Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
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  − 1.45·2-s + 0.119·4-s − 2.38·5-s − 7-s + 2.73·8-s + 3.47·10-s − 0.911·13-s + 1.45·14-s − 4.22·16-s + 2.18·17-s + 0.711·19-s − 0.285·20-s + 7.80·23-s + 0.698·25-s + 1.32·26-s − 0.119·28-s − 8.86·29-s − 6.62·31-s + 0.675·32-s − 3.18·34-s + 2.38·35-s − 4.20·37-s − 1.03·38-s − 6.53·40-s + 11.9·41-s + 7.51·43-s − 11.3·46-s + ⋯
L(s)  = 1  − 1.02·2-s + 0.0598·4-s − 1.06·5-s − 0.377·7-s + 0.967·8-s + 1.09·10-s − 0.252·13-s + 0.389·14-s − 1.05·16-s + 0.530·17-s + 0.163·19-s − 0.0638·20-s + 1.62·23-s + 0.139·25-s + 0.260·26-s − 0.0226·28-s − 1.64·29-s − 1.19·31-s + 0.119·32-s − 0.545·34-s + 0.403·35-s − 0.691·37-s − 0.168·38-s − 1.03·40-s + 1.86·41-s + 1.14·43-s − 1.67·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7623} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5011162347$
$L(\frac12)$  $\approx$  $0.5011162347$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 1.45T + 2T^{2} \)
5 \( 1 + 2.38T + 5T^{2} \)
13 \( 1 + 0.911T + 13T^{2} \)
17 \( 1 - 2.18T + 17T^{2} \)
19 \( 1 - 0.711T + 19T^{2} \)
23 \( 1 - 7.80T + 23T^{2} \)
29 \( 1 + 8.86T + 29T^{2} \)
31 \( 1 + 6.62T + 31T^{2} \)
37 \( 1 + 4.20T + 37T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 - 7.51T + 43T^{2} \)
47 \( 1 + 6.27T + 47T^{2} \)
53 \( 1 - 6.56T + 53T^{2} \)
59 \( 1 - 9.62T + 59T^{2} \)
61 \( 1 + 0.0627T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 - 5.96T + 71T^{2} \)
73 \( 1 + 4.38T + 73T^{2} \)
79 \( 1 + 4.85T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 0.261T + 89T^{2} \)
97 \( 1 - 9.88T + 97T^{2} \)
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

−7.73095395624374122080173828001, −7.45324710477697369747599340173, −6.95215029912810885454184068770, −5.74631074076542939348774610369, −5.09039809734285007109952613905, −4.12693570707946889029657033695, −3.64554719061296354498366202123, −2.62226773967796975621425380241, −1.42677117157122029804410031917, −0.44435449608172592586083596473, 0.44435449608172592586083596473, 1.42677117157122029804410031917, 2.62226773967796975621425380241, 3.64554719061296354498366202123, 4.12693570707946889029657033695, 5.09039809734285007109952613905, 5.74631074076542939348774610369, 6.95215029912810885454184068770, 7.45324710477697369747599340173, 7.73095395624374122080173828001

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