −9.379331712425866, −8.418249547107686, −7.973434483804963, −6.958085146091432, −6.614182143990548, −5.989669593114979, −5.329915559818094, −4.355339483779706, −3.764822904115286, −3.491626692861573, −2.220811329224649, −1.180436988547250, 0, 1.180436988547250, 2.220811329224649, 3.491626692861573, 3.764822904115286, 4.355339483779706, 5.329915559818094, 5.989669593114979, 6.614182143990548, 6.958085146091432, 7.973434483804963, 8.418249547107686, 9.379331712425866, 9.906543858714935, 10.80478954682357, 11.22525300748871, 11.72733302239579, 12.32101825072007, 12.80783301676970, 13.36809151162345, 14.07501502586285, 14.61817049209255, 15.35943558381796, 15.78952141938611, 16.25795428384356, 16.67391534978190

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