Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.51·3-s + 4-s − 0.563·5-s + 1.51·6-s + 2.76·7-s + 8-s − 0.689·9-s − 0.563·10-s − 5.11·11-s + 1.51·12-s + 3.17·13-s + 2.76·14-s − 0.855·15-s + 16-s − 4.89·17-s − 0.689·18-s + 19-s − 0.563·20-s + 4.20·21-s − 5.11·22-s + 3.22·23-s + 1.51·24-s − 4.68·25-s + 3.17·26-s − 5.60·27-s + 2.76·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.877·3-s + 0.5·4-s − 0.251·5-s + 0.620·6-s + 1.04·7-s + 0.353·8-s − 0.229·9-s − 0.178·10-s − 1.54·11-s + 0.438·12-s + 0.880·13-s + 0.740·14-s − 0.220·15-s + 0.250·16-s − 1.18·17-s − 0.162·18-s + 0.229·19-s − 0.125·20-s + 0.918·21-s − 1.09·22-s + 0.672·23-s + 0.310·24-s − 0.936·25-s + 0.622·26-s − 1.07·27-s + 0.523·28-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  =  1
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 1.51T + 3T^{2} \)
5 \( 1 + 0.563T + 5T^{2} \)
7 \( 1 - 2.76T + 7T^{2} \)
11 \( 1 + 5.11T + 11T^{2} \)
13 \( 1 - 3.17T + 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
23 \( 1 - 3.22T + 23T^{2} \)
29 \( 1 + 9.42T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + 4.23T + 41T^{2} \)
43 \( 1 + 7.71T + 43T^{2} \)
47 \( 1 - 1.83T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 - 9.35T + 59T^{2} \)
61 \( 1 + 3.59T + 61T^{2} \)
67 \( 1 - 1.15T + 67T^{2} \)
71 \( 1 - 4.49T + 71T^{2} \)
73 \( 1 + 6.10T + 73T^{2} \)
79 \( 1 + 6.05T + 79T^{2} \)
83 \( 1 + 5.53T + 83T^{2} \)
89 \( 1 - 2.94T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−7.36630056813812330853148770116, −7.12592997589832848942943868249, −5.72362035461037794082346615426, −5.44038309700538399494814111315, −4.67105283828919698657490289295, −3.68859421928351085030768298664, −3.32959630972763468531158895095, −2.13014705338856190643900394615, −1.88330243898225286044459748733, 0, 1.88330243898225286044459748733, 2.13014705338856190643900394615, 3.32959630972763468531158895095, 3.68859421928351085030768298664, 4.67105283828919698657490289295, 5.44038309700538399494814111315, 5.72362035461037794082346615426, 7.12592997589832848942943868249, 7.36630056813812330853148770116

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