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  − 2.58·2-s − 2.87·3-s + 4.68·4-s − 1.51·5-s + 7.44·6-s + 0.161·7-s − 6.93·8-s + 5.28·9-s + 3.91·10-s + 11-s − 13.4·12-s − 6.99·13-s − 0.417·14-s + 4.36·15-s + 8.56·16-s − 4.66·17-s − 13.6·18-s + 4.03·19-s − 7.09·20-s − 0.464·21-s − 2.58·22-s + 4.97·23-s + 19.9·24-s − 2.70·25-s + 18.0·26-s − 6.58·27-s + 0.756·28-s + ⋯
L(s)  = 1  − 1.82·2-s − 1.66·3-s + 2.34·4-s − 0.677·5-s + 3.03·6-s + 0.0610·7-s − 2.45·8-s + 1.76·9-s + 1.23·10-s + 0.301·11-s − 3.89·12-s − 1.94·13-s − 0.111·14-s + 1.12·15-s + 2.14·16-s − 1.13·17-s − 3.22·18-s + 0.926·19-s − 1.58·20-s − 0.101·21-s − 0.551·22-s + 1.03·23-s + 4.07·24-s − 0.540·25-s + 3.54·26-s − 1.26·27-s + 0.142·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad11 \( 1 - T \)
547 \( 1 - T \)
good2 \( 1 + 2.58T + 2T^{2} \)
3 \( 1 + 2.87T + 3T^{2} \)
5 \( 1 + 1.51T + 5T^{2} \)
7 \( 1 - 0.161T + 7T^{2} \)
13 \( 1 + 6.99T + 13T^{2} \)
17 \( 1 + 4.66T + 17T^{2} \)
19 \( 1 - 4.03T + 19T^{2} \)
23 \( 1 - 4.97T + 23T^{2} \)
29 \( 1 + 1.95T + 29T^{2} \)
31 \( 1 + 0.546T + 31T^{2} \)
37 \( 1 + 3.94T + 37T^{2} \)
41 \( 1 - 4.07T + 41T^{2} \)
43 \( 1 - 5.47T + 43T^{2} \)
47 \( 1 + 2.72T + 47T^{2} \)
53 \( 1 - 13.7T + 53T^{2} \)
59 \( 1 + 0.114T + 59T^{2} \)
61 \( 1 + 2.36T + 61T^{2} \)
67 \( 1 - 4.13T + 67T^{2} \)
71 \( 1 - 5.67T + 71T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 + 7.56T + 79T^{2} \)
83 \( 1 + 9.11T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 6.91T + 97T^{2} \)
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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

−7.38994461031121956642871599622, −7.30299229980785341424858487116, −6.67892452174389750580776204959, −5.78040641329260006557218507305, −5.02832947363877595040718129775, −4.24326848942294284168840130482, −2.83919895121863082737995581876, −1.83118850833484500051265882160, −0.72761826427934126468404841149, 0, 0.72761826427934126468404841149, 1.83118850833484500051265882160, 2.83919895121863082737995581876, 4.24326848942294284168840130482, 5.02832947363877595040718129775, 5.78040641329260006557218507305, 6.67892452174389750580776204959, 7.30299229980785341424858487116, 7.38994461031121956642871599622

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