Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.704·2-s + 3-s − 1.50·4-s − 1.50·5-s − 0.704·6-s + 2.46·8-s + 9-s + 1.05·10-s − 4.26·11-s − 1.50·12-s − 2.30·13-s − 1.50·15-s + 1.26·16-s − 0.905·17-s − 0.704·18-s − 1.17·19-s + 2.26·20-s + 3.00·22-s − 0.893·23-s + 2.46·24-s − 2.73·25-s + 1.62·26-s + 27-s + 3.32·29-s + 1.05·30-s + 3.09·31-s − 5.83·32-s + ⋯
L(s)  = 1  − 0.498·2-s + 0.577·3-s − 0.751·4-s − 0.672·5-s − 0.287·6-s + 0.872·8-s + 0.333·9-s + 0.335·10-s − 1.28·11-s − 0.434·12-s − 0.638·13-s − 0.388·15-s + 0.316·16-s − 0.219·17-s − 0.166·18-s − 0.269·19-s + 0.505·20-s + 0.641·22-s − 0.186·23-s + 0.503·24-s − 0.547·25-s + 0.318·26-s + 0.192·27-s + 0.617·29-s + 0.193·30-s + 0.555·31-s − 1.03·32-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  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6642630577$
$L(\frac12)$  $\approx$  $0.6642630577$
$L(\frac{3}{2})$   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 \( 1 + 0.704T + 2T^{2} \)
5 \( 1 + 1.50T + 5T^{2} \)
11 \( 1 + 4.26T + 11T^{2} \)
13 \( 1 + 2.30T + 13T^{2} \)
17 \( 1 + 0.905T + 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 + 0.893T + 23T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 - 3.09T + 31T^{2} \)
37 \( 1 + 2.36T + 37T^{2} \)
43 \( 1 + 7.84T + 43T^{2} \)
47 \( 1 - 1.30T + 47T^{2} \)
53 \( 1 + 6.64T + 53T^{2} \)
59 \( 1 - 0.638T + 59T^{2} \)
61 \( 1 - 9.84T + 61T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + 0.245T + 71T^{2} \)
73 \( 1 - 2.08T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 6.41T + 83T^{2} \)
89 \( 1 - 1.81T + 89T^{2} \)
97 \( 1 - 6.29T + 97T^{2} \)
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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

−8.205587441701624177590607663774, −7.60645603274251702812783199026, −7.03797192964273824858895415357, −5.89869234828497478361617429720, −4.93254017753831758949180927288, −4.51917019306950125394227445550, −3.62742135017520801864893846074, −2.79849874987979233633920068320, −1.82293284791531446133331586147, −0.43968734951597055735551428549, 0.43968734951597055735551428549, 1.82293284791531446133331586147, 2.79849874987979233633920068320, 3.62742135017520801864893846074, 4.51917019306950125394227445550, 4.93254017753831758949180927288, 5.89869234828497478361617429720, 7.03797192964273824858895415357, 7.60645603274251702812783199026, 8.205587441701624177590607663774

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