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 + 2.23·3-s + 4-s + 0.607·5-s + 2.23·6-s − 1.94·7-s + 8-s + 2.00·9-s + 0.607·10-s − 1.03·11-s + 2.23·12-s − 4.02·13-s − 1.94·14-s + 1.35·15-s + 16-s − 3.02·17-s + 2.00·18-s + 19-s + 0.607·20-s − 4.36·21-s − 1.03·22-s − 4.16·23-s + 2.23·24-s − 4.63·25-s − 4.02·26-s − 2.22·27-s − 1.94·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.271·5-s + 0.913·6-s − 0.736·7-s + 0.353·8-s + 0.668·9-s + 0.191·10-s − 0.312·11-s + 0.645·12-s − 1.11·13-s − 0.520·14-s + 0.350·15-s + 0.250·16-s − 0.733·17-s + 0.472·18-s + 0.229·19-s + 0.135·20-s − 0.951·21-s − 0.221·22-s − 0.868·23-s + 0.456·24-s − 0.926·25-s − 0.788·26-s − 0.428·27-s − 0.368·28-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 \( 1 - 2.23T + 3T^{2} \)
5 \( 1 - 0.607T + 5T^{2} \)
7 \( 1 + 1.94T + 7T^{2} \)
11 \( 1 + 1.03T + 11T^{2} \)
13 \( 1 + 4.02T + 13T^{2} \)
17 \( 1 + 3.02T + 17T^{2} \)
23 \( 1 + 4.16T + 23T^{2} \)
29 \( 1 - 0.615T + 29T^{2} \)
31 \( 1 + 9.20T + 31T^{2} \)
37 \( 1 - 3.82T + 37T^{2} \)
41 \( 1 + 0.126T + 41T^{2} \)
43 \( 1 - 5.11T + 43T^{2} \)
47 \( 1 - 1.69T + 47T^{2} \)
53 \( 1 - 3.69T + 53T^{2} \)
59 \( 1 - 6.56T + 59T^{2} \)
61 \( 1 + 5.49T + 61T^{2} \)
67 \( 1 + 5.99T + 67T^{2} \)
71 \( 1 + 6.59T + 71T^{2} \)
73 \( 1 - 6.60T + 73T^{2} \)
79 \( 1 - 0.650T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + 4.78T + 89T^{2} \)
97 \( 1 - 0.168T + 97T^{2} \)
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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

−7.49176351060178272838441389361, −6.91545399200286288939564465358, −5.98722681629676099198501506134, −5.45921719617154474285548074448, −4.42527073319741075655929266374, −3.83693006281422663596729435621, −3.04579666434937690250060394677, −2.42132995695630793489246353393, −1.84619456252831327977844355175, 0, 1.84619456252831327977844355175, 2.42132995695630793489246353393, 3.04579666434937690250060394677, 3.83693006281422663596729435621, 4.42527073319741075655929266374, 5.45921719617154474285548074448, 5.98722681629676099198501506134, 6.91545399200286288939564465358, 7.49176351060178272838441389361

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