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  + 2.27·2-s + 3-s + 3.19·4-s + 3.19·5-s + 2.27·6-s + 2.71·8-s + 9-s + 7.27·10-s − 2.80·11-s + 3.19·12-s + 4.10·13-s + 3.19·15-s − 0.198·16-s + 0.365·17-s + 2.27·18-s + 1.56·19-s + 10.1·20-s − 6.38·22-s − 0.452·23-s + 2.71·24-s + 5.18·25-s + 9.35·26-s + 27-s + 8.07·29-s + 7.27·30-s + 4.36·31-s − 5.88·32-s + ⋯
L(s)  = 1  + 1.61·2-s + 0.577·3-s + 1.59·4-s + 1.42·5-s + 0.930·6-s + 0.959·8-s + 0.333·9-s + 2.29·10-s − 0.844·11-s + 0.921·12-s + 1.13·13-s + 0.823·15-s − 0.0496·16-s + 0.0886·17-s + 0.537·18-s + 0.358·19-s + 2.27·20-s − 1.36·22-s − 0.0944·23-s + 0.553·24-s + 1.03·25-s + 1.83·26-s + 0.192·27-s + 1.49·29-s + 1.32·30-s + 0.784·31-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.728883238$
$L(\frac12)$  $\approx$  $8.728883238$
$L(\frac{3}{2})$   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 \( 1 - 2.27T + 2T^{2} \)
5 \( 1 - 3.19T + 5T^{2} \)
11 \( 1 + 2.80T + 11T^{2} \)
13 \( 1 - 4.10T + 13T^{2} \)
17 \( 1 - 0.365T + 17T^{2} \)
19 \( 1 - 1.56T + 19T^{2} \)
23 \( 1 + 0.452T + 23T^{2} \)
29 \( 1 - 8.07T + 29T^{2} \)
31 \( 1 - 4.36T + 31T^{2} \)
37 \( 1 + 2.16T + 37T^{2} \)
43 \( 1 + 7.06T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 - 5.23T + 53T^{2} \)
59 \( 1 - 0.833T + 59T^{2} \)
61 \( 1 - 9.06T + 61T^{2} \)
67 \( 1 - 0.145T + 67T^{2} \)
71 \( 1 + 2.88T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 4.01T + 79T^{2} \)
83 \( 1 + 8.93T + 83T^{2} \)
89 \( 1 + 0.731T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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

−8.146545358822609714400420727932, −6.92025791592736556764595918338, −6.47900282218327087364870093347, −5.77107501573137888590400039178, −5.20940055477422901691465283945, −4.55064061691604467599709184044, −3.57039981056327012749947034668, −2.90221058852194751988939217289, −2.26228361192809710772194810035, −1.34199826485016237070942909885, 1.34199826485016237070942909885, 2.26228361192809710772194810035, 2.90221058852194751988939217289, 3.57039981056327012749947034668, 4.55064061691604467599709184044, 5.20940055477422901691465283945, 5.77107501573137888590400039178, 6.47900282218327087364870093347, 6.92025791592736556764595918338, 8.146545358822609714400420727932

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