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 − 3.09·3-s + 4-s + 2.68·5-s − 3.09·6-s + 2.69·7-s + 8-s + 6.57·9-s + 2.68·10-s − 5.80·11-s − 3.09·12-s − 1.02·13-s + 2.69·14-s − 8.31·15-s + 16-s − 5.21·17-s + 6.57·18-s + 19-s + 2.68·20-s − 8.34·21-s − 5.80·22-s + 4.91·23-s − 3.09·24-s + 2.22·25-s − 1.02·26-s − 11.0·27-s + 2.69·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.78·3-s + 0.5·4-s + 1.20·5-s − 1.26·6-s + 1.01·7-s + 0.353·8-s + 2.19·9-s + 0.850·10-s − 1.75·11-s − 0.893·12-s − 0.282·13-s + 0.721·14-s − 2.14·15-s + 0.250·16-s − 1.26·17-s + 1.54·18-s + 0.229·19-s + 0.601·20-s − 1.82·21-s − 1.23·22-s + 1.02·23-s − 0.631·24-s + 0.445·25-s − 0.200·26-s − 2.12·27-s + 0.509·28-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 \( 1 + 3.09T + 3T^{2} \)
5 \( 1 - 2.68T + 5T^{2} \)
7 \( 1 - 2.69T + 7T^{2} \)
11 \( 1 + 5.80T + 11T^{2} \)
13 \( 1 + 1.02T + 13T^{2} \)
17 \( 1 + 5.21T + 17T^{2} \)
23 \( 1 - 4.91T + 23T^{2} \)
29 \( 1 - 3.23T + 29T^{2} \)
31 \( 1 - 6.99T + 31T^{2} \)
37 \( 1 + 9.62T + 37T^{2} \)
41 \( 1 + 5.63T + 41T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 + 2.14T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 5.68T + 59T^{2} \)
61 \( 1 + 8.48T + 61T^{2} \)
67 \( 1 + 4.70T + 67T^{2} \)
71 \( 1 - 6.37T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 - 2.49T + 79T^{2} \)
83 \( 1 + 6.10T + 83T^{2} \)
89 \( 1 + 2.99T + 89T^{2} \)
97 \( 1 - 4.49T + 97T^{2} \)
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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

−7.12746335252625328147529171037, −6.55014780853296401509137397998, −5.95823663908774545708324233250, −5.12145842521884219161221879968, −5.03884936908399574943597612838, −4.52459318902343777914712911863, −3.00689700281529067623892084836, −2.08895279891090164096940795010, −1.37546159903023060944484672814, 0, 1.37546159903023060944484672814, 2.08895279891090164096940795010, 3.00689700281529067623892084836, 4.52459318902343777914712911863, 5.03884936908399574943597612838, 5.12145842521884219161221879968, 5.95823663908774545708324233250, 6.55014780853296401509137397998, 7.12746335252625328147529171037

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