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  − 2.74·2-s + 2.83·3-s + 5.55·4-s − 0.268·5-s − 7.77·6-s + 0.663·7-s − 9.75·8-s + 5.01·9-s + 0.736·10-s − 0.384·11-s + 15.7·12-s − 5.76·13-s − 1.82·14-s − 0.758·15-s + 15.7·16-s + 4.35·17-s − 13.7·18-s − 0.449·19-s − 1.48·20-s + 1.87·21-s + 1.05·22-s − 6.98·23-s − 27.6·24-s − 4.92·25-s + 15.8·26-s + 5.70·27-s + 3.68·28-s + ⋯
L(s)  = 1  − 1.94·2-s + 1.63·3-s + 2.77·4-s − 0.119·5-s − 3.17·6-s + 0.250·7-s − 3.44·8-s + 1.67·9-s + 0.232·10-s − 0.115·11-s + 4.53·12-s − 1.59·13-s − 0.487·14-s − 0.195·15-s + 3.92·16-s + 1.05·17-s − 3.24·18-s − 0.103·19-s − 0.332·20-s + 0.409·21-s + 0.225·22-s − 1.45·23-s − 5.63·24-s − 0.985·25-s + 3.10·26-s + 1.09·27-s + 0.695·28-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 \( 1 + 2.74T + 2T^{2} \)
3 \( 1 - 2.83T + 3T^{2} \)
5 \( 1 + 0.268T + 5T^{2} \)
7 \( 1 - 0.663T + 7T^{2} \)
11 \( 1 + 0.384T + 11T^{2} \)
13 \( 1 + 5.76T + 13T^{2} \)
17 \( 1 - 4.35T + 17T^{2} \)
19 \( 1 + 0.449T + 19T^{2} \)
23 \( 1 + 6.98T + 23T^{2} \)
29 \( 1 - 8.69T + 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 + 7.57T + 37T^{2} \)
41 \( 1 + 2.18T + 41T^{2} \)
43 \( 1 - 2.89T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 - 8.78T + 53T^{2} \)
59 \( 1 + 2.66T + 59T^{2} \)
61 \( 1 + 7.78T + 61T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 9.56T + 73T^{2} \)
79 \( 1 - 2.68T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−7.84858563335744838127324404373, −7.59844801939859060074948717073, −6.83991538334675612639001639021, −5.96454028012809822166682378698, −4.73276187638374316500637373590, −3.50672731139345032896076043275, −2.81072581368836440535531681646, −2.15067824472739682377503682563, −1.45193083575464778218828812993, 0, 1.45193083575464778218828812993, 2.15067824472739682377503682563, 2.81072581368836440535531681646, 3.50672731139345032896076043275, 4.73276187638374316500637373590, 5.96454028012809822166682378698, 6.83991538334675612639001639021, 7.59844801939859060074948717073, 7.84858563335744838127324404373

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