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  + 0.871·2-s − 1.24·4-s + 4.06·5-s + 7-s − 2.82·8-s + 3.54·10-s + 6.09·13-s + 0.871·14-s + 0.0189·16-s + 1.70·17-s − 5.45·19-s − 5.04·20-s + 6.39·23-s + 11.5·25-s + 5.30·26-s − 1.24·28-s + 4.74·29-s − 4.36·31-s + 5.66·32-s + 1.48·34-s + 4.06·35-s + 2.43·37-s − 4.75·38-s − 11.4·40-s − 3.21·41-s − 0.127·43-s + 5.57·46-s + ⋯
L(s)  = 1  + 0.616·2-s − 0.620·4-s + 1.81·5-s + 0.377·7-s − 0.998·8-s + 1.12·10-s + 1.68·13-s + 0.232·14-s + 0.00474·16-s + 0.412·17-s − 1.25·19-s − 1.12·20-s + 1.33·23-s + 2.30·25-s + 1.04·26-s − 0.234·28-s + 0.881·29-s − 0.783·31-s + 1.00·32-s + 0.254·34-s + 0.687·35-s + 0.401·37-s − 0.771·38-s − 1.81·40-s − 0.501·41-s − 0.0193·43-s + 0.821·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.959720370$
$L(\frac12)$  $\approx$  $3.959720370$
$L(\frac{3}{2})$   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 \( 1 - 0.871T + 2T^{2} \)
5 \( 1 - 4.06T + 5T^{2} \)
13 \( 1 - 6.09T + 13T^{2} \)
17 \( 1 - 1.70T + 17T^{2} \)
19 \( 1 + 5.45T + 19T^{2} \)
23 \( 1 - 6.39T + 23T^{2} \)
29 \( 1 - 4.74T + 29T^{2} \)
31 \( 1 + 4.36T + 31T^{2} \)
37 \( 1 - 2.43T + 37T^{2} \)
41 \( 1 + 3.21T + 41T^{2} \)
43 \( 1 + 0.127T + 43T^{2} \)
47 \( 1 + 8.46T + 47T^{2} \)
53 \( 1 - 4.71T + 53T^{2} \)
59 \( 1 + 4.63T + 59T^{2} \)
61 \( 1 + 5.31T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 + 2.21T + 71T^{2} \)
73 \( 1 - 5.57T + 73T^{2} \)
79 \( 1 + 6.27T + 79T^{2} \)
83 \( 1 + 0.127T + 83T^{2} \)
89 \( 1 - 8.12T + 89T^{2} \)
97 \( 1 + 2.59T + 97T^{2} \)
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

−8.131046586107142216052303122081, −6.73111372385331555279094351468, −6.35565770996684121979008903282, −5.67865316618704059274674890555, −5.17223184701496458744104791567, −4.46924582314452759382966850914, −3.54874866485497132819468109527, −2.78516884139379287868070721395, −1.77953597960529035707312861298, −0.985139180003050969382983945775, 0.985139180003050969382983945775, 1.77953597960529035707312861298, 2.78516884139379287868070721395, 3.54874866485497132819468109527, 4.46924582314452759382966850914, 5.17223184701496458744104791567, 5.67865316618704059274674890555, 6.35565770996684121979008903282, 6.73111372385331555279094351468, 8.131046586107142216052303122081

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