Properties

Degree 2
Conductor 6047
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 + 0.109·3-s + 5.06·4-s + 0.753·5-s − 0.292·6-s + 3.98·7-s − 8.13·8-s − 2.98·9-s − 2.00·10-s + 0.383·11-s + 0.556·12-s + 2.64·13-s − 10.5·14-s + 0.0828·15-s + 11.4·16-s + 8.14·17-s + 7.93·18-s − 5.99·19-s + 3.81·20-s + 0.438·21-s − 1.02·22-s − 1.73·23-s − 0.894·24-s − 4.43·25-s − 7.02·26-s − 0.658·27-s + 20.1·28-s + ⋯
L(s)  = 1  − 1.87·2-s + 0.0634·3-s + 2.53·4-s + 0.337·5-s − 0.119·6-s + 1.50·7-s − 2.87·8-s − 0.995·9-s − 0.633·10-s + 0.115·11-s + 0.160·12-s + 0.733·13-s − 2.83·14-s + 0.0213·15-s + 2.87·16-s + 1.97·17-s + 1.87·18-s − 1.37·19-s + 0.852·20-s + 0.0956·21-s − 0.217·22-s − 0.362·23-s − 0.182·24-s − 0.886·25-s − 1.37·26-s − 0.126·27-s + 3.81·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6047\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6047} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6047,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 6047$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 6047$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad6047 \( 1+O(T) \)
good2 \( 1 + 2.65T + 2T^{2} \)
3 \( 1 - 0.109T + 3T^{2} \)
5 \( 1 - 0.753T + 5T^{2} \)
7 \( 1 - 3.98T + 7T^{2} \)
11 \( 1 - 0.383T + 11T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 - 8.14T + 17T^{2} \)
19 \( 1 + 5.99T + 19T^{2} \)
23 \( 1 + 1.73T + 23T^{2} \)
29 \( 1 - 5.51T + 29T^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 - 2.46T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 9.53T + 43T^{2} \)
47 \( 1 - 9.71T + 47T^{2} \)
53 \( 1 + 6.80T + 53T^{2} \)
59 \( 1 + 1.90T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 6.26T + 67T^{2} \)
71 \( 1 - 3.63T + 71T^{2} \)
73 \( 1 - 4.38T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 4.10T + 83T^{2} \)
89 \( 1 + 1.79T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−7.933180842993486636053689900985, −7.48535310026595035190253044283, −6.42145951600170746805184007588, −5.84062578521542378014186966210, −5.15084151826288012512729373615, −3.77340203483779707257087560783, −2.80969564588285458756602271391, −1.77437622653453381002132735862, −1.40602793794636103038764758252, 0, 1.40602793794636103038764758252, 1.77437622653453381002132735862, 2.80969564588285458756602271391, 3.77340203483779707257087560783, 5.15084151826288012512729373615, 5.84062578521542378014186966210, 6.42145951600170746805184007588, 7.48535310026595035190253044283, 7.933180842993486636053689900985

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