Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.190·2-s − 3-s − 1.96·4-s − 2.28·5-s + 0.190·6-s + 0.753·8-s + 9-s + 0.434·10-s + 1.72·11-s + 1.96·12-s − 1.61·13-s + 2.28·15-s + 3.78·16-s + 3.96·17-s − 0.190·18-s − 1.10·19-s + 4.48·20-s − 0.328·22-s − 2.20·23-s − 0.753·24-s + 0.213·25-s + 0.307·26-s − 27-s + 7.83·29-s − 0.434·30-s − 3.24·31-s − 2.22·32-s + ⋯
L(s)  = 1  − 0.134·2-s − 0.577·3-s − 0.981·4-s − 1.02·5-s + 0.0776·6-s + 0.266·8-s + 0.333·9-s + 0.137·10-s + 0.521·11-s + 0.566·12-s − 0.448·13-s + 0.589·15-s + 0.946·16-s + 0.962·17-s − 0.0448·18-s − 0.254·19-s + 1.00·20-s − 0.0701·22-s − 0.460·23-s − 0.153·24-s + 0.0427·25-s + 0.0602·26-s − 0.192·27-s + 1.45·29-s − 0.0792·30-s − 0.581·31-s − 0.393·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6278288221$
$L(\frac12)$  $\approx$  $0.6278288221$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 + 0.190T + 2T^{2} \)
5 \( 1 + 2.28T + 5T^{2} \)
11 \( 1 - 1.72T + 11T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 - 3.96T + 17T^{2} \)
19 \( 1 + 1.10T + 19T^{2} \)
23 \( 1 + 2.20T + 23T^{2} \)
29 \( 1 - 7.83T + 29T^{2} \)
31 \( 1 + 3.24T + 31T^{2} \)
37 \( 1 - 1.90T + 37T^{2} \)
43 \( 1 - 2.91T + 43T^{2} \)
47 \( 1 + 5.86T + 47T^{2} \)
53 \( 1 + 2.97T + 53T^{2} \)
59 \( 1 + 5.61T + 59T^{2} \)
61 \( 1 + 4.86T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 - 0.406T + 73T^{2} \)
79 \( 1 + 2.23T + 79T^{2} \)
83 \( 1 - 1.25T + 83T^{2} \)
89 \( 1 + 8.84T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−7.932165895488492649956202424690, −7.63360301341795669126372733874, −6.65971420514771274243070148745, −5.89554202538460311987121765227, −5.06423826215688276732236500228, −4.43524439724640159749912432642, −3.84950040051976753608808939666, −3.03504558777575910235490997944, −1.49214575989798564887943244507, −0.46976463337647541018749586675, 0.46976463337647541018749586675, 1.49214575989798564887943244507, 3.03504558777575910235490997944, 3.84950040051976753608808939666, 4.43524439724640159749912432642, 5.06423826215688276732236500228, 5.89554202538460311987121765227, 6.65971420514771274243070148745, 7.63360301341795669126372733874, 7.932165895488492649956202424690

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