Properties

Degree 2
Conductor 8011
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.57·2-s + 0.178·3-s + 4.63·4-s − 1.43·5-s − 0.458·6-s + 4.00·7-s − 6.77·8-s − 2.96·9-s + 3.69·10-s + 3.24·11-s + 0.825·12-s + 0.109·13-s − 10.3·14-s − 0.255·15-s + 8.19·16-s + 4.02·17-s + 7.64·18-s + 0.800·19-s − 6.64·20-s + 0.714·21-s − 8.36·22-s + 2.16·23-s − 1.20·24-s − 2.94·25-s − 0.281·26-s − 1.06·27-s + 18.5·28-s + ⋯
L(s)  = 1  − 1.82·2-s + 0.102·3-s + 2.31·4-s − 0.641·5-s − 0.187·6-s + 1.51·7-s − 2.39·8-s − 0.989·9-s + 1.16·10-s + 0.979·11-s + 0.238·12-s + 0.0302·13-s − 2.75·14-s − 0.0659·15-s + 2.04·16-s + 0.976·17-s + 1.80·18-s + 0.183·19-s − 1.48·20-s + 0.155·21-s − 1.78·22-s + 0.451·23-s − 0.246·24-s − 0.588·25-s − 0.0551·26-s − 0.204·27-s + 3.50·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8011\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8011} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8011,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8929251134$
$L(\frac12)$  $\approx$  $0.8929251134$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 8011$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 8011$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad8011 \( 1+O(T) \)
good2 \( 1 + 2.57T + 2T^{2} \)
3 \( 1 - 0.178T + 3T^{2} \)
5 \( 1 + 1.43T + 5T^{2} \)
7 \( 1 - 4.00T + 7T^{2} \)
11 \( 1 - 3.24T + 11T^{2} \)
13 \( 1 - 0.109T + 13T^{2} \)
17 \( 1 - 4.02T + 17T^{2} \)
19 \( 1 - 0.800T + 19T^{2} \)
23 \( 1 - 2.16T + 23T^{2} \)
29 \( 1 - 1.12T + 29T^{2} \)
31 \( 1 + 9.40T + 31T^{2} \)
37 \( 1 + 9.59T + 37T^{2} \)
41 \( 1 - 2.07T + 41T^{2} \)
43 \( 1 - 4.11T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 - 8.93T + 53T^{2} \)
59 \( 1 - 3.20T + 59T^{2} \)
61 \( 1 + 8.18T + 61T^{2} \)
67 \( 1 - 5.25T + 67T^{2} \)
71 \( 1 + 6.13T + 71T^{2} \)
73 \( 1 - 2.04T + 73T^{2} \)
79 \( 1 - 1.93T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 1.83T + 89T^{2} \)
97 \( 1 - 1.53T + 97T^{2} \)
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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

−7.955832044397266142591445233025, −7.45451385185826145747022767879, −6.94254139168113348535551193301, −5.83673597652599929803313042848, −5.32048370366491354666604071315, −4.10249070516448996569643950309, −3.30728294947911823173982427531, −2.23127578790135731187464758787, −1.51059624643766903486312810052, −0.65060793016409744618030857586, 0.65060793016409744618030857586, 1.51059624643766903486312810052, 2.23127578790135731187464758787, 3.30728294947911823173982427531, 4.10249070516448996569643950309, 5.32048370366491354666604071315, 5.83673597652599929803313042848, 6.94254139168113348535551193301, 7.45451385185826145747022767879, 7.955832044397266142591445233025

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