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  − 1.32·2-s + 3-s − 0.246·4-s + 4.03·5-s − 1.32·6-s + 2.97·8-s + 9-s − 5.34·10-s + 4.32·11-s − 0.246·12-s + 1.67·13-s + 4.03·15-s − 3.44·16-s − 1.53·17-s − 1.32·18-s + 4.63·19-s − 0.995·20-s − 5.72·22-s + 1.68·23-s + 2.97·24-s + 11.2·25-s − 2.22·26-s + 27-s + 6.19·29-s − 5.34·30-s − 7.82·31-s − 1.38·32-s + ⋯
L(s)  = 1  − 0.936·2-s + 0.577·3-s − 0.123·4-s + 1.80·5-s − 0.540·6-s + 1.05·8-s + 0.333·9-s − 1.68·10-s + 1.30·11-s − 0.0712·12-s + 0.465·13-s + 1.04·15-s − 0.861·16-s − 0.372·17-s − 0.312·18-s + 1.06·19-s − 0.222·20-s − 1.21·22-s + 0.350·23-s + 0.607·24-s + 2.25·25-s − 0.435·26-s + 0.192·27-s + 1.14·29-s − 0.975·30-s − 1.40·31-s − 0.245·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.506109034$
$L(\frac12)$  $\approx$  $2.506109034$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 + 1.32T + 2T^{2} \)
5 \( 1 - 4.03T + 5T^{2} \)
11 \( 1 - 4.32T + 11T^{2} \)
13 \( 1 - 1.67T + 13T^{2} \)
17 \( 1 + 1.53T + 17T^{2} \)
19 \( 1 - 4.63T + 19T^{2} \)
23 \( 1 - 1.68T + 23T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 + 7.82T + 31T^{2} \)
37 \( 1 - 5.18T + 37T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
47 \( 1 + 7.25T + 47T^{2} \)
53 \( 1 + 1.52T + 53T^{2} \)
59 \( 1 - 6.75T + 59T^{2} \)
61 \( 1 + 1.19T + 61T^{2} \)
67 \( 1 - 3.40T + 67T^{2} \)
71 \( 1 + 16.5T + 71T^{2} \)
73 \( 1 + 5.99T + 73T^{2} \)
79 \( 1 - 5.84T + 79T^{2} \)
83 \( 1 + 0.0883T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 2.82T + 97T^{2} \)
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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

−8.379485815431346758445804080859, −7.43766040012280070579372095064, −6.74946028228645232686984193866, −6.10965723469070294307326706153, −5.26393747790032848418764169923, −4.48165415012821944415885819100, −3.51187985278840866041913146501, −2.48928087312513173790389010865, −1.55966390217817631281731593454, −1.09352384669276846253458790048, 1.09352384669276846253458790048, 1.55966390217817631281731593454, 2.48928087312513173790389010865, 3.51187985278840866041913146501, 4.48165415012821944415885819100, 5.26393747790032848418764169923, 6.10965723469070294307326706153, 6.74946028228645232686984193866, 7.43766040012280070579372095064, 8.379485815431346758445804080859

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