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 + 0.846·3-s + 4-s + 2.51·5-s + 0.846·6-s − 3.73·7-s + 8-s − 2.28·9-s + 2.51·10-s − 0.0753·11-s + 0.846·12-s − 5.73·13-s − 3.73·14-s + 2.13·15-s + 16-s + 3.44·17-s − 2.28·18-s + 19-s + 2.51·20-s − 3.15·21-s − 0.0753·22-s + 6.79·23-s + 0.846·24-s + 1.32·25-s − 5.73·26-s − 4.47·27-s − 3.73·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.488·3-s + 0.5·4-s + 1.12·5-s + 0.345·6-s − 1.41·7-s + 0.353·8-s − 0.760·9-s + 0.795·10-s − 0.0227·11-s + 0.244·12-s − 1.59·13-s − 0.997·14-s + 0.550·15-s + 0.250·16-s + 0.835·17-s − 0.538·18-s + 0.229·19-s + 0.562·20-s − 0.689·21-s − 0.0160·22-s + 1.41·23-s + 0.172·24-s + 0.265·25-s − 1.12·26-s − 0.860·27-s − 0.705·28-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 \( 1 - 0.846T + 3T^{2} \)
5 \( 1 - 2.51T + 5T^{2} \)
7 \( 1 + 3.73T + 7T^{2} \)
11 \( 1 + 0.0753T + 11T^{2} \)
13 \( 1 + 5.73T + 13T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
23 \( 1 - 6.79T + 23T^{2} \)
29 \( 1 - 1.05T + 29T^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 - 0.0136T + 37T^{2} \)
41 \( 1 + 5.23T + 41T^{2} \)
43 \( 1 + 8.50T + 43T^{2} \)
47 \( 1 + 6.20T + 47T^{2} \)
53 \( 1 - 3.31T + 53T^{2} \)
59 \( 1 - 8.85T + 59T^{2} \)
61 \( 1 + 1.05T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 - 9.08T + 79T^{2} \)
83 \( 1 - 0.661T + 83T^{2} \)
89 \( 1 + 4.15T + 89T^{2} \)
97 \( 1 + 0.682T + 97T^{2} \)
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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

−7.21395759099017216202265453089, −6.77663201084079715701622640717, −5.97833997267334507862105124315, −5.41737227304553566632450630498, −4.88502440677348297941863783983, −3.64495642709694972996330662939, −2.94733898534302993298914074871, −2.64378237268762622507407862976, −1.59409285758841356847971979027, 0, 1.59409285758841356847971979027, 2.64378237268762622507407862976, 2.94733898534302993298914074871, 3.64495642709694972996330662939, 4.88502440677348297941863783983, 5.41737227304553566632450630498, 5.97833997267334507862105124315, 6.77663201084079715701622640717, 7.21395759099017216202265453089

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