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  + 1.67·2-s + 3-s + 0.795·4-s + 2.68·5-s + 1.67·6-s − 2.01·8-s + 9-s + 4.48·10-s − 5.74·11-s + 0.795·12-s − 4.92·13-s + 2.68·15-s − 4.95·16-s − 2.19·17-s + 1.67·18-s + 3.26·19-s + 2.13·20-s − 9.61·22-s − 6.02·23-s − 2.01·24-s + 2.19·25-s − 8.23·26-s + 27-s − 4.01·29-s + 4.48·30-s + 7.54·31-s − 4.26·32-s + ⋯
L(s)  = 1  + 1.18·2-s + 0.577·3-s + 0.397·4-s + 1.19·5-s + 0.682·6-s − 0.712·8-s + 0.333·9-s + 1.41·10-s − 1.73·11-s + 0.229·12-s − 1.36·13-s + 0.692·15-s − 1.23·16-s − 0.531·17-s + 0.394·18-s + 0.748·19-s + 0.477·20-s − 2.04·22-s − 1.25·23-s − 0.411·24-s + 0.439·25-s − 1.61·26-s + 0.192·27-s − 0.744·29-s + 0.819·30-s + 1.35·31-s − 0.753·32-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 \( 1 - 1.67T + 2T^{2} \)
5 \( 1 - 2.68T + 5T^{2} \)
11 \( 1 + 5.74T + 11T^{2} \)
13 \( 1 + 4.92T + 13T^{2} \)
17 \( 1 + 2.19T + 17T^{2} \)
19 \( 1 - 3.26T + 19T^{2} \)
23 \( 1 + 6.02T + 23T^{2} \)
29 \( 1 + 4.01T + 29T^{2} \)
31 \( 1 - 7.54T + 31T^{2} \)
37 \( 1 - 7.78T + 37T^{2} \)
43 \( 1 + 4.98T + 43T^{2} \)
47 \( 1 + 4.57T + 47T^{2} \)
53 \( 1 - 1.25T + 53T^{2} \)
59 \( 1 + 6.36T + 59T^{2} \)
61 \( 1 + 6.05T + 61T^{2} \)
67 \( 1 - 1.90T + 67T^{2} \)
71 \( 1 + 9.71T + 71T^{2} \)
73 \( 1 - 6.59T + 73T^{2} \)
79 \( 1 - 3.22T + 79T^{2} \)
83 \( 1 + 9.82T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 7.28T + 97T^{2} \)
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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

−7.73523068252753218272464144334, −6.85355626196598751749644444732, −6.01132616391271459380083912681, −5.46573150348553535912210344882, −4.85795821201146021626440623222, −4.23972017825933061143878101532, −2.99140254605097366197796000491, −2.62425712259383727581337264029, −1.90121731472281249403182227332, 0, 1.90121731472281249403182227332, 2.62425712259383727581337264029, 2.99140254605097366197796000491, 4.23972017825933061143878101532, 4.85795821201146021626440623222, 5.46573150348553535912210344882, 6.01132616391271459380083912681, 6.85355626196598751749644444732, 7.73523068252753218272464144334

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