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  − 1.98·2-s + 1.93·4-s + 0.0269·5-s + 7-s + 0.122·8-s − 0.0534·10-s − 4.88·13-s − 1.98·14-s − 4.11·16-s − 1.67·17-s + 1.37·19-s + 0.0521·20-s − 8.06·23-s − 4.99·25-s + 9.68·26-s + 1.93·28-s + 6.39·29-s + 4.01·31-s + 7.93·32-s + 3.33·34-s + 0.0269·35-s + 0.521·37-s − 2.72·38-s + 0.00329·40-s + 10.5·41-s + 3.73·43-s + 16.0·46-s + ⋯
L(s)  = 1  − 1.40·2-s + 0.969·4-s + 0.0120·5-s + 0.377·7-s + 0.0432·8-s − 0.0168·10-s − 1.35·13-s − 0.530·14-s − 1.02·16-s − 0.407·17-s + 0.315·19-s + 0.0116·20-s − 1.68·23-s − 0.999·25-s + 1.89·26-s + 0.366·28-s + 1.18·29-s + 0.720·31-s + 1.40·32-s + 0.571·34-s + 0.00455·35-s + 0.0856·37-s − 0.442·38-s + 0.000521·40-s + 1.65·41-s + 0.570·43-s + 2.35·46-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 \( 1 + 1.98T + 2T^{2} \)
5 \( 1 - 0.0269T + 5T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 - 1.37T + 19T^{2} \)
23 \( 1 + 8.06T + 23T^{2} \)
29 \( 1 - 6.39T + 29T^{2} \)
31 \( 1 - 4.01T + 31T^{2} \)
37 \( 1 - 0.521T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 3.73T + 43T^{2} \)
47 \( 1 - 8.95T + 47T^{2} \)
53 \( 1 - 3.96T + 53T^{2} \)
59 \( 1 + 9.73T + 59T^{2} \)
61 \( 1 - 8.46T + 61T^{2} \)
67 \( 1 - 2.81T + 67T^{2} \)
71 \( 1 + 2.04T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 5.85T + 79T^{2} \)
83 \( 1 + 2.60T + 83T^{2} \)
89 \( 1 + 1.21T + 89T^{2} \)
97 \( 1 - 3.39T + 97T^{2} \)
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

−7.74393139119881470042606485788, −7.18544858521687424797020322240, −6.33918791419430626287698363453, −5.57006128654203523431771221232, −4.56244287811666354746431004779, −4.10582482168397100557023076288, −2.60610662173162115953789937561, −2.14861279764480641867310182757, −1.03555223875678682698222945060, 0, 1.03555223875678682698222945060, 2.14861279764480641867310182757, 2.60610662173162115953789937561, 4.10582482168397100557023076288, 4.56244287811666354746431004779, 5.57006128654203523431771221232, 6.33918791419430626287698363453, 7.18544858521687424797020322240, 7.74393139119881470042606485788

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