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  + 1.77·2-s + 1.13·4-s − 1.38·5-s + 7-s − 1.53·8-s − 2.45·10-s − 3.51·11-s + 4.71·13-s + 1.77·14-s − 4.98·16-s + 4.68·17-s − 3.74·19-s − 1.57·20-s − 6.22·22-s + 2.00·23-s − 3.08·25-s + 8.34·26-s + 1.13·28-s + 4.37·29-s − 0.875·31-s − 5.76·32-s + 8.29·34-s − 1.38·35-s − 9.02·37-s − 6.63·38-s + 2.11·40-s − 1.57·41-s + ⋯
L(s)  = 1  + 1.25·2-s + 0.567·4-s − 0.618·5-s + 0.377·7-s − 0.541·8-s − 0.774·10-s − 1.05·11-s + 1.30·13-s + 0.473·14-s − 1.24·16-s + 1.13·17-s − 0.859·19-s − 0.351·20-s − 1.32·22-s + 0.417·23-s − 0.617·25-s + 1.63·26-s + 0.214·28-s + 0.811·29-s − 0.157·31-s − 1.01·32-s + 1.42·34-s − 0.233·35-s − 1.48·37-s − 1.07·38-s + 0.334·40-s − 0.245·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.063473281$
$L(\frac12)$  $\approx$  $3.063473281$
$L(\frac{3}{2})$   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 \( 1 - 1.77T + 2T^{2} \)
5 \( 1 + 1.38T + 5T^{2} \)
11 \( 1 + 3.51T + 11T^{2} \)
13 \( 1 - 4.71T + 13T^{2} \)
17 \( 1 - 4.68T + 17T^{2} \)
19 \( 1 + 3.74T + 19T^{2} \)
23 \( 1 - 2.00T + 23T^{2} \)
29 \( 1 - 4.37T + 29T^{2} \)
31 \( 1 + 0.875T + 31T^{2} \)
37 \( 1 + 9.02T + 37T^{2} \)
41 \( 1 + 1.57T + 41T^{2} \)
43 \( 1 + 2.16T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 - 0.801T + 59T^{2} \)
61 \( 1 - 3.10T + 61T^{2} \)
67 \( 1 - 6.63T + 67T^{2} \)
71 \( 1 - 8.91T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 3.05T + 79T^{2} \)
83 \( 1 + 4.04T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + 8.60T + 97T^{2} \)
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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

−7.84027751423313253178876369611, −6.98060499421262676447568539040, −6.28317136802977571892806226409, −5.39481739972564486148451189829, −5.22229821597916534479593922985, −4.10636640003628164879611873461, −3.76656273351828497569216230036, −2.97511540210777780391486860823, −2.07197502041433505216774095577, −0.70513061146608351961765347578, 0.70513061146608351961765347578, 2.07197502041433505216774095577, 2.97511540210777780391486860823, 3.76656273351828497569216230036, 4.10636640003628164879611873461, 5.22229821597916534479593922985, 5.39481739972564486148451189829, 6.28317136802977571892806226409, 6.98060499421262676447568539040, 7.84027751423313253178876369611

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