Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.81·2-s + 3-s + 1.28·4-s + 1.10·5-s − 1.81·6-s + 1.28·8-s + 9-s − 2·10-s + 0.813·11-s + 1.28·12-s − 5.10·13-s + 1.10·15-s − 4.91·16-s − 3.39·17-s − 1.81·18-s − 3.10·19-s + 1.42·20-s − 1.47·22-s − 0.897·23-s + 1.28·24-s − 3.78·25-s + 9.25·26-s + 27-s + 4.44·29-s − 2·30-s + 8.96·31-s + 6.33·32-s + ⋯
L(s)  = 1  − 1.28·2-s + 0.577·3-s + 0.644·4-s + 0.493·5-s − 0.740·6-s + 0.455·8-s + 0.333·9-s − 0.632·10-s + 0.245·11-s + 0.372·12-s − 1.41·13-s + 0.284·15-s − 1.22·16-s − 0.822·17-s − 0.427·18-s − 0.711·19-s + 0.317·20-s − 0.314·22-s − 0.187·23-s + 0.263·24-s − 0.756·25-s + 1.81·26-s + 0.192·27-s + 0.824·29-s − 0.365·30-s + 1.61·31-s + 1.12·32-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 + 1.81T + 2T^{2} \)
5 \( 1 - 1.10T + 5T^{2} \)
11 \( 1 - 0.813T + 11T^{2} \)
13 \( 1 + 5.10T + 13T^{2} \)
17 \( 1 + 3.39T + 17T^{2} \)
19 \( 1 + 3.10T + 19T^{2} \)
23 \( 1 + 0.897T + 23T^{2} \)
29 \( 1 - 4.44T + 29T^{2} \)
31 \( 1 - 8.96T + 31T^{2} \)
37 \( 1 - 2.08T + 37T^{2} \)
43 \( 1 - 9.07T + 43T^{2} \)
47 \( 1 - 0.235T + 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + 1.04T + 59T^{2} \)
61 \( 1 + 1.91T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 4.75T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 - 7.68T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 2.72T + 97T^{2} \)
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

−7.87300271648913760931621048517, −7.24513233272592872275306932976, −6.62785510414224473596620669061, −5.76165040722357987498405445808, −4.55456438716255186101379185128, −4.26276476813171114000745602022, −2.68790432287775649397997189871, −2.27315507031838323711502170154, −1.25011270715050356204940204496, 0, 1.25011270715050356204940204496, 2.27315507031838323711502170154, 2.68790432287775649397997189871, 4.26276476813171114000745602022, 4.55456438716255186101379185128, 5.76165040722357987498405445808, 6.62785510414224473596620669061, 7.24513233272592872275306932976, 7.87300271648913760931621048517

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