Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.64·2-s + 3-s + 0.710·4-s + 0.0457·5-s − 1.64·6-s + 2.12·8-s + 9-s − 0.0753·10-s + 0.153·11-s + 0.710·12-s + 3.46·13-s + 0.0457·15-s − 4.91·16-s − 0.713·17-s − 1.64·18-s − 6.38·19-s + 0.0325·20-s − 0.252·22-s − 1.26·23-s + 2.12·24-s − 4.99·25-s − 5.69·26-s + 27-s + 1.30·29-s − 0.0753·30-s + 0.735·31-s + 3.84·32-s + ⋯
L(s)  = 1  − 1.16·2-s + 0.577·3-s + 0.355·4-s + 0.0204·5-s − 0.672·6-s + 0.750·8-s + 0.333·9-s − 0.0238·10-s + 0.0462·11-s + 0.205·12-s + 0.959·13-s + 0.0118·15-s − 1.22·16-s − 0.172·17-s − 0.388·18-s − 1.46·19-s + 0.00726·20-s − 0.0538·22-s − 0.263·23-s + 0.433·24-s − 0.999·25-s − 1.11·26-s + 0.192·27-s + 0.243·29-s − 0.0137·30-s + 0.132·31-s + 0.680·32-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  =  1
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 + 1.64T + 2T^{2} \)
5 \( 1 - 0.0457T + 5T^{2} \)
11 \( 1 - 0.153T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + 0.713T + 17T^{2} \)
19 \( 1 + 6.38T + 19T^{2} \)
23 \( 1 + 1.26T + 23T^{2} \)
29 \( 1 - 1.30T + 29T^{2} \)
31 \( 1 - 0.735T + 31T^{2} \)
37 \( 1 - 9.12T + 37T^{2} \)
43 \( 1 - 2.39T + 43T^{2} \)
47 \( 1 - 9.00T + 47T^{2} \)
53 \( 1 + 14.1T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + 6.60T + 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 + 6.18T + 79T^{2} \)
83 \( 1 + 2.20T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 8.42T + 97T^{2} \)
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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

−8.007987255755291226991279234198, −7.35752052777333238993070761838, −6.40300606554222119453603575944, −5.89185097432867949723877755093, −4.44260734677625031659184930356, −4.22121219148035519296869516968, −3.02807572551165628349549256799, −2.04510722738900492988176063710, −1.28537426146931059884669644762, 0, 1.28537426146931059884669644762, 2.04510722738900492988176063710, 3.02807572551165628349549256799, 4.22121219148035519296869516968, 4.44260734677625031659184930356, 5.89185097432867949723877755093, 6.40300606554222119453603575944, 7.35752052777333238993070761838, 8.007987255755291226991279234198

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