Properties

Degree 2
Conductor $ 3^{2} $
Sign $1$
Motivic weight 37
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 7.17e5·2-s + 3.77e11·4-s + 9.63e12·5-s + 1.74e15·7-s − 1.71e17·8-s − 6.91e18·10-s − 2.16e19·11-s + 4.06e20·13-s − 1.25e21·14-s + 7.14e22·16-s − 4.16e21·17-s − 4.03e23·19-s + 3.63e24·20-s + 1.55e25·22-s + 3.07e25·23-s + 2.01e25·25-s − 2.91e26·26-s + 6.57e26·28-s + 3.72e26·29-s − 1.69e27·31-s − 2.76e28·32-s + 2.98e27·34-s + 1.68e28·35-s + 3.17e28·37-s + 2.89e29·38-s − 1.65e30·40-s + 1.14e29·41-s + ⋯
L(s)  = 1  − 1.93·2-s + 2.74·4-s + 1.12·5-s + 0.404·7-s − 3.37·8-s − 2.18·10-s − 1.17·11-s + 1.00·13-s − 0.783·14-s + 3.78·16-s − 0.0718·17-s − 0.889·19-s + 3.09·20-s + 2.27·22-s + 1.97·23-s + 0.276·25-s − 1.94·26-s + 1.11·28-s + 0.328·29-s − 0.435·31-s − 3.94·32-s + 0.139·34-s + 0.457·35-s + 0.309·37-s + 1.72·38-s − 3.81·40-s + 0.166·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9\)    =    \(3^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(37\)
character  :  $\chi_{9} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 9,\ (\ :37/2),\ 1)$
$L(19)$  $\approx$  $1.120924462$
$L(\frac12)$  $\approx$  $1.120924462$
$L(\frac{39}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + 7.17e5T + 1.37e11T^{2} \)
5 \( 1 - 9.63e12T + 7.27e25T^{2} \)
7 \( 1 - 1.74e15T + 1.85e31T^{2} \)
11 \( 1 + 2.16e19T + 3.40e38T^{2} \)
13 \( 1 - 4.06e20T + 1.64e41T^{2} \)
17 \( 1 + 4.16e21T + 3.36e45T^{2} \)
19 \( 1 + 4.03e23T + 2.06e47T^{2} \)
23 \( 1 - 3.07e25T + 2.42e50T^{2} \)
29 \( 1 - 3.72e26T + 1.28e54T^{2} \)
31 \( 1 + 1.69e27T + 1.51e55T^{2} \)
37 \( 1 - 3.17e28T + 1.05e58T^{2} \)
41 \( 1 - 1.14e29T + 4.70e59T^{2} \)
43 \( 1 - 7.17e29T + 2.74e60T^{2} \)
47 \( 1 + 3.03e29T + 7.37e61T^{2} \)
53 \( 1 - 6.59e31T + 6.28e63T^{2} \)
59 \( 1 - 8.50e32T + 3.32e65T^{2} \)
61 \( 1 - 6.05e32T + 1.14e66T^{2} \)
67 \( 1 + 2.62e33T + 3.67e67T^{2} \)
71 \( 1 + 1.38e34T + 3.13e68T^{2} \)
73 \( 1 + 1.07e34T + 8.76e68T^{2} \)
79 \( 1 - 2.43e35T + 1.63e70T^{2} \)
83 \( 1 - 3.57e34T + 1.01e71T^{2} \)
89 \( 1 + 1.28e36T + 1.34e72T^{2} \)
97 \( 1 + 8.70e36T + 3.24e73T^{2} \)
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

−13.07080562489545957930454802082, −11.12224034742114512185069459794, −10.38023460729680869488477941344, −9.159857007178715406475287336762, −8.198214539228176137496746117651, −6.81802612383089184368840713825, −5.59795792675660459153752481295, −2.75345715904718220366226058916, −1.78182685151025830588465770767, −0.73282083784995701345709659954, 0.73282083784995701345709659954, 1.78182685151025830588465770767, 2.75345715904718220366226058916, 5.59795792675660459153752481295, 6.81802612383089184368840713825, 8.198214539228176137496746117651, 9.159857007178715406475287336762, 10.38023460729680869488477941344, 11.12224034742114512185069459794, 13.07080562489545957930454802082

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