Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 503 $
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  + 3-s + 0.0349·5-s + 0.637·7-s + 9-s − 0.863·11-s − 4.36·13-s + 0.0349·15-s − 2.27·17-s − 4.02·19-s + 0.637·21-s + 8.23·23-s − 4.99·25-s + 27-s + 1.29·29-s − 1.21·31-s − 0.863·33-s + 0.0223·35-s + 7.14·37-s − 4.36·39-s + 2.42·41-s + 0.00158·43-s + 0.0349·45-s − 3.98·47-s − 6.59·49-s − 2.27·51-s + 7.67·53-s − 0.0302·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.0156·5-s + 0.240·7-s + 0.333·9-s − 0.260·11-s − 1.21·13-s + 0.00903·15-s − 0.552·17-s − 0.922·19-s + 0.139·21-s + 1.71·23-s − 0.999·25-s + 0.192·27-s + 0.239·29-s − 0.217·31-s − 0.150·33-s + 0.00377·35-s + 1.17·37-s − 0.699·39-s + 0.378·41-s + 0.000241·43-s + 0.00521·45-s − 0.581·47-s − 0.941·49-s − 0.318·51-s + 1.05·53-s − 0.00407·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6036\)    =    \(2^{2} \cdot 3 \cdot 503\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
503 \( 1 + T \)
good5 \( 1 - 0.0349T + 5T^{2} \)
7 \( 1 - 0.637T + 7T^{2} \)
11 \( 1 + 0.863T + 11T^{2} \)
13 \( 1 + 4.36T + 13T^{2} \)
17 \( 1 + 2.27T + 17T^{2} \)
19 \( 1 + 4.02T + 19T^{2} \)
23 \( 1 - 8.23T + 23T^{2} \)
29 \( 1 - 1.29T + 29T^{2} \)
31 \( 1 + 1.21T + 31T^{2} \)
37 \( 1 - 7.14T + 37T^{2} \)
41 \( 1 - 2.42T + 41T^{2} \)
43 \( 1 - 0.00158T + 43T^{2} \)
47 \( 1 + 3.98T + 47T^{2} \)
53 \( 1 - 7.67T + 53T^{2} \)
59 \( 1 + 1.59T + 59T^{2} \)
61 \( 1 + 3.50T + 61T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 2.85T + 73T^{2} \)
79 \( 1 - 7.20T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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

−7.69010305312449316980640645788, −7.17503262569630470375737757660, −6.41462221651789010864291967929, −5.50946642474330710851837620110, −4.66232668471564391027938280786, −4.19924904010278338762196204265, −2.99583655317649717031099644203, −2.46627572500099346173864119336, −1.48262556560617263098500624049, 0, 1.48262556560617263098500624049, 2.46627572500099346173864119336, 2.99583655317649717031099644203, 4.19924904010278338762196204265, 4.66232668471564391027938280786, 5.50946642474330710851837620110, 6.41462221651789010864291967929, 7.17503262569630470375737757660, 7.69010305312449316980640645788

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