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  + 2.59·2-s + 4.73·4-s + 1.63·5-s + 7-s + 7.11·8-s + 4.23·10-s − 4.79·11-s + 0.263·13-s + 2.59·14-s + 8.98·16-s + 5.12·17-s + 5.93·19-s + 7.73·20-s − 12.4·22-s + 4.10·23-s − 2.33·25-s + 0.684·26-s + 4.73·28-s − 0.459·29-s − 9.24·31-s + 9.09·32-s + 13.2·34-s + 1.63·35-s + 6.64·37-s + 15.4·38-s + 11.6·40-s + 3.43·41-s + ⋯
L(s)  = 1  + 1.83·2-s + 2.36·4-s + 0.730·5-s + 0.377·7-s + 2.51·8-s + 1.34·10-s − 1.44·11-s + 0.0731·13-s + 0.693·14-s + 2.24·16-s + 1.24·17-s + 1.36·19-s + 1.72·20-s − 2.65·22-s + 0.855·23-s − 0.467·25-s + 0.134·26-s + 0.895·28-s − 0.0853·29-s − 1.66·31-s + 1.60·32-s + 2.27·34-s + 0.275·35-s + 1.09·37-s + 2.49·38-s + 1.83·40-s + 0.536·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.671078905$
$L(\frac12)$  $\approx$  $8.671078905$
$L(\frac{3}{2})$   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 \( 1 - 2.59T + 2T^{2} \)
5 \( 1 - 1.63T + 5T^{2} \)
11 \( 1 + 4.79T + 11T^{2} \)
13 \( 1 - 0.263T + 13T^{2} \)
17 \( 1 - 5.12T + 17T^{2} \)
19 \( 1 - 5.93T + 19T^{2} \)
23 \( 1 - 4.10T + 23T^{2} \)
29 \( 1 + 0.459T + 29T^{2} \)
31 \( 1 + 9.24T + 31T^{2} \)
37 \( 1 - 6.64T + 37T^{2} \)
41 \( 1 - 3.43T + 41T^{2} \)
43 \( 1 - 2.33T + 43T^{2} \)
47 \( 1 - 5.69T + 47T^{2} \)
53 \( 1 + 2.55T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 1.34T + 67T^{2} \)
71 \( 1 + 5.05T + 71T^{2} \)
73 \( 1 + 16.4T + 73T^{2} \)
79 \( 1 + 0.111T + 79T^{2} \)
83 \( 1 + 6.65T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + 5.51T + 97T^{2} \)
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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

−7.36043319725588326034376182068, −7.22716287872792873403923710558, −5.93601460531352146363187943585, −5.50405444436146237381237586860, −5.31579176656238177351670664277, −4.39148999076527276159056265080, −3.52372078539257202478194789020, −2.84145987745374769042751815387, −2.21164548132781392140877729912, −1.18933920074968271592727519936, 1.18933920074968271592727519936, 2.21164548132781392140877729912, 2.84145987745374769042751815387, 3.52372078539257202478194789020, 4.39148999076527276159056265080, 5.31579176656238177351670664277, 5.50405444436146237381237586860, 5.93601460531352146363187943585, 7.22716287872792873403923710558, 7.36043319725588326034376182068

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