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  − 2.56·2-s − 0.512·3-s + 4.60·4-s − 4.01·5-s + 1.31·6-s + 0.174·7-s − 6.69·8-s − 2.73·9-s + 10.3·10-s − 2.58·11-s − 2.36·12-s + 0.398·13-s − 0.449·14-s + 2.05·15-s + 7.99·16-s + 2.25·17-s + 7.03·18-s + 4.68·19-s − 18.4·20-s − 0.0895·21-s + 6.65·22-s − 6.94·23-s + 3.43·24-s + 11.0·25-s − 1.02·26-s + 2.94·27-s + 0.804·28-s + ⋯
L(s)  = 1  − 1.81·2-s − 0.295·3-s + 2.30·4-s − 1.79·5-s + 0.537·6-s + 0.0660·7-s − 2.36·8-s − 0.912·9-s + 3.26·10-s − 0.780·11-s − 0.681·12-s + 0.110·13-s − 0.120·14-s + 0.530·15-s + 1.99·16-s + 0.547·17-s + 1.65·18-s + 1.07·19-s − 4.13·20-s − 0.0195·21-s + 1.41·22-s − 1.44·23-s + 0.700·24-s + 2.21·25-s − 0.201·26-s + 0.565·27-s + 0.152·28-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  =  $(2,\ 4019,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.07913326432$
$L(\frac12)$  $\approx$  $0.07913326432$
$L(\frac{3}{2})$   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 \( 1 + 2.56T + 2T^{2} \)
3 \( 1 + 0.512T + 3T^{2} \)
5 \( 1 + 4.01T + 5T^{2} \)
7 \( 1 - 0.174T + 7T^{2} \)
11 \( 1 + 2.58T + 11T^{2} \)
13 \( 1 - 0.398T + 13T^{2} \)
17 \( 1 - 2.25T + 17T^{2} \)
19 \( 1 - 4.68T + 19T^{2} \)
23 \( 1 + 6.94T + 23T^{2} \)
29 \( 1 - 0.630T + 29T^{2} \)
31 \( 1 + 1.13T + 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
41 \( 1 + 0.895T + 41T^{2} \)
43 \( 1 + 7.33T + 43T^{2} \)
47 \( 1 + 5.63T + 47T^{2} \)
53 \( 1 + 4.48T + 53T^{2} \)
59 \( 1 + 1.04T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 0.160T + 67T^{2} \)
71 \( 1 + 1.67T + 71T^{2} \)
73 \( 1 + 4.74T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + 5.97T + 89T^{2} \)
97 \( 1 - 19.2T + 97T^{2} \)
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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

−8.298321039363364689168114086724, −7.900611095810507049011956545335, −7.44136790927635943034660254452, −6.61578621773474099308670072786, −5.69866777618084841745258743998, −4.73512405140416793358118394377, −3.43374358566989160972212269965, −2.90716426435038381032054137745, −1.50695865036746605952357561964, −0.21731875526645175043336423202, 0.21731875526645175043336423202, 1.50695865036746605952357561964, 2.90716426435038381032054137745, 3.43374358566989160972212269965, 4.73512405140416793358118394377, 5.69866777618084841745258743998, 6.61578621773474099308670072786, 7.44136790927635943034660254452, 7.900611095810507049011956545335, 8.298321039363364689168114086724

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