Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 61 $
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  − 2.54·2-s + 4.46·4-s − 0.329·5-s + 0.595·7-s − 6.27·8-s + 0.838·10-s + 11-s + 0.0829·13-s − 1.51·14-s + 7.03·16-s − 6.96·17-s + 1.08·19-s − 1.47·20-s − 2.54·22-s − 3.20·23-s − 4.89·25-s − 0.211·26-s + 2.66·28-s − 8.72·29-s − 3.05·31-s − 5.32·32-s + 17.7·34-s − 0.196·35-s − 5.63·37-s − 2.75·38-s + 2.07·40-s + 0.0715·41-s + ⋯
L(s)  = 1  − 1.79·2-s + 2.23·4-s − 0.147·5-s + 0.225·7-s − 2.21·8-s + 0.265·10-s + 0.301·11-s + 0.0230·13-s − 0.404·14-s + 1.75·16-s − 1.68·17-s + 0.248·19-s − 0.329·20-s − 0.542·22-s − 0.667·23-s − 0.978·25-s − 0.0413·26-s + 0.502·28-s − 1.62·29-s − 0.549·31-s − 0.941·32-s + 3.03·34-s − 0.0331·35-s − 0.926·37-s − 0.446·38-s + 0.327·40-s + 0.0111·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6039} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4647081788$
$L(\frac12)$  $\approx$  $0.4647081788$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;61\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;61\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
11 \( 1 - T \)
61 \( 1 - T \)
good2 \( 1 + 2.54T + 2T^{2} \)
5 \( 1 + 0.329T + 5T^{2} \)
7 \( 1 - 0.595T + 7T^{2} \)
13 \( 1 - 0.0829T + 13T^{2} \)
17 \( 1 + 6.96T + 17T^{2} \)
19 \( 1 - 1.08T + 19T^{2} \)
23 \( 1 + 3.20T + 23T^{2} \)
29 \( 1 + 8.72T + 29T^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 + 5.63T + 37T^{2} \)
41 \( 1 - 0.0715T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 - 8.24T + 53T^{2} \)
59 \( 1 - 0.674T + 59T^{2} \)
67 \( 1 - 0.140T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 1.10T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 - 1.08T + 83T^{2} \)
89 \( 1 - 1.53T + 89T^{2} \)
97 \( 1 + 8.74T + 97T^{2} \)
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

−8.197473359062292791554268999431, −7.54260415719775860397483449942, −6.94902417681888655503682637570, −6.29353658629589947751675585417, −5.47731203169963096263976363700, −4.31900323469839738883621809163, −3.45848821460427428771627256563, −2.16619497659845561753887030185, −1.81510747478904852956750515620, −0.45410945173783718289007838765, 0.45410945173783718289007838765, 1.81510747478904852956750515620, 2.16619497659845561753887030185, 3.45848821460427428771627256563, 4.31900323469839738883621809163, 5.47731203169963096263976363700, 6.29353658629589947751675585417, 6.94902417681888655503682637570, 7.54260415719775860397483449942, 8.197473359062292791554268999431

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