Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
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-s − 3.07·3-s + 4-s − 3.30·5-s − 3.07·6-s − 2.62·7-s + 8-s + 6.48·9-s − 3.30·10-s + 3.59·11-s − 3.07·12-s − 3.17·13-s − 2.62·14-s + 10.1·15-s + 16-s + 4.85·17-s + 6.48·18-s − 19-s − 3.30·20-s + 8.07·21-s + 3.59·22-s − 3.16·23-s − 3.07·24-s + 5.93·25-s − 3.17·26-s − 10.7·27-s − 2.62·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.77·3-s + 0.5·4-s − 1.47·5-s − 1.25·6-s − 0.990·7-s + 0.353·8-s + 2.16·9-s − 1.04·10-s + 1.08·11-s − 0.888·12-s − 0.879·13-s − 0.700·14-s + 2.62·15-s + 0.250·16-s + 1.17·17-s + 1.52·18-s − 0.229·19-s − 0.739·20-s + 1.76·21-s + 0.766·22-s − 0.660·23-s − 0.628·24-s + 1.18·25-s − 0.622·26-s − 2.06·27-s − 0.495·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8018\)    =    \(2 \cdot 19 \cdot 211\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8018} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5726602054$
$L(\frac12)$  $\approx$  $0.5726602054$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
19 \( 1 + T \)
211 \( 1 + T \)
good3 \( 1 + 3.07T + 3T^{2} \)
5 \( 1 + 3.30T + 5T^{2} \)
7 \( 1 + 2.62T + 7T^{2} \)
11 \( 1 - 3.59T + 11T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 - 4.85T + 17T^{2} \)
23 \( 1 + 3.16T + 23T^{2} \)
29 \( 1 + 1.52T + 29T^{2} \)
31 \( 1 + 6.15T + 31T^{2} \)
37 \( 1 - 7.44T + 37T^{2} \)
41 \( 1 - 0.725T + 41T^{2} \)
43 \( 1 + 1.68T + 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 - 9.64T + 53T^{2} \)
59 \( 1 + 9.95T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 5.58T + 67T^{2} \)
71 \( 1 - 3.62T + 71T^{2} \)
73 \( 1 - 7.67T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 4.32T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + 9.41T + 97T^{2} \)
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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

−7.42516246598545169651670517767, −7.01476905813939927283630026235, −6.32725965956562526383376284303, −5.79857630245533194357235148870, −5.03204515348639892729591899058, −4.27348777040628447640018281717, −3.83958897802089496999911159533, −3.05221820785067812586565283312, −1.50018851239033205838018208598, −0.38205663054728895015924175010, 0.38205663054728895015924175010, 1.50018851239033205838018208598, 3.05221820785067812586565283312, 3.83958897802089496999911159533, 4.27348777040628447640018281717, 5.03204515348639892729591899058, 5.79857630245533194357235148870, 6.32725965956562526383376284303, 7.01476905813939927283630026235, 7.42516246598545169651670517767

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