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  − 0.304·2-s + 3-s − 1.90·4-s − 0.824·5-s − 0.304·6-s + 1.18·8-s + 9-s + 0.250·10-s + 1.95·11-s − 1.90·12-s + 3.83·13-s − 0.824·15-s + 3.45·16-s − 6.15·17-s − 0.304·18-s + 0.304·19-s + 1.57·20-s − 0.594·22-s − 2.69·23-s + 1.18·24-s − 4.32·25-s − 1.16·26-s + 27-s − 8.02·29-s + 0.250·30-s + 0.213·31-s − 3.42·32-s + ⋯
L(s)  = 1  − 0.214·2-s + 0.577·3-s − 0.953·4-s − 0.368·5-s − 0.124·6-s + 0.420·8-s + 0.333·9-s + 0.0792·10-s + 0.589·11-s − 0.550·12-s + 1.06·13-s − 0.212·15-s + 0.863·16-s − 1.49·17-s − 0.0716·18-s + 0.0697·19-s + 0.351·20-s − 0.126·22-s − 0.562·23-s + 0.242·24-s − 0.864·25-s − 0.228·26-s + 0.192·27-s − 1.49·29-s + 0.0457·30-s + 0.0384·31-s − 0.605·32-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 \( 1 + 0.304T + 2T^{2} \)
5 \( 1 + 0.824T + 5T^{2} \)
11 \( 1 - 1.95T + 11T^{2} \)
13 \( 1 - 3.83T + 13T^{2} \)
17 \( 1 + 6.15T + 17T^{2} \)
19 \( 1 - 0.304T + 19T^{2} \)
23 \( 1 + 2.69T + 23T^{2} \)
29 \( 1 + 8.02T + 29T^{2} \)
31 \( 1 - 0.213T + 31T^{2} \)
37 \( 1 - 1.79T + 37T^{2} \)
43 \( 1 - 9.13T + 43T^{2} \)
47 \( 1 - 5.90T + 47T^{2} \)
53 \( 1 - 3.40T + 53T^{2} \)
59 \( 1 + 6.19T + 59T^{2} \)
61 \( 1 - 2.00T + 61T^{2} \)
67 \( 1 + 0.370T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 + 0.0753T + 79T^{2} \)
83 \( 1 - 0.111T + 83T^{2} \)
89 \( 1 - 4.92T + 89T^{2} \)
97 \( 1 - 5.06T + 97T^{2} \)
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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

−7.76060629854831994970072639390, −7.32393717403791356778388586231, −6.22861945870452879449602582274, −5.67337390577982064450977629514, −4.45422107640125875989245316844, −4.08526879567592385085837279448, −3.47999681121074937908360393429, −2.24738914736679851867079834515, −1.27153890651061397911835891039, 0, 1.27153890651061397911835891039, 2.24738914736679851867079834515, 3.47999681121074937908360393429, 4.08526879567592385085837279448, 4.45422107640125875989245316844, 5.67337390577982064450977629514, 6.22861945870452879449602582274, 7.32393717403791356778388586231, 7.76060629854831994970072639390

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