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  − 2.79·2-s + 3.22·3-s + 5.82·4-s − 0.295·5-s − 9.01·6-s + 1.71·7-s − 10.7·8-s + 7.38·9-s + 0.827·10-s + 4.76·11-s + 18.7·12-s − 0.569·13-s − 4.81·14-s − 0.953·15-s + 18.2·16-s + 2.37·17-s − 20.6·18-s + 4.95·19-s − 1.72·20-s + 5.54·21-s − 13.3·22-s + 6.53·23-s − 34.4·24-s − 4.91·25-s + 1.59·26-s + 14.1·27-s + 10.0·28-s + ⋯
L(s)  = 1  − 1.97·2-s + 1.86·3-s + 2.91·4-s − 0.132·5-s − 3.68·6-s + 0.649·7-s − 3.78·8-s + 2.46·9-s + 0.261·10-s + 1.43·11-s + 5.41·12-s − 0.157·13-s − 1.28·14-s − 0.246·15-s + 4.57·16-s + 0.576·17-s − 4.86·18-s + 1.13·19-s − 0.385·20-s + 1.20·21-s − 2.83·22-s + 1.36·23-s − 7.03·24-s − 0.982·25-s + 0.312·26-s + 2.71·27-s + 1.89·28-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  =  $(2,\ 8011,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.486599526$
$L(\frac12)$  $\approx$  $2.486599526$
$L(\frac{3}{2})$   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 \( 1 + 2.79T + 2T^{2} \)
3 \( 1 - 3.22T + 3T^{2} \)
5 \( 1 + 0.295T + 5T^{2} \)
7 \( 1 - 1.71T + 7T^{2} \)
11 \( 1 - 4.76T + 11T^{2} \)
13 \( 1 + 0.569T + 13T^{2} \)
17 \( 1 - 2.37T + 17T^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
23 \( 1 - 6.53T + 23T^{2} \)
29 \( 1 - 9.48T + 29T^{2} \)
31 \( 1 + 5.25T + 31T^{2} \)
37 \( 1 - 2.25T + 37T^{2} \)
41 \( 1 + 2.41T + 41T^{2} \)
43 \( 1 + 6.28T + 43T^{2} \)
47 \( 1 - 3.48T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 - 6.78T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 6.95T + 71T^{2} \)
73 \( 1 - 9.90T + 73T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 - 8.47T + 83T^{2} \)
89 \( 1 - 2.21T + 89T^{2} \)
97 \( 1 + 4.85T + 97T^{2} \)
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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

−8.040024229070305789811593700299, −7.48628311938805113115253320511, −6.99628144077731126424605659669, −6.32705422930376457008857152426, −5.01988404795515520552381599948, −3.73387259305929619909159550071, −3.18939582620609527046517565319, −2.42284947964799045253667401450, −1.48533525792125206237881939163, −1.13104149932765199321446033321, 1.13104149932765199321446033321, 1.48533525792125206237881939163, 2.42284947964799045253667401450, 3.18939582620609527046517565319, 3.73387259305929619909159550071, 5.01988404795515520552381599948, 6.32705422930376457008857152426, 6.99628144077731126424605659669, 7.48628311938805113115253320511, 8.040024229070305789811593700299

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