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  − 2.73·2-s + 3-s + 5.45·4-s − 3.67·5-s − 2.73·6-s − 9.43·8-s + 9-s + 10.0·10-s − 5.55·11-s + 5.45·12-s − 1.35·13-s − 3.67·15-s + 14.8·16-s + 5.33·17-s − 2.73·18-s + 3.13·19-s − 20.0·20-s + 15.1·22-s + 6.55·23-s − 9.43·24-s + 8.54·25-s + 3.68·26-s + 27-s − 8.35·29-s + 10.0·30-s − 3.83·31-s − 21.6·32-s + ⋯
L(s)  = 1  − 1.93·2-s + 0.577·3-s + 2.72·4-s − 1.64·5-s − 1.11·6-s − 3.33·8-s + 0.333·9-s + 3.17·10-s − 1.67·11-s + 1.57·12-s − 0.374·13-s − 0.950·15-s + 3.71·16-s + 1.29·17-s − 0.643·18-s + 0.718·19-s − 4.48·20-s + 3.23·22-s + 1.36·23-s − 1.92·24-s + 1.70·25-s + 0.723·26-s + 0.192·27-s − 1.55·29-s + 1.83·30-s − 0.688·31-s − 3.83·32-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 + 2.73T + 2T^{2} \)
5 \( 1 + 3.67T + 5T^{2} \)
11 \( 1 + 5.55T + 11T^{2} \)
13 \( 1 + 1.35T + 13T^{2} \)
17 \( 1 - 5.33T + 17T^{2} \)
19 \( 1 - 3.13T + 19T^{2} \)
23 \( 1 - 6.55T + 23T^{2} \)
29 \( 1 + 8.35T + 29T^{2} \)
31 \( 1 + 3.83T + 31T^{2} \)
37 \( 1 - 0.883T + 37T^{2} \)
43 \( 1 + 3.78T + 43T^{2} \)
47 \( 1 + 2.25T + 47T^{2} \)
53 \( 1 - 4.13T + 53T^{2} \)
59 \( 1 - 4.43T + 59T^{2} \)
61 \( 1 + 4.50T + 61T^{2} \)
67 \( 1 + 9.88T + 67T^{2} \)
71 \( 1 - 3.25T + 71T^{2} \)
73 \( 1 - 6.37T + 73T^{2} \)
79 \( 1 - 4.67T + 79T^{2} \)
83 \( 1 - 0.851T + 83T^{2} \)
89 \( 1 - 9.60T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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

−7.81230819952215485858293981020, −7.36449952570998505212236085802, −7.11370534454640076735595426524, −5.72213299868610471486591121902, −4.92879086362063599631045991477, −3.42084133616423795961306868468, −3.15823493522522571958720469906, −2.15404130062600625421871918208, −0.935801847183346865681483677671, 0, 0.935801847183346865681483677671, 2.15404130062600625421871918208, 3.15823493522522571958720469906, 3.42084133616423795961306868468, 4.92879086362063599631045991477, 5.72213299868610471486591121902, 7.11370534454640076735595426524, 7.36449952570998505212236085802, 7.81230819952215485858293981020

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