Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
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-s − 4-s − 5-s − 7-s + 3·8-s + 10-s − 5·13-s + 14-s − 16-s − 7·17-s + 6·19-s + 20-s + 4·23-s − 4·25-s + 5·26-s + 28-s + 9·29-s − 2·31-s − 5·32-s + 7·34-s + 35-s + 9·37-s − 6·38-s − 3·40-s − 7·41-s + 6·43-s − 4·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s + 1.06·8-s + 0.316·10-s − 1.38·13-s + 0.267·14-s − 1/4·16-s − 1.69·17-s + 1.37·19-s + 0.223·20-s + 0.834·23-s − 4/5·25-s + 0.980·26-s + 0.188·28-s + 1.67·29-s − 0.359·31-s − 0.883·32-s + 1.20·34-s + 0.169·35-s + 1.47·37-s − 0.973·38-s − 0.474·40-s − 1.09·41-s + 0.914·43-s − 0.589·46-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  :  $\chi_{7623} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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

−17.51252957118975, −16.86480685880187, −16.25230438642691, −15.69283147157108, −15.13301930514676, −14.41724515496870, −13.69946256279218, −13.35967240025027, −12.55851965284596, −12.01685008711882, −11.28590927162350, −10.69607797957320, −9.925168763987323, −9.446143827663995, −9.031840001286681, −8.158473418162883, −7.667188494343371, −7.028566829478870, −6.366548355779013, −5.183073272821284, −4.738204155828831, −4.039163921704895, −3.036273753075497, −2.197202414087177, −0.9269748978524627, 0, 0.9269748978524627, 2.197202414087177, 3.036273753075497, 4.039163921704895, 4.738204155828831, 5.183073272821284, 6.366548355779013, 7.028566829478870, 7.667188494343371, 8.158473418162883, 9.031840001286681, 9.446143827663995, 9.925168763987323, 10.69607797957320, 11.28590927162350, 12.01685008711882, 12.55851965284596, 13.35967240025027, 13.69946256279218, 14.41724515496870, 15.13301930514676, 15.69283147157108, 16.25230438642691, 16.86480685880187, 17.51252957118975

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