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  + 2.13·2-s + 3-s + 2.54·4-s − 0.889·5-s + 2.13·6-s + 1.16·8-s + 9-s − 1.89·10-s + 3.41·11-s + 2.54·12-s − 6.24·13-s − 0.889·15-s − 2.60·16-s − 7.99·17-s + 2.13·18-s + 0.0729·19-s − 2.26·20-s + 7.28·22-s + 1.57·23-s + 1.16·24-s − 4.20·25-s − 13.3·26-s + 27-s − 0.756·29-s − 1.89·30-s − 2.56·31-s − 7.88·32-s + ⋯
L(s)  = 1  + 1.50·2-s + 0.577·3-s + 1.27·4-s − 0.398·5-s + 0.870·6-s + 0.413·8-s + 0.333·9-s − 0.600·10-s + 1.03·11-s + 0.735·12-s − 1.73·13-s − 0.229·15-s − 0.650·16-s − 1.94·17-s + 0.502·18-s + 0.0167·19-s − 0.507·20-s + 1.55·22-s + 0.328·23-s + 0.238·24-s − 0.841·25-s − 2.61·26-s + 0.192·27-s − 0.140·29-s − 0.346·30-s − 0.460·31-s − 1.39·32-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 \( 1 - 2.13T + 2T^{2} \)
5 \( 1 + 0.889T + 5T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 + 6.24T + 13T^{2} \)
17 \( 1 + 7.99T + 17T^{2} \)
19 \( 1 - 0.0729T + 19T^{2} \)
23 \( 1 - 1.57T + 23T^{2} \)
29 \( 1 + 0.756T + 29T^{2} \)
31 \( 1 + 2.56T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
43 \( 1 + 6.17T + 43T^{2} \)
47 \( 1 + 1.59T + 47T^{2} \)
53 \( 1 - 9.15T + 53T^{2} \)
59 \( 1 - 0.919T + 59T^{2} \)
61 \( 1 - 8.51T + 61T^{2} \)
67 \( 1 - 2.41T + 67T^{2} \)
71 \( 1 - 8.88T + 71T^{2} \)
73 \( 1 + 3.64T + 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 + 4.03T + 83T^{2} \)
89 \( 1 - 8.44T + 89T^{2} \)
97 \( 1 - 9.05T + 97T^{2} \)
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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

−7.34247312784942439669777315565, −6.92254558699231814492126163892, −6.36200255074335341331009181890, −5.21699979241784106958343639376, −4.78825690804523023469155663355, −3.96643422946446131189854750171, −3.55591946136099686831507104684, −2.45711544894388942401829934563, −1.96453248231044472765949995746, 0, 1.96453248231044472765949995746, 2.45711544894388942401829934563, 3.55591946136099686831507104684, 3.96643422946446131189854750171, 4.78825690804523023469155663355, 5.21699979241784106958343639376, 6.36200255074335341331009181890, 6.92254558699231814492126163892, 7.34247312784942439669777315565

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