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 − 2.58·3-s + 5.07·4-s + 0.599·5-s + 6.87·6-s − 3.90·7-s − 8.17·8-s + 3.67·9-s − 1.59·10-s − 2.24·11-s − 13.1·12-s + 13-s + 10.3·14-s − 1.54·15-s + 11.6·16-s − 5.38·17-s − 9.76·18-s − 1.97·19-s + 3.04·20-s + 10.0·21-s + 5.96·22-s + 6.80·23-s + 21.1·24-s − 4.64·25-s − 2.65·26-s − 1.73·27-s − 19.8·28-s + ⋯
L(s)  = 1  − 1.88·2-s − 1.49·3-s + 2.53·4-s + 0.268·5-s + 2.80·6-s − 1.47·7-s − 2.89·8-s + 1.22·9-s − 0.504·10-s − 0.676·11-s − 3.78·12-s + 0.277·13-s + 2.77·14-s − 0.399·15-s + 2.90·16-s − 1.30·17-s − 2.30·18-s − 0.452·19-s + 0.680·20-s + 2.20·21-s + 1.27·22-s + 1.41·23-s + 4.31·24-s − 0.928·25-s − 0.521·26-s − 0.333·27-s − 3.74·28-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 \( 1 + 2.58T + 3T^{2} \)
5 \( 1 - 0.599T + 5T^{2} \)
7 \( 1 + 3.90T + 7T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
17 \( 1 + 5.38T + 17T^{2} \)
19 \( 1 + 1.97T + 19T^{2} \)
23 \( 1 - 6.80T + 23T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 - 1.00T + 31T^{2} \)
37 \( 1 + 0.135T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 + 7.37T + 47T^{2} \)
53 \( 1 + 13.0T + 53T^{2} \)
59 \( 1 - 5.66T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 + 8.28T + 67T^{2} \)
71 \( 1 - 5.77T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 2.74T + 79T^{2} \)
83 \( 1 - 9.98T + 83T^{2} \)
89 \( 1 - 0.0258T + 89T^{2} \)
97 \( 1 + 8.17T + 97T^{2} \)
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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

−7.38659465368981256383899120146, −6.78566508449271362415801360429, −6.33625062563799631587178629134, −5.85902928248299977673485015703, −5.01039453054262082839448283808, −3.72051037704930043580033433355, −2.69636437323222573111975153382, −1.87631436296697441402941560421, −0.65344389828480060914112398326, 0, 0.65344389828480060914112398326, 1.87631436296697441402941560421, 2.69636437323222573111975153382, 3.72051037704930043580033433355, 5.01039453054262082839448283808, 5.85902928248299977673485015703, 6.33625062563799631587178629134, 6.78566508449271362415801360429, 7.38659465368981256383899120146

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