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 − 2.47·3-s + 4-s − 0.567·5-s − 2.47·6-s − 4.82·7-s + 8-s + 3.13·9-s − 0.567·10-s + 1.83·11-s − 2.47·12-s − 5.69·13-s − 4.82·14-s + 1.40·15-s + 16-s + 4.15·17-s + 3.13·18-s + 19-s − 0.567·20-s + 11.9·21-s + 1.83·22-s + 3.32·23-s − 2.47·24-s − 4.67·25-s − 5.69·26-s − 0.333·27-s − 4.82·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.42·3-s + 0.5·4-s − 0.253·5-s − 1.01·6-s − 1.82·7-s + 0.353·8-s + 1.04·9-s − 0.179·10-s + 0.552·11-s − 0.714·12-s − 1.58·13-s − 1.28·14-s + 0.362·15-s + 0.250·16-s + 1.00·17-s + 0.738·18-s + 0.229·19-s − 0.126·20-s + 2.60·21-s + 0.390·22-s + 0.693·23-s − 0.505·24-s − 0.935·25-s − 1.11·26-s − 0.0641·27-s − 0.911·28-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 \( 1 + 2.47T + 3T^{2} \)
5 \( 1 + 0.567T + 5T^{2} \)
7 \( 1 + 4.82T + 7T^{2} \)
11 \( 1 - 1.83T + 11T^{2} \)
13 \( 1 + 5.69T + 13T^{2} \)
17 \( 1 - 4.15T + 17T^{2} \)
23 \( 1 - 3.32T + 23T^{2} \)
29 \( 1 - 0.666T + 29T^{2} \)
31 \( 1 - 1.19T + 31T^{2} \)
37 \( 1 + 1.66T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 + 0.920T + 43T^{2} \)
47 \( 1 - 5.33T + 47T^{2} \)
53 \( 1 + 6.95T + 53T^{2} \)
59 \( 1 + 9.75T + 59T^{2} \)
61 \( 1 - 3.51T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 0.562T + 73T^{2} \)
79 \( 1 - 7.21T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + 8.31T + 89T^{2} \)
97 \( 1 - 6.85T + 97T^{2} \)
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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

−7.29598724657043021987111758467, −6.47734861547441512344195495295, −6.15039859091400527183776548689, −5.44916939344132358396237643653, −4.81994362431783415170646619748, −3.98832653693840574849692369065, −3.24098293013627034149577299388, −2.47729176210984530926167934155, −0.959177492867094051376393720103, 0, 0.959177492867094051376393720103, 2.47729176210984530926167934155, 3.24098293013627034149577299388, 3.98832653693840574849692369065, 4.81994362431783415170646619748, 5.44916939344132358396237643653, 6.15039859091400527183776548689, 6.47734861547441512344195495295, 7.29598724657043021987111758467

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