Properties

Degree 2
Conductor $ 3 \cdot 2671 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.75·2-s + 3-s + 5.59·4-s + 0.110·5-s − 2.75·6-s + 1.43·7-s − 9.91·8-s + 9-s − 0.303·10-s − 0.0788·11-s + 5.59·12-s − 1.19·13-s − 3.96·14-s + 0.110·15-s + 16.1·16-s + 0.781·17-s − 2.75·18-s − 2.26·19-s + 0.616·20-s + 1.43·21-s + 0.217·22-s − 5.36·23-s − 9.91·24-s − 4.98·25-s + 3.28·26-s + 27-s + 8.05·28-s + ⋯
L(s)  = 1  − 1.94·2-s + 0.577·3-s + 2.79·4-s + 0.0492·5-s − 1.12·6-s + 0.544·7-s − 3.50·8-s + 0.333·9-s − 0.0959·10-s − 0.0237·11-s + 1.61·12-s − 0.330·13-s − 1.06·14-s + 0.0284·15-s + 4.03·16-s + 0.189·17-s − 0.649·18-s − 0.520·19-s + 0.137·20-s + 0.314·21-s + 0.0463·22-s − 1.11·23-s − 2.02·24-s − 0.997·25-s + 0.644·26-s + 0.192·27-s + 1.52·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8013\)    =    \(3 \cdot 2671\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8013} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8013,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9351535336$
$L(\frac12)$  $\approx$  $0.9351535336$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;2671\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;2671\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
2671 \( 1 + T \)
good2 \( 1 + 2.75T + 2T^{2} \)
5 \( 1 - 0.110T + 5T^{2} \)
7 \( 1 - 1.43T + 7T^{2} \)
11 \( 1 + 0.0788T + 11T^{2} \)
13 \( 1 + 1.19T + 13T^{2} \)
17 \( 1 - 0.781T + 17T^{2} \)
19 \( 1 + 2.26T + 19T^{2} \)
23 \( 1 + 5.36T + 23T^{2} \)
29 \( 1 + 0.140T + 29T^{2} \)
31 \( 1 - 6.90T + 31T^{2} \)
37 \( 1 - 0.903T + 37T^{2} \)
41 \( 1 + 1.92T + 41T^{2} \)
43 \( 1 + 2.72T + 43T^{2} \)
47 \( 1 + 0.0476T + 47T^{2} \)
53 \( 1 - 3.74T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 6.72T + 61T^{2} \)
67 \( 1 - 9.41T + 67T^{2} \)
71 \( 1 + 2.32T + 71T^{2} \)
73 \( 1 - 15.4T + 73T^{2} \)
79 \( 1 - 7.02T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + 3.16T + 97T^{2} \)
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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

−8.010745493913503074779451931597, −7.59021826786325434794744274337, −6.67346758099427269048753911401, −6.21541999600003264505692579535, −5.22497974853414777460822701749, −4.06336167776244082957594371838, −3.10151667225805821379647292169, −2.20136765838752944619168968632, −1.75385636642239565950742537010, −0.61426258404235204964213459127, 0.61426258404235204964213459127, 1.75385636642239565950742537010, 2.20136765838752944619168968632, 3.10151667225805821379647292169, 4.06336167776244082957594371838, 5.22497974853414777460822701749, 6.21541999600003264505692579535, 6.67346758099427269048753911401, 7.59021826786325434794744274337, 8.010745493913503074779451931597

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