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  + 2.23·2-s + 3.00·4-s − 1.61·5-s + 7-s + 2.23·8-s − 3.61·10-s − 1.23·13-s + 2.23·14-s − 0.999·16-s + 1.85·17-s − 3.85·19-s − 4.85·20-s − 6.61·23-s − 2.38·25-s − 2.76·26-s + 3.00·28-s + 6·29-s + 8.09·31-s − 6.70·32-s + 4.14·34-s − 1.61·35-s + 2.38·37-s − 8.61·38-s − 3.61·40-s − 9.61·41-s − 3.70·43-s − 14.7·46-s + ⋯
L(s)  = 1  + 1.58·2-s + 1.50·4-s − 0.723·5-s + 0.377·7-s + 0.790·8-s − 1.14·10-s − 0.342·13-s + 0.597·14-s − 0.249·16-s + 0.449·17-s − 0.884·19-s − 1.08·20-s − 1.37·23-s − 0.476·25-s − 0.542·26-s + 0.566·28-s + 1.11·29-s + 1.45·31-s − 1.18·32-s + 0.711·34-s − 0.273·35-s + 0.391·37-s − 1.39·38-s − 0.572·40-s − 1.50·41-s − 0.565·43-s − 2.18·46-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 \( 1 - 2.23T + 2T^{2} \)
5 \( 1 + 1.61T + 5T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 - 1.85T + 17T^{2} \)
19 \( 1 + 3.85T + 19T^{2} \)
23 \( 1 + 6.61T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8.09T + 31T^{2} \)
37 \( 1 - 2.38T + 37T^{2} \)
41 \( 1 + 9.61T + 41T^{2} \)
43 \( 1 + 3.70T + 43T^{2} \)
47 \( 1 - 4.47T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 1.23T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 - 0.291T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 7.52T + 83T^{2} \)
89 \( 1 + 3.38T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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

−7.35397365692341344944584635532, −6.58703213407774613901806172964, −6.04239184318070980121968899899, −5.28582640359934451469141683785, −4.40790285380491469658246984458, −4.26126498243089773543364637937, −3.28855809197973904528713209183, −2.60712715457247415344601303352, −1.62929801711823748842356098569, 0, 1.62929801711823748842356098569, 2.60712715457247415344601303352, 3.28855809197973904528713209183, 4.26126498243089773543364637937, 4.40790285380491469658246984458, 5.28582640359934451469141683785, 6.04239184318070980121968899899, 6.58703213407774613901806172964, 7.35397365692341344944584635532

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