Properties

Label 2-6047-1.1-c1-0-446
Degree $2$
Conductor $6047$
Sign $-1$
Analytic cond. $48.2855$
Root an. cond. $6.94877$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.50·2-s + 2.01·3-s + 4.27·4-s + 3.19·5-s − 5.03·6-s + 0.790·7-s − 5.68·8-s + 1.04·9-s − 7.99·10-s − 2.85·11-s + 8.59·12-s − 2.42·13-s − 1.98·14-s + 6.41·15-s + 5.70·16-s + 2.08·17-s − 2.61·18-s + 0.238·19-s + 13.6·20-s + 1.59·21-s + 7.13·22-s + 0.00126·23-s − 11.4·24-s + 5.17·25-s + 6.06·26-s − 3.93·27-s + 3.37·28-s + ⋯
L(s)  = 1  − 1.77·2-s + 1.16·3-s + 2.13·4-s + 1.42·5-s − 2.05·6-s + 0.298·7-s − 2.01·8-s + 0.347·9-s − 2.52·10-s − 0.859·11-s + 2.47·12-s − 0.671·13-s − 0.529·14-s + 1.65·15-s + 1.42·16-s + 0.504·17-s − 0.615·18-s + 0.0547·19-s + 3.04·20-s + 0.347·21-s + 1.52·22-s + 0.000264·23-s − 2.33·24-s + 1.03·25-s + 1.18·26-s − 0.757·27-s + 0.638·28-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

Degree: \(2\)
Conductor: \(6047\)
Sign: $-1$
Analytic conductor: \(48.2855\)
Root analytic conductor: \(6.94877\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6047,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6047 \( 1+O(T) \)
good2 \( 1 + 2.50T + 2T^{2} \)
3 \( 1 - 2.01T + 3T^{2} \)
5 \( 1 - 3.19T + 5T^{2} \)
7 \( 1 - 0.790T + 7T^{2} \)
11 \( 1 + 2.85T + 11T^{2} \)
13 \( 1 + 2.42T + 13T^{2} \)
17 \( 1 - 2.08T + 17T^{2} \)
19 \( 1 - 0.238T + 19T^{2} \)
23 \( 1 - 0.00126T + 23T^{2} \)
29 \( 1 + 9.31T + 29T^{2} \)
31 \( 1 + 9.00T + 31T^{2} \)
37 \( 1 + 0.877T + 37T^{2} \)
41 \( 1 + 5.36T + 41T^{2} \)
43 \( 1 + 1.10T + 43T^{2} \)
47 \( 1 + 9.52T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + 1.84T + 59T^{2} \)
61 \( 1 - 1.35T + 61T^{2} \)
67 \( 1 + 9.34T + 67T^{2} \)
71 \( 1 - 3.61T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 8.15T + 79T^{2} \)
83 \( 1 - 5.00T + 83T^{2} \)
89 \( 1 + 18.2T + 89T^{2} \)
97 \( 1 - 6.52T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.86698751406218639863147351667, −7.43526718075550367685090146268, −6.67256476836332357061528651738, −5.67212096373183390767123669667, −5.16927632005411074686832603365, −3.56014404357042658192403950508, −2.69695438820202691650022348910, −2.02150824620541957741016502003, −1.61661755526275070942037211099, 0, 1.61661755526275070942037211099, 2.02150824620541957741016502003, 2.69695438820202691650022348910, 3.56014404357042658192403950508, 5.16927632005411074686832603365, 5.67212096373183390767123669667, 6.67256476836332357061528651738, 7.43526718075550367685090146268, 7.86698751406218639863147351667

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