Properties

Degree 2
Conductor $ 13 \cdot 619 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.65·2-s + 2.65·3-s + 5.03·4-s − 0.675·5-s − 7.04·6-s − 2.02·7-s − 8.04·8-s + 4.04·9-s + 1.79·10-s + 4.17·11-s + 13.3·12-s + 13-s + 5.37·14-s − 1.79·15-s + 11.2·16-s − 7.58·17-s − 10.7·18-s + 6.75·19-s − 3.40·20-s − 5.38·21-s − 11.0·22-s − 6.60·23-s − 21.3·24-s − 4.54·25-s − 2.65·26-s + 2.78·27-s − 10.2·28-s + ⋯
L(s)  = 1  − 1.87·2-s + 1.53·3-s + 2.51·4-s − 0.302·5-s − 2.87·6-s − 0.766·7-s − 2.84·8-s + 1.34·9-s + 0.566·10-s + 1.25·11-s + 3.85·12-s + 0.277·13-s + 1.43·14-s − 0.463·15-s + 2.81·16-s − 1.83·17-s − 2.53·18-s + 1.54·19-s − 0.760·20-s − 1.17·21-s − 2.35·22-s − 1.37·23-s − 4.36·24-s − 0.908·25-s − 0.520·26-s + 0.535·27-s − 1.92·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8047\)    =    \(13 \cdot 619\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8047} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8047,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{13,\;619\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{13,\;619\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 - T \)
619 \( 1 - T \)
good2 \( 1 + 2.65T + 2T^{2} \)
3 \( 1 - 2.65T + 3T^{2} \)
5 \( 1 + 0.675T + 5T^{2} \)
7 \( 1 + 2.02T + 7T^{2} \)
11 \( 1 - 4.17T + 11T^{2} \)
17 \( 1 + 7.58T + 17T^{2} \)
19 \( 1 - 6.75T + 19T^{2} \)
23 \( 1 + 6.60T + 23T^{2} \)
29 \( 1 - 1.94T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 + 5.32T + 37T^{2} \)
41 \( 1 + 3.19T + 41T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 - 5.32T + 47T^{2} \)
53 \( 1 - 0.0678T + 53T^{2} \)
59 \( 1 - 6.08T + 59T^{2} \)
61 \( 1 - 3.16T + 61T^{2} \)
67 \( 1 - 5.30T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 3.17T + 73T^{2} \)
79 \( 1 - 2.37T + 79T^{2} \)
83 \( 1 + 17.5T + 83T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 + 6.42T + 97T^{2} \)
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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

−7.78804038032169183156665607248, −6.99471793800257933787082243284, −6.68826210226418355559640979865, −5.85128830968704062912264677389, −4.20061076738076143106265180806, −3.57456099183427027512872391144, −2.80110127560840403563888528477, −2.06305348137516494155336209150, −1.30273254856256233070230659707, 0, 1.30273254856256233070230659707, 2.06305348137516494155336209150, 2.80110127560840403563888528477, 3.57456099183427027512872391144, 4.20061076738076143106265180806, 5.85128830968704062912264677389, 6.68826210226418355559640979865, 6.99471793800257933787082243284, 7.78804038032169183156665607248

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