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  − 2.73·2-s + 1.50·3-s + 5.45·4-s − 3.60·5-s − 4.10·6-s + 2.19·7-s − 9.43·8-s − 0.737·9-s + 9.84·10-s + 3.45·11-s + 8.20·12-s + 13-s − 5.98·14-s − 5.42·15-s + 14.8·16-s + 0.177·17-s + 2.01·18-s + 2.34·19-s − 19.6·20-s + 3.29·21-s − 9.44·22-s − 6.25·23-s − 14.1·24-s + 8.00·25-s − 2.73·26-s − 5.62·27-s + 11.9·28-s + ⋯
L(s)  = 1  − 1.93·2-s + 0.868·3-s + 2.72·4-s − 1.61·5-s − 1.67·6-s + 0.828·7-s − 3.33·8-s − 0.245·9-s + 3.11·10-s + 1.04·11-s + 2.36·12-s + 0.277·13-s − 1.59·14-s − 1.40·15-s + 3.71·16-s + 0.0430·17-s + 0.474·18-s + 0.537·19-s − 4.39·20-s + 0.719·21-s − 2.01·22-s − 1.30·23-s − 2.89·24-s + 1.60·25-s − 0.535·26-s − 1.08·27-s + 2.26·28-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 \( 1 + 2.73T + 2T^{2} \)
3 \( 1 - 1.50T + 3T^{2} \)
5 \( 1 + 3.60T + 5T^{2} \)
7 \( 1 - 2.19T + 7T^{2} \)
11 \( 1 - 3.45T + 11T^{2} \)
17 \( 1 - 0.177T + 17T^{2} \)
19 \( 1 - 2.34T + 19T^{2} \)
23 \( 1 + 6.25T + 23T^{2} \)
29 \( 1 + 6.36T + 29T^{2} \)
31 \( 1 + 0.718T + 31T^{2} \)
37 \( 1 - 7.99T + 37T^{2} \)
41 \( 1 - 9.73T + 41T^{2} \)
43 \( 1 - 4.13T + 43T^{2} \)
47 \( 1 + 2.61T + 47T^{2} \)
53 \( 1 + 9.07T + 53T^{2} \)
59 \( 1 - 4.03T + 59T^{2} \)
61 \( 1 + 0.809T + 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 + 6.47T + 79T^{2} \)
83 \( 1 - 5.29T + 83T^{2} \)
89 \( 1 - 8.97T + 89T^{2} \)
97 \( 1 + 5.00T + 97T^{2} \)
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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

−7.76768834233504241453079238424, −7.48797746550365996515155985098, −6.46533636646376643832522364622, −5.75554912880446836845586681673, −4.30183818887180853215898456257, −3.65428406072062455164812748327, −2.88624351877326581004465517029, −1.94705491456334666542461836145, −1.10129529354204782222211950231, 0, 1.10129529354204782222211950231, 1.94705491456334666542461836145, 2.88624351877326581004465517029, 3.65428406072062455164812748327, 4.30183818887180853215898456257, 5.75554912880446836845586681673, 6.46533636646376643832522364622, 7.48797746550365996515155985098, 7.76768834233504241453079238424

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