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  − 2.69·2-s + 3-s + 5.24·4-s − 3.62·5-s − 2.69·6-s − 8.73·8-s + 9-s + 9.76·10-s + 3.79·11-s + 5.24·12-s + 0.986·13-s − 3.62·15-s + 13.0·16-s + 1.05·17-s − 2.69·18-s − 2.81·19-s − 19.0·20-s − 10.2·22-s − 5.66·23-s − 8.73·24-s + 8.16·25-s − 2.65·26-s + 27-s − 2.21·29-s + 9.76·30-s + 10.4·31-s − 17.5·32-s + ⋯
L(s)  = 1  − 1.90·2-s + 0.577·3-s + 2.62·4-s − 1.62·5-s − 1.09·6-s − 3.08·8-s + 0.333·9-s + 3.08·10-s + 1.14·11-s + 1.51·12-s + 0.273·13-s − 0.936·15-s + 3.25·16-s + 0.256·17-s − 0.634·18-s − 0.644·19-s − 4.25·20-s − 2.17·22-s − 1.18·23-s − 1.78·24-s + 1.63·25-s − 0.521·26-s + 0.192·27-s − 0.411·29-s + 1.78·30-s + 1.87·31-s − 3.10·32-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 \( 1 + 2.69T + 2T^{2} \)
5 \( 1 + 3.62T + 5T^{2} \)
11 \( 1 - 3.79T + 11T^{2} \)
13 \( 1 - 0.986T + 13T^{2} \)
17 \( 1 - 1.05T + 17T^{2} \)
19 \( 1 + 2.81T + 19T^{2} \)
23 \( 1 + 5.66T + 23T^{2} \)
29 \( 1 + 2.21T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + 7.86T + 37T^{2} \)
43 \( 1 - 4.16T + 43T^{2} \)
47 \( 1 + 5.08T + 47T^{2} \)
53 \( 1 + 9.17T + 53T^{2} \)
59 \( 1 + 7.58T + 59T^{2} \)
61 \( 1 - 5.96T + 61T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 3.75T + 73T^{2} \)
79 \( 1 + 4.83T + 79T^{2} \)
83 \( 1 - 7.04T + 83T^{2} \)
89 \( 1 - 9.53T + 89T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−7.84397447795423038587734532440, −7.48717094888349937637536748559, −6.61503457997488093439260969885, −6.18837642504643737919320132087, −4.56336982885777432837862479847, −3.73602822678437426898397782788, −3.09981346064062993582552235857, −1.98847315638187500452609744865, −1.07161100136134824749903281745, 0, 1.07161100136134824749903281745, 1.98847315638187500452609744865, 3.09981346064062993582552235857, 3.73602822678437426898397782788, 4.56336982885777432837862479847, 6.18837642504643737919320132087, 6.61503457997488093439260969885, 7.48717094888349937637536748559, 7.84397447795423038587734532440

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