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  − 2.52·2-s + 0.978·3-s + 4.39·4-s − 0.161·5-s − 2.47·6-s − 2.77·7-s − 6.06·8-s − 2.04·9-s + 0.409·10-s − 5.90·11-s + 4.30·12-s − 4.84·13-s + 7.01·14-s − 0.158·15-s + 6.54·16-s − 7.33·17-s + 5.16·18-s + 4.31·19-s − 0.712·20-s − 2.71·21-s + 14.9·22-s + 0.421·23-s − 5.93·24-s − 4.97·25-s + 12.2·26-s − 4.93·27-s − 12.2·28-s + ⋯
L(s)  = 1  − 1.78·2-s + 0.565·3-s + 2.19·4-s − 0.0724·5-s − 1.01·6-s − 1.04·7-s − 2.14·8-s − 0.680·9-s + 0.129·10-s − 1.78·11-s + 1.24·12-s − 1.34·13-s + 1.87·14-s − 0.0409·15-s + 1.63·16-s − 1.77·17-s + 1.21·18-s + 0.988·19-s − 0.159·20-s − 0.592·21-s + 3.18·22-s + 0.0879·23-s − 1.21·24-s − 0.994·25-s + 2.40·26-s − 0.949·27-s − 2.30·28-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  =  $(2,\ 4019,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.03824110125$
$L(\frac12)$  $\approx$  $0.03824110125$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad4019 \( 1+O(T) \)
good2 \( 1 + 2.52T + 2T^{2} \)
3 \( 1 - 0.978T + 3T^{2} \)
5 \( 1 + 0.161T + 5T^{2} \)
7 \( 1 + 2.77T + 7T^{2} \)
11 \( 1 + 5.90T + 11T^{2} \)
13 \( 1 + 4.84T + 13T^{2} \)
17 \( 1 + 7.33T + 17T^{2} \)
19 \( 1 - 4.31T + 19T^{2} \)
23 \( 1 - 0.421T + 23T^{2} \)
29 \( 1 + 4.33T + 29T^{2} \)
31 \( 1 - 6.27T + 31T^{2} \)
37 \( 1 + 2.79T + 37T^{2} \)
41 \( 1 - 0.785T + 41T^{2} \)
43 \( 1 + 2.17T + 43T^{2} \)
47 \( 1 + 7.48T + 47T^{2} \)
53 \( 1 + 5.34T + 53T^{2} \)
59 \( 1 + 0.387T + 59T^{2} \)
61 \( 1 - 8.26T + 61T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + 6.31T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 2.64T + 83T^{2} \)
89 \( 1 + 3.89T + 89T^{2} \)
97 \( 1 + 6.96T + 97T^{2} \)
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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

−8.477513215788354352332352243944, −7.84833660539634551883316633309, −7.35975902561114310991845156603, −6.61896374673470509022749637891, −5.75196994783415466189245862298, −4.77939599126737991471457803021, −3.23116209979495923133910772307, −2.62736948044999091404332271437, −2.05772602428720830633685419868, −0.12769246294974464188213845307, 0.12769246294974464188213845307, 2.05772602428720830633685419868, 2.62736948044999091404332271437, 3.23116209979495923133910772307, 4.77939599126737991471457803021, 5.75196994783415466189245862298, 6.61896374673470509022749637891, 7.35975902561114310991845156603, 7.84833660539634551883316633309, 8.477513215788354352332352243944

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