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  + 2.14·2-s + 3-s + 2.60·4-s − 2.36·5-s + 2.14·6-s + 1.30·8-s + 9-s − 5.07·10-s − 2.70·11-s + 2.60·12-s − 3.97·13-s − 2.36·15-s − 2.41·16-s + 6.48·17-s + 2.14·18-s + 7.54·19-s − 6.16·20-s − 5.79·22-s − 1.03·23-s + 1.30·24-s + 0.587·25-s − 8.52·26-s + 27-s − 8.83·29-s − 5.07·30-s − 2.51·31-s − 7.79·32-s + ⋯
L(s)  = 1  + 1.51·2-s + 0.577·3-s + 1.30·4-s − 1.05·5-s + 0.876·6-s + 0.460·8-s + 0.333·9-s − 1.60·10-s − 0.814·11-s + 0.752·12-s − 1.10·13-s − 0.610·15-s − 0.604·16-s + 1.57·17-s + 0.505·18-s + 1.73·19-s − 1.37·20-s − 1.23·22-s − 0.216·23-s + 0.265·24-s + 0.117·25-s − 1.67·26-s + 0.192·27-s − 1.64·29-s − 0.926·30-s − 0.452·31-s − 1.37·32-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 \( 1 - 2.14T + 2T^{2} \)
5 \( 1 + 2.36T + 5T^{2} \)
11 \( 1 + 2.70T + 11T^{2} \)
13 \( 1 + 3.97T + 13T^{2} \)
17 \( 1 - 6.48T + 17T^{2} \)
19 \( 1 - 7.54T + 19T^{2} \)
23 \( 1 + 1.03T + 23T^{2} \)
29 \( 1 + 8.83T + 29T^{2} \)
31 \( 1 + 2.51T + 31T^{2} \)
37 \( 1 + 6.41T + 37T^{2} \)
43 \( 1 + 0.547T + 43T^{2} \)
47 \( 1 + 3.62T + 47T^{2} \)
53 \( 1 + 8.41T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + 4.26T + 61T^{2} \)
67 \( 1 + 9.56T + 67T^{2} \)
71 \( 1 + 6.72T + 71T^{2} \)
73 \( 1 - 7.72T + 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 - 8.73T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 - 7.45T + 97T^{2} \)
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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

−7.56165395506026952916030406400, −7.25350648529688470630568925400, −6.03829619436180578271778370200, −5.24127225757662518980758508574, −4.92361690343460291417291667601, −3.89596672632467287991911004029, −3.33135417791800884988832775597, −2.89416711648602318226886373255, −1.72333876467311491672337357711, 0, 1.72333876467311491672337357711, 2.89416711648602318226886373255, 3.33135417791800884988832775597, 3.89596672632467287991911004029, 4.92361690343460291417291667601, 5.24127225757662518980758508574, 6.03829619436180578271778370200, 7.25350648529688470630568925400, 7.56165395506026952916030406400

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