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  − 2.26·2-s + 3.13·4-s + 0.263·5-s + 7-s − 2.56·8-s − 0.596·10-s − 2.11·11-s + 5.02·13-s − 2.26·14-s − 0.448·16-s − 0.600·17-s − 0.662·19-s + 0.825·20-s + 4.79·22-s + 0.862·23-s − 4.93·25-s − 11.3·26-s + 3.13·28-s − 7.49·29-s − 6.27·31-s + 6.15·32-s + 1.36·34-s + 0.263·35-s + 1.88·37-s + 1.50·38-s − 0.676·40-s − 9.92·41-s + ⋯
L(s)  = 1  − 1.60·2-s + 1.56·4-s + 0.117·5-s + 0.377·7-s − 0.908·8-s − 0.188·10-s − 0.637·11-s + 1.39·13-s − 0.605·14-s − 0.112·16-s − 0.145·17-s − 0.151·19-s + 0.184·20-s + 1.02·22-s + 0.179·23-s − 0.986·25-s − 2.23·26-s + 0.592·28-s − 1.39·29-s − 1.12·31-s + 1.08·32-s + 0.233·34-s + 0.0445·35-s + 0.310·37-s + 0.243·38-s − 0.106·40-s − 1.54·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7636009457$
$L(\frac12)$  $\approx$  $0.7636009457$
$L(\frac{3}{2})$   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 \( 1 + 2.26T + 2T^{2} \)
5 \( 1 - 0.263T + 5T^{2} \)
11 \( 1 + 2.11T + 11T^{2} \)
13 \( 1 - 5.02T + 13T^{2} \)
17 \( 1 + 0.600T + 17T^{2} \)
19 \( 1 + 0.662T + 19T^{2} \)
23 \( 1 - 0.862T + 23T^{2} \)
29 \( 1 + 7.49T + 29T^{2} \)
31 \( 1 + 6.27T + 31T^{2} \)
37 \( 1 - 1.88T + 37T^{2} \)
41 \( 1 + 9.92T + 41T^{2} \)
43 \( 1 + 0.837T + 43T^{2} \)
47 \( 1 - 6.32T + 47T^{2} \)
53 \( 1 + 2.22T + 53T^{2} \)
59 \( 1 - 8.92T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 + 6.63T + 67T^{2} \)
71 \( 1 + 3.04T + 71T^{2} \)
73 \( 1 - 3.58T + 73T^{2} \)
79 \( 1 + 6.24T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−7.979686218406969879947428209586, −7.39969620697137300394199693519, −6.74051975729620273136208408588, −5.89185567193853185365689128831, −5.29034545127872856704031103249, −4.15041958035353041420345833943, −3.37317386894169226145739586446, −2.15790301366867370295441003729, −1.66037284783311726600813647603, −0.55824042645876188061607035632, 0.55824042645876188061607035632, 1.66037284783311726600813647603, 2.15790301366867370295441003729, 3.37317386894169226145739586446, 4.15041958035353041420345833943, 5.29034545127872856704031103249, 5.89185567193853185365689128831, 6.74051975729620273136208408588, 7.39969620697137300394199693519, 7.979686218406969879947428209586

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