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  + 1.48·2-s + 0.216·4-s + 2.91·5-s + 7-s − 2.65·8-s + 4.33·10-s − 4.14·11-s + 3.36·13-s + 1.48·14-s − 4.38·16-s − 4.79·17-s + 0.147·19-s + 0.631·20-s − 6.17·22-s + 5.23·23-s + 3.47·25-s + 5.01·26-s + 0.216·28-s + 1.23·29-s + 6.14·31-s − 1.22·32-s − 7.14·34-s + 2.91·35-s + 7.83·37-s + 0.220·38-s − 7.73·40-s − 0.0490·41-s + ⋯
L(s)  = 1  + 1.05·2-s + 0.108·4-s + 1.30·5-s + 0.377·7-s − 0.938·8-s + 1.37·10-s − 1.25·11-s + 0.933·13-s + 0.397·14-s − 1.09·16-s − 1.16·17-s + 0.0339·19-s + 0.141·20-s − 1.31·22-s + 1.09·23-s + 0.695·25-s + 0.983·26-s + 0.0409·28-s + 0.228·29-s + 1.10·31-s − 0.215·32-s − 1.22·34-s + 0.492·35-s + 1.28·37-s + 0.0357·38-s − 1.22·40-s − 0.00766·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.028049084$
$L(\frac12)$  $\approx$  $4.028049084$
$L(\frac{3}{2})$   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 \( 1 - 1.48T + 2T^{2} \)
5 \( 1 - 2.91T + 5T^{2} \)
11 \( 1 + 4.14T + 11T^{2} \)
13 \( 1 - 3.36T + 13T^{2} \)
17 \( 1 + 4.79T + 17T^{2} \)
19 \( 1 - 0.147T + 19T^{2} \)
23 \( 1 - 5.23T + 23T^{2} \)
29 \( 1 - 1.23T + 29T^{2} \)
31 \( 1 - 6.14T + 31T^{2} \)
37 \( 1 - 7.83T + 37T^{2} \)
41 \( 1 + 0.0490T + 41T^{2} \)
43 \( 1 + 8.18T + 43T^{2} \)
47 \( 1 - 3.35T + 47T^{2} \)
53 \( 1 + 4.90T + 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + 1.83T + 67T^{2} \)
71 \( 1 - 9.72T + 71T^{2} \)
73 \( 1 - 7.33T + 73T^{2} \)
79 \( 1 - 6.96T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + 0.471T + 97T^{2} \)
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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

−7.916656351645875397370161080064, −6.62292948183095868729963872981, −6.43190587186325270531226296260, −5.49872945201142467497668289047, −5.13104077364198694826783739931, −4.49661902068026009616723541662, −3.56749936270012538000073012891, −2.63800565215346419372200704552, −2.17149393072623322823613206309, −0.844986161602685709467057718943, 0.844986161602685709467057718943, 2.17149393072623322823613206309, 2.63800565215346419372200704552, 3.56749936270012538000073012891, 4.49661902068026009616723541662, 5.13104077364198694826783739931, 5.49872945201142467497668289047, 6.43190587186325270531226296260, 6.62292948183095868729963872981, 7.916656351645875397370161080064

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