Properties

Degree 2
Conductor $ 13 \cdot 619 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·2-s − 2.61·3-s + 0.618·4-s − 1.23·5-s − 4.23·6-s − 0.381·7-s − 2.23·8-s + 3.85·9-s − 2.00·10-s + 5·11-s − 1.61·12-s − 13-s − 0.618·14-s + 3.23·15-s − 4.85·16-s − 0.236·17-s + 6.23·18-s − 3.76·19-s − 0.763·20-s + 21-s + 8.09·22-s − 4·23-s + 5.85·24-s − 3.47·25-s − 1.61·26-s − 2.23·27-s − 0.236·28-s + ⋯
L(s)  = 1  + 1.14·2-s − 1.51·3-s + 0.309·4-s − 0.552·5-s − 1.72·6-s − 0.144·7-s − 0.790·8-s + 1.28·9-s − 0.632·10-s + 1.50·11-s − 0.467·12-s − 0.277·13-s − 0.165·14-s + 0.835·15-s − 1.21·16-s − 0.0572·17-s + 1.46·18-s − 0.863·19-s − 0.170·20-s + 0.218·21-s + 1.72·22-s − 0.834·23-s + 1.19·24-s − 0.694·25-s − 0.317·26-s − 0.430·27-s − 0.0446·28-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  =  0
Selberg data  =  $(2,\ 8047,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9900264295$
$L(\frac12)$  $\approx$  $0.9900264295$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad13 \( 1 + T \)
619 \( 1 - T \)
good2 \( 1 - 1.61T + 2T^{2} \)
3 \( 1 + 2.61T + 3T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 + 0.381T + 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
17 \( 1 + 0.236T + 17T^{2} \)
19 \( 1 + 3.76T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2.76T + 29T^{2} \)
31 \( 1 + 1.76T + 31T^{2} \)
37 \( 1 - 6.47T + 37T^{2} \)
41 \( 1 + 0.236T + 41T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + 7.70T + 53T^{2} \)
59 \( 1 + 0.0901T + 59T^{2} \)
61 \( 1 + 5.94T + 61T^{2} \)
67 \( 1 - 6.70T + 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 + 0.527T + 79T^{2} \)
83 \( 1 + 6.23T + 83T^{2} \)
89 \( 1 - 6.85T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−7.51765719916662807588965463678, −6.70378590707603140027602997971, −6.18954150855916691256307382640, −5.85944248714376720987578440052, −4.94258001466775102999026884635, −4.25175162268397730553425788011, −4.03306415257982623498696513385, −2.99403507139816978495714122321, −1.71092222731916602807218821990, −0.44544847473242977688641858282, 0.44544847473242977688641858282, 1.71092222731916602807218821990, 2.99403507139816978495714122321, 4.03306415257982623498696513385, 4.25175162268397730553425788011, 4.94258001466775102999026884635, 5.85944248714376720987578440052, 6.18954150855916691256307382640, 6.70378590707603140027602997971, 7.51765719916662807588965463678

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