Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.375·2-s − 1.85·4-s − 2.49·5-s + 7-s + 1.44·8-s + 0.937·10-s + 0.0439·11-s + 1.23·13-s − 0.375·14-s + 3.17·16-s + 4.66·17-s − 4.55·19-s + 4.64·20-s − 0.0165·22-s + 5.31·23-s + 1.23·25-s − 0.464·26-s − 1.85·28-s + 5.23·29-s + 0.715·31-s − 4.08·32-s − 1.75·34-s − 2.49·35-s + 7.90·37-s + 1.71·38-s − 3.61·40-s + 4.97·41-s + ⋯
L(s)  = 1  − 0.265·2-s − 0.929·4-s − 1.11·5-s + 0.377·7-s + 0.512·8-s + 0.296·10-s + 0.0132·11-s + 0.343·13-s − 0.100·14-s + 0.793·16-s + 1.13·17-s − 1.04·19-s + 1.03·20-s − 0.00352·22-s + 1.10·23-s + 0.246·25-s − 0.0910·26-s − 0.351·28-s + 0.972·29-s + 0.128·31-s − 0.722·32-s − 0.300·34-s − 0.422·35-s + 1.29·37-s + 0.277·38-s − 0.571·40-s + 0.776·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8001} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.044874412$
$L(\frac12)$  $\approx$  $1.044874412$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;127\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;127\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - T \)
127 \( 1 - T \)
good2 \( 1 + 0.375T + 2T^{2} \)
5 \( 1 + 2.49T + 5T^{2} \)
11 \( 1 - 0.0439T + 11T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 - 4.66T + 17T^{2} \)
19 \( 1 + 4.55T + 19T^{2} \)
23 \( 1 - 5.31T + 23T^{2} \)
29 \( 1 - 5.23T + 29T^{2} \)
31 \( 1 - 0.715T + 31T^{2} \)
37 \( 1 - 7.90T + 37T^{2} \)
41 \( 1 - 4.97T + 41T^{2} \)
43 \( 1 - 0.341T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 - 4.41T + 53T^{2} \)
59 \( 1 + 9.78T + 59T^{2} \)
61 \( 1 + 9.42T + 61T^{2} \)
67 \( 1 - 0.00811T + 67T^{2} \)
71 \( 1 - 4.91T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + 1.44T + 79T^{2} \)
83 \( 1 - 7.62T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 8.97T + 97T^{2} \)
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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

−7.943896249985666134069360278077, −7.47021372755660764272389679624, −6.49256729561760140429733346089, −5.69735795254282918864311380570, −4.75555200675748272651887489255, −4.40587130108324231817481704279, −3.60925332306397224581786097317, −2.88106038226220591508230866474, −1.43598822903198532335978528267, −0.58210249404207636364466936864, 0.58210249404207636364466936864, 1.43598822903198532335978528267, 2.88106038226220591508230866474, 3.60925332306397224581786097317, 4.40587130108324231817481704279, 4.75555200675748272651887489255, 5.69735795254282918864311380570, 6.49256729561760140429733346089, 7.47021372755660764272389679624, 7.943896249985666134069360278077

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