Properties

Degree 2
Conductor $ 13 \cdot 463 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·2-s + 0.516·3-s + 4.83·4-s − 3.96·5-s − 1.35·6-s + 3.66·7-s − 7.41·8-s − 2.73·9-s + 10.3·10-s − 4.05·11-s + 2.49·12-s − 13-s − 9.58·14-s − 2.04·15-s + 9.71·16-s + 6.83·17-s + 7.14·18-s + 5.65·19-s − 19.1·20-s + 1.89·21-s + 10.6·22-s − 4.75·23-s − 3.82·24-s + 10.7·25-s + 2.61·26-s − 2.96·27-s + 17.7·28-s + ⋯
L(s)  = 1  − 1.84·2-s + 0.298·3-s + 2.41·4-s − 1.77·5-s − 0.551·6-s + 1.38·7-s − 2.62·8-s − 0.911·9-s + 3.28·10-s − 1.22·11-s + 0.721·12-s − 0.277·13-s − 2.56·14-s − 0.529·15-s + 2.42·16-s + 1.65·17-s + 1.68·18-s + 1.29·19-s − 4.29·20-s + 0.412·21-s + 2.26·22-s − 0.992·23-s − 0.781·24-s + 2.14·25-s + 0.512·26-s − 0.569·27-s + 3.34·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6019\)    =    \(13 \cdot 463\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6019} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6019,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;463\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{13,\;463\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad13 \( 1 + T \)
463 \( 1 + T \)
good2 \( 1 + 2.61T + 2T^{2} \)
3 \( 1 - 0.516T + 3T^{2} \)
5 \( 1 + 3.96T + 5T^{2} \)
7 \( 1 - 3.66T + 7T^{2} \)
11 \( 1 + 4.05T + 11T^{2} \)
17 \( 1 - 6.83T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 + 4.75T + 23T^{2} \)
29 \( 1 - 2.57T + 29T^{2} \)
31 \( 1 + 3.95T + 31T^{2} \)
37 \( 1 + 7.21T + 37T^{2} \)
41 \( 1 + 9.50T + 41T^{2} \)
43 \( 1 + 5.45T + 43T^{2} \)
47 \( 1 - 9.77T + 47T^{2} \)
53 \( 1 - 2.55T + 53T^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 - 6.44T + 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 - 9.66T + 73T^{2} \)
79 \( 1 - 17.0T + 79T^{2} \)
83 \( 1 + 5.34T + 83T^{2} \)
89 \( 1 - 1.55T + 89T^{2} \)
97 \( 1 + 8.63T + 97T^{2} \)
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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

−7.79291668845610789882393426316, −7.68106575457857118276500889648, −6.92082007553453172426555905649, −5.49880540451018679660748644220, −5.08203614237073897906257079334, −3.64884661951060600444294943629, −3.04856759359229206723009250237, −2.06960104453330681247205019735, −0.959869424300801309885742343224, 0, 0.959869424300801309885742343224, 2.06960104453330681247205019735, 3.04856759359229206723009250237, 3.64884661951060600444294943629, 5.08203614237073897906257079334, 5.49880540451018679660748644220, 6.92082007553453172426555905649, 7.68106575457857118276500889648, 7.79291668845610789882393426316

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