Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.95·2-s − 3-s + 1.83·4-s + 0.513·5-s + 1.95·6-s + 0.318·8-s + 9-s − 1.00·10-s + 5.65·11-s − 1.83·12-s + 0.337·13-s − 0.513·15-s − 4.29·16-s − 4.63·17-s − 1.95·18-s + 7.43·19-s + 0.942·20-s − 11.0·22-s + 3.51·23-s − 0.318·24-s − 4.73·25-s − 0.660·26-s − 27-s + 0.00385·29-s + 1.00·30-s − 2.90·31-s + 7.78·32-s + ⋯
L(s)  = 1  − 1.38·2-s − 0.577·3-s + 0.918·4-s + 0.229·5-s + 0.799·6-s + 0.112·8-s + 0.333·9-s − 0.317·10-s + 1.70·11-s − 0.530·12-s + 0.0935·13-s − 0.132·15-s − 1.07·16-s − 1.12·17-s − 0.461·18-s + 1.70·19-s + 0.210·20-s − 2.36·22-s + 0.732·23-s − 0.0650·24-s − 0.947·25-s − 0.129·26-s − 0.192·27-s + 0.000715·29-s + 0.183·30-s − 0.522·31-s + 1.37·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9428914600$
$L(\frac12)$  $\approx$  $0.9428914600$
$L(\frac{3}{2})$   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 \( 1 + 1.95T + 2T^{2} \)
5 \( 1 - 0.513T + 5T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 - 0.337T + 13T^{2} \)
17 \( 1 + 4.63T + 17T^{2} \)
19 \( 1 - 7.43T + 19T^{2} \)
23 \( 1 - 3.51T + 23T^{2} \)
29 \( 1 - 0.00385T + 29T^{2} \)
31 \( 1 + 2.90T + 31T^{2} \)
37 \( 1 - 5.69T + 37T^{2} \)
43 \( 1 - 6.32T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + 1.59T + 53T^{2} \)
59 \( 1 + 1.42T + 59T^{2} \)
61 \( 1 + 2.50T + 61T^{2} \)
67 \( 1 - 9.61T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 0.539T + 73T^{2} \)
79 \( 1 - 7.84T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + 2.56T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.139455440278391868911235094800, −7.36792249528019144354653968442, −6.85829611833452050523500308001, −6.20410888645911382922914858794, −5.36073926127098446519041625395, −4.40886456328392104059088000107, −3.71124954793088503663117972923, −2.37016070939859386023541339324, −1.40396202301518336003023647582, −0.73366716070138387665287298533, 0.73366716070138387665287298533, 1.40396202301518336003023647582, 2.37016070939859386023541339324, 3.71124954793088503663117972923, 4.40886456328392104059088000107, 5.36073926127098446519041625395, 6.20410888645911382922914858794, 6.85829611833452050523500308001, 7.36792249528019144354653968442, 8.139455440278391868911235094800

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