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  + 2.09·2-s + 2.40·4-s + 2.53·5-s − 7-s + 0.839·8-s + 5.31·10-s − 5.69·13-s − 2.09·14-s − 3.03·16-s − 0.996·17-s − 6.02·19-s + 6.07·20-s + 1.25·23-s + 1.41·25-s − 11.9·26-s − 2.40·28-s − 1.33·29-s − 1.85·31-s − 8.05·32-s − 2.09·34-s − 2.53·35-s + 9.65·37-s − 12.6·38-s + 2.12·40-s − 10.0·41-s + 5.04·43-s + 2.63·46-s + ⋯
L(s)  = 1  + 1.48·2-s + 1.20·4-s + 1.13·5-s − 0.377·7-s + 0.296·8-s + 1.67·10-s − 1.58·13-s − 0.560·14-s − 0.759·16-s − 0.241·17-s − 1.38·19-s + 1.35·20-s + 0.262·23-s + 0.282·25-s − 2.34·26-s − 0.453·28-s − 0.248·29-s − 0.332·31-s − 1.42·32-s − 0.358·34-s − 0.428·35-s + 1.58·37-s − 2.05·38-s + 0.336·40-s − 1.56·41-s + 0.769·43-s + 0.389·46-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  =  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 \( 1 - 2.09T + 2T^{2} \)
5 \( 1 - 2.53T + 5T^{2} \)
13 \( 1 + 5.69T + 13T^{2} \)
17 \( 1 + 0.996T + 17T^{2} \)
19 \( 1 + 6.02T + 19T^{2} \)
23 \( 1 - 1.25T + 23T^{2} \)
29 \( 1 + 1.33T + 29T^{2} \)
31 \( 1 + 1.85T + 31T^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 5.04T + 43T^{2} \)
47 \( 1 - 1.03T + 47T^{2} \)
53 \( 1 + 7.43T + 53T^{2} \)
59 \( 1 + 6.55T + 59T^{2} \)
61 \( 1 + 3.76T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 + 6.41T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 2.05T + 97T^{2} \)
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

−7.16716833316206160599855344703, −6.52694101379962394867899241566, −6.02800526681075931084268524070, −5.37762162614188663823772322229, −4.69875653039092936833254190430, −4.15762538865902976828928188839, −3.08460589518154013908782156516, −2.46009245322973443884706117277, −1.83585691466481492731517289442, 0, 1.83585691466481492731517289442, 2.46009245322973443884706117277, 3.08460589518154013908782156516, 4.15762538865902976828928188839, 4.69875653039092936833254190430, 5.37762162614188663823772322229, 6.02800526681075931084268524070, 6.52694101379962394867899241566, 7.16716833316206160599855344703

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