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  − 2.59·2-s + 3.24·3-s + 4.72·4-s − 1.12·5-s − 8.40·6-s − 0.119·7-s − 7.05·8-s + 7.51·9-s + 2.91·10-s − 1.92·11-s + 15.3·12-s + 3.49·13-s + 0.309·14-s − 3.64·15-s + 8.85·16-s + 0.738·17-s − 19.4·18-s − 7.27·19-s − 5.30·20-s − 0.386·21-s + 4.98·22-s + 7.57·23-s − 22.8·24-s − 3.73·25-s − 9.07·26-s + 14.6·27-s − 0.563·28-s + ⋯
L(s)  = 1  − 1.83·2-s + 1.87·3-s + 2.36·4-s − 0.502·5-s − 3.43·6-s − 0.0450·7-s − 2.49·8-s + 2.50·9-s + 0.921·10-s − 0.579·11-s + 4.41·12-s + 0.970·13-s + 0.0826·14-s − 0.941·15-s + 2.21·16-s + 0.179·17-s − 4.59·18-s − 1.66·19-s − 1.18·20-s − 0.0843·21-s + 1.06·22-s + 1.57·23-s − 4.67·24-s − 0.747·25-s − 1.77·26-s + 2.81·27-s − 0.106·28-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 \( 1 + 2.59T + 2T^{2} \)
3 \( 1 - 3.24T + 3T^{2} \)
5 \( 1 + 1.12T + 5T^{2} \)
7 \( 1 + 0.119T + 7T^{2} \)
11 \( 1 + 1.92T + 11T^{2} \)
13 \( 1 - 3.49T + 13T^{2} \)
17 \( 1 - 0.738T + 17T^{2} \)
19 \( 1 + 7.27T + 19T^{2} \)
23 \( 1 - 7.57T + 23T^{2} \)
29 \( 1 + 3.31T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 1.50T + 41T^{2} \)
43 \( 1 + 0.406T + 43T^{2} \)
47 \( 1 - 1.47T + 47T^{2} \)
53 \( 1 + 8.19T + 53T^{2} \)
59 \( 1 + 12.0T + 59T^{2} \)
61 \( 1 + 3.82T + 61T^{2} \)
67 \( 1 + 5.32T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + 4.10T + 73T^{2} \)
79 \( 1 + 4.08T + 79T^{2} \)
83 \( 1 - 3.38T + 83T^{2} \)
89 \( 1 - 2.04T + 89T^{2} \)
97 \( 1 + 1.22T + 97T^{2} \)
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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

−7.950589807777121024327325038045, −7.41759611551183669848270844847, −6.93110816912913928327839393837, −5.98836740980496589331718261154, −4.54645199301175423456994272894, −3.50025891751203679552727405507, −3.06203480485483695308012826134, −1.97855599679975364759655057507, −1.55090095468504953602593138910, 0, 1.55090095468504953602593138910, 1.97855599679975364759655057507, 3.06203480485483695308012826134, 3.50025891751203679552727405507, 4.54645199301175423456994272894, 5.98836740980496589331718261154, 6.93110816912913928327839393837, 7.41759611551183669848270844847, 7.950589807777121024327325038045

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