Properties

Degree 2
Conductor $ 11 \cdot 17 \cdot 43 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.34·2-s + 3.28·3-s + 3.50·4-s + 3.73·5-s − 7.71·6-s + 3.13·7-s − 3.53·8-s + 7.79·9-s − 8.77·10-s + 11-s + 11.5·12-s + 0.157·13-s − 7.35·14-s + 12.2·15-s + 1.28·16-s + 17-s − 18.3·18-s − 5.83·19-s + 13.1·20-s + 10.3·21-s − 2.34·22-s + 0.802·23-s − 11.6·24-s + 8.98·25-s − 0.369·26-s + 15.7·27-s + 10.9·28-s + ⋯
L(s)  = 1  − 1.65·2-s + 1.89·3-s + 1.75·4-s + 1.67·5-s − 3.14·6-s + 1.18·7-s − 1.24·8-s + 2.59·9-s − 2.77·10-s + 0.301·11-s + 3.32·12-s + 0.0436·13-s − 1.96·14-s + 3.17·15-s + 0.320·16-s + 0.242·17-s − 4.31·18-s − 1.33·19-s + 2.93·20-s + 2.24·21-s − 0.500·22-s + 0.167·23-s − 2.37·24-s + 1.79·25-s − 0.0724·26-s + 3.03·27-s + 2.07·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8041\)    =    \(11 \cdot 17 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8041} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8041,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.550414689\)
\(L(\frac12)\)  \(\approx\)  \(3.550414689\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{11,\;17,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{11,\;17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad11 \( 1 - T \)
17 \( 1 - T \)
43 \( 1 - T \)
good2 \( 1 + 2.34T + 2T^{2} \)
3 \( 1 - 3.28T + 3T^{2} \)
5 \( 1 - 3.73T + 5T^{2} \)
7 \( 1 - 3.13T + 7T^{2} \)
13 \( 1 - 0.157T + 13T^{2} \)
19 \( 1 + 5.83T + 19T^{2} \)
23 \( 1 - 0.802T + 23T^{2} \)
29 \( 1 - 1.68T + 29T^{2} \)
31 \( 1 - 0.923T + 31T^{2} \)
37 \( 1 - 7.42T + 37T^{2} \)
41 \( 1 + 5.49T + 41T^{2} \)
47 \( 1 + 6.58T + 47T^{2} \)
53 \( 1 + 4.16T + 53T^{2} \)
59 \( 1 + 0.00333T + 59T^{2} \)
61 \( 1 - 7.96T + 61T^{2} \)
67 \( 1 + 2.69T + 67T^{2} \)
71 \( 1 + 0.778T + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 + 7.37T + 79T^{2} \)
83 \( 1 - 6.85T + 83T^{2} \)
89 \( 1 + 9.62T + 89T^{2} \)
97 \( 1 - 5.96T + 97T^{2} \)
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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

−8.215028977206536624337043453497, −7.49952620186974967704031180317, −6.78402307655172693653886423880, −6.16395918489297097223055807309, −4.94345086299236171321799423489, −4.18547598012274198717090418884, −2.88877813168523752347315748993, −2.31995333427357147683455362429, −1.66893526541161428302077675760, −1.29065314398540561233596851331, 1.29065314398540561233596851331, 1.66893526541161428302077675760, 2.31995333427357147683455362429, 2.88877813168523752347315748993, 4.18547598012274198717090418884, 4.94345086299236171321799423489, 6.16395918489297097223055807309, 6.78402307655172693653886423880, 7.49952620186974967704031180317, 8.215028977206536624337043453497

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