Properties

Degree 2
Conductor 4019
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.43·2-s + 2.49·3-s + 3.92·4-s + 3.69·5-s − 6.07·6-s + 0.601·7-s − 4.69·8-s + 3.21·9-s − 8.98·10-s − 5.31·11-s + 9.79·12-s + 3.09·13-s − 1.46·14-s + 9.20·15-s + 3.56·16-s + 0.734·17-s − 7.83·18-s + 2.83·19-s + 14.4·20-s + 1.49·21-s + 12.9·22-s − 1.79·23-s − 11.6·24-s + 8.62·25-s − 7.53·26-s + 0.543·27-s + 2.36·28-s + ⋯
L(s)  = 1  − 1.72·2-s + 1.43·3-s + 1.96·4-s + 1.65·5-s − 2.47·6-s + 0.227·7-s − 1.65·8-s + 1.07·9-s − 2.84·10-s − 1.60·11-s + 2.82·12-s + 0.858·13-s − 0.391·14-s + 2.37·15-s + 0.891·16-s + 0.178·17-s − 1.84·18-s + 0.650·19-s + 3.24·20-s + 0.327·21-s + 2.76·22-s − 0.374·23-s − 2.38·24-s + 1.72·25-s − 1.47·26-s + 0.104·27-s + 0.446·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4019\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4019} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4019,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.076007204$
$L(\frac12)$  $\approx$  $2.076007204$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 4019$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 4019$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad4019 \( 1+O(T) \)
good2 \( 1 + 2.43T + 2T^{2} \)
3 \( 1 - 2.49T + 3T^{2} \)
5 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 - 0.601T + 7T^{2} \)
11 \( 1 + 5.31T + 11T^{2} \)
13 \( 1 - 3.09T + 13T^{2} \)
17 \( 1 - 0.734T + 17T^{2} \)
19 \( 1 - 2.83T + 19T^{2} \)
23 \( 1 + 1.79T + 23T^{2} \)
29 \( 1 - 7.08T + 29T^{2} \)
31 \( 1 - 2.28T + 31T^{2} \)
37 \( 1 - 9.08T + 37T^{2} \)
41 \( 1 - 1.45T + 41T^{2} \)
43 \( 1 - 0.573T + 43T^{2} \)
47 \( 1 + 7.18T + 47T^{2} \)
53 \( 1 - 1.63T + 53T^{2} \)
59 \( 1 - 0.375T + 59T^{2} \)
61 \( 1 + 3.59T + 61T^{2} \)
67 \( 1 - 9.13T + 67T^{2} \)
71 \( 1 + 3.82T + 71T^{2} \)
73 \( 1 - 6.13T + 73T^{2} \)
79 \( 1 - 5.08T + 79T^{2} \)
83 \( 1 + 1.18T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 5.67T + 97T^{2} \)
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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

−8.380381345457406740909385409090, −8.132765193345328736666816633432, −7.42665535766548810734434334162, −6.46699816320083339813784395050, −5.79083956387425816786865196711, −4.76417839846769634682157089707, −3.14978981803843182756010857547, −2.55896759642306460501294242006, −1.94883536874987214775897634421, −1.04529591821678887259058565954, 1.04529591821678887259058565954, 1.94883536874987214775897634421, 2.55896759642306460501294242006, 3.14978981803843182756010857547, 4.76417839846769634682157089707, 5.79083956387425816786865196711, 6.46699816320083339813784395050, 7.42665535766548810734434334162, 8.132765193345328736666816633432, 8.380381345457406740909385409090

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