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  − 0.381·2-s − 1.85·4-s − 5-s − 7-s + 1.47·8-s + 0.381·10-s + 1.23·13-s + 0.381·14-s + 3.14·16-s + 3.09·17-s + 1.76·19-s + 1.85·20-s − 5.09·23-s − 4·25-s − 0.472·26-s + 1.85·28-s − 4.61·29-s − 4.23·31-s − 4.14·32-s − 1.18·34-s + 35-s + 6.47·37-s − 0.673·38-s − 1.47·40-s − 11.1·41-s + 12.5·43-s + 1.94·46-s + ⋯
L(s)  = 1  − 0.270·2-s − 0.927·4-s − 0.447·5-s − 0.377·7-s + 0.520·8-s + 0.120·10-s + 0.342·13-s + 0.102·14-s + 0.786·16-s + 0.749·17-s + 0.404·19-s + 0.414·20-s − 1.06·23-s − 0.800·25-s − 0.0925·26-s + 0.350·28-s − 0.857·29-s − 0.760·31-s − 0.732·32-s − 0.202·34-s + 0.169·35-s + 1.06·37-s − 0.109·38-s − 0.232·40-s − 1.74·41-s + 1.91·43-s + 0.286·46-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 \( 1 + 0.381T + 2T^{2} \)
5 \( 1 + T + 5T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 - 3.09T + 17T^{2} \)
19 \( 1 - 1.76T + 19T^{2} \)
23 \( 1 + 5.09T + 23T^{2} \)
29 \( 1 + 4.61T + 29T^{2} \)
31 \( 1 + 4.23T + 31T^{2} \)
37 \( 1 - 6.47T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 - 6.61T + 47T^{2} \)
53 \( 1 - 2.38T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 - 7.61T + 61T^{2} \)
67 \( 1 + 8.32T + 67T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 + 6.38T + 79T^{2} \)
83 \( 1 - 2.70T + 83T^{2} \)
89 \( 1 + 6.85T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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

−7.76154565758793658606021479162, −7.03150306474627187014297388811, −5.90075080593352903463373853332, −5.57970956095861238270863247684, −4.57890786804700912309586076732, −3.82840143272932226708281518862, −3.44217108486327928255452659176, −2.15715976539399520472391528167, −1.02296854151087814089041589982, 0, 1.02296854151087814089041589982, 2.15715976539399520472391528167, 3.44217108486327928255452659176, 3.82840143272932226708281518862, 4.57890786804700912309586076732, 5.57970956095861238270863247684, 5.90075080593352903463373853332, 7.03150306474627187014297388811, 7.76154565758793658606021479162

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