Properties

Degree 2
Conductor $ 11 \cdot 547 $
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 + 1.81·3-s + 4.84·4-s + 3.74·5-s − 4.75·6-s + 0.406·7-s − 7.43·8-s + 0.304·9-s − 9.80·10-s + 11-s + 8.80·12-s − 3.01·13-s − 1.06·14-s + 6.81·15-s + 9.76·16-s − 6.49·17-s − 0.796·18-s − 1.64·19-s + 18.1·20-s + 0.738·21-s − 2.61·22-s − 1.93·23-s − 13.5·24-s + 9.04·25-s + 7.88·26-s − 4.90·27-s + 1.96·28-s + ⋯
L(s)  = 1  − 1.84·2-s + 1.04·3-s + 2.42·4-s + 1.67·5-s − 1.94·6-s + 0.153·7-s − 2.62·8-s + 0.101·9-s − 3.09·10-s + 0.301·11-s + 2.54·12-s − 0.835·13-s − 0.284·14-s + 1.75·15-s + 2.44·16-s − 1.57·17-s − 0.187·18-s − 0.377·19-s + 4.05·20-s + 0.161·21-s − 0.557·22-s − 0.404·23-s − 2.75·24-s + 1.80·25-s + 1.54·26-s − 0.943·27-s + 0.371·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6017} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6017,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{11,\;547\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{11,\;547\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 - T \)
547 \( 1 - T \)
good2 \( 1 + 2.61T + 2T^{2} \)
3 \( 1 - 1.81T + 3T^{2} \)
5 \( 1 - 3.74T + 5T^{2} \)
7 \( 1 - 0.406T + 7T^{2} \)
13 \( 1 + 3.01T + 13T^{2} \)
17 \( 1 + 6.49T + 17T^{2} \)
19 \( 1 + 1.64T + 19T^{2} \)
23 \( 1 + 1.93T + 23T^{2} \)
29 \( 1 + 7.98T + 29T^{2} \)
31 \( 1 + 2.09T + 31T^{2} \)
37 \( 1 + 4.51T + 37T^{2} \)
41 \( 1 - 8.72T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + 1.52T + 47T^{2} \)
53 \( 1 + 5.40T + 53T^{2} \)
59 \( 1 - 5.97T + 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 - 7.92T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + 6.94T + 73T^{2} \)
79 \( 1 - 7.28T + 79T^{2} \)
83 \( 1 - 8.02T + 83T^{2} \)
89 \( 1 - 0.679T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−8.126380968778194264196630845132, −7.11347292213751743671705512329, −6.68450912713972960012147859771, −5.92445436005406152603643017000, −5.05340535273736906660314039042, −3.64502653965387600636356347904, −2.44328479189190905102015523941, −2.21013020336724721534198203479, −1.57364847846550094244004760861, 0, 1.57364847846550094244004760861, 2.21013020336724721534198203479, 2.44328479189190905102015523941, 3.64502653965387600636356347904, 5.05340535273736906660314039042, 5.92445436005406152603643017000, 6.68450912713972960012147859771, 7.11347292213751743671705512329, 8.126380968778194264196630845132

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