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 + 0.460·3-s + 4-s + 0.645·5-s + 0.460·6-s + 1.69·7-s + 8-s − 2.78·9-s + 0.645·10-s + 0.878·11-s + 0.460·12-s − 2.21·13-s + 1.69·14-s + 0.297·15-s + 16-s − 3.24·17-s − 2.78·18-s + 19-s + 0.645·20-s + 0.782·21-s + 0.878·22-s − 0.250·23-s + 0.460·24-s − 4.58·25-s − 2.21·26-s − 2.66·27-s + 1.69·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.266·3-s + 0.5·4-s + 0.288·5-s + 0.188·6-s + 0.641·7-s + 0.353·8-s − 0.929·9-s + 0.204·10-s + 0.264·11-s + 0.133·12-s − 0.613·13-s + 0.453·14-s + 0.0767·15-s + 0.250·16-s − 0.787·17-s − 0.657·18-s + 0.229·19-s + 0.144·20-s + 0.170·21-s + 0.187·22-s − 0.0522·23-s + 0.0940·24-s − 0.916·25-s − 0.433·26-s − 0.513·27-s + 0.320·28-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 \( 1 - 0.460T + 3T^{2} \)
5 \( 1 - 0.645T + 5T^{2} \)
7 \( 1 - 1.69T + 7T^{2} \)
11 \( 1 - 0.878T + 11T^{2} \)
13 \( 1 + 2.21T + 13T^{2} \)
17 \( 1 + 3.24T + 17T^{2} \)
23 \( 1 + 0.250T + 23T^{2} \)
29 \( 1 + 6.95T + 29T^{2} \)
31 \( 1 + 5.17T + 31T^{2} \)
37 \( 1 - 4.12T + 37T^{2} \)
41 \( 1 + 4.51T + 41T^{2} \)
43 \( 1 - 0.863T + 43T^{2} \)
47 \( 1 + 8.45T + 47T^{2} \)
53 \( 1 - 1.10T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 7.83T + 61T^{2} \)
67 \( 1 + 4.12T + 67T^{2} \)
71 \( 1 + 6.15T + 71T^{2} \)
73 \( 1 + 0.0206T + 73T^{2} \)
79 \( 1 - 3.73T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 0.944T + 89T^{2} \)
97 \( 1 + 8.08T + 97T^{2} \)
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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

−7.56886674093581785274509066700, −6.67600075612039521709149611847, −5.98076308140867186976706231710, −5.35370989681612695296676369319, −4.73939562608908792029248037810, −3.89161979351856341261839978238, −3.16132366299433694306930238454, −2.23967873457407165815545158129, −1.68366339972686948143125319175, 0, 1.68366339972686948143125319175, 2.23967873457407165815545158129, 3.16132366299433694306930238454, 3.89161979351856341261839978238, 4.73939562608908792029248037810, 5.35370989681612695296676369319, 5.98076308140867186976706231710, 6.67600075612039521709149611847, 7.56886674093581785274509066700

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