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  + 2.34·2-s + 3.48·4-s + 0.146·5-s − 7-s + 3.48·8-s + 0.342·10-s − 4.34·13-s − 2.34·14-s + 1.19·16-s − 0.146·17-s + 1.83·19-s + 0.510·20-s − 8.81·23-s − 4.97·25-s − 10.1·26-s − 3.48·28-s + 4.34·29-s + 0.292·31-s − 4.17·32-s − 0.342·34-s − 0.146·35-s − 3.48·37-s + 4.29·38-s + 0.510·40-s + 2.80·41-s + 7.86·43-s − 20.6·46-s + ⋯
L(s)  = 1  + 1.65·2-s + 1.74·4-s + 0.0654·5-s − 0.377·7-s + 1.23·8-s + 0.108·10-s − 1.20·13-s − 0.626·14-s + 0.299·16-s − 0.0354·17-s + 0.420·19-s + 0.114·20-s − 1.83·23-s − 0.995·25-s − 1.99·26-s − 0.659·28-s + 0.806·29-s + 0.0525·31-s − 0.738·32-s − 0.0588·34-s − 0.0247·35-s − 0.573·37-s + 0.696·38-s + 0.0807·40-s + 0.437·41-s + 1.19·43-s − 3.04·46-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 \( 1 - 2.34T + 2T^{2} \)
5 \( 1 - 0.146T + 5T^{2} \)
13 \( 1 + 4.34T + 13T^{2} \)
17 \( 1 + 0.146T + 17T^{2} \)
19 \( 1 - 1.83T + 19T^{2} \)
23 \( 1 + 8.81T + 23T^{2} \)
29 \( 1 - 4.34T + 29T^{2} \)
31 \( 1 - 0.292T + 31T^{2} \)
37 \( 1 + 3.48T + 37T^{2} \)
41 \( 1 - 2.80T + 41T^{2} \)
43 \( 1 - 7.86T + 43T^{2} \)
47 \( 1 - 0.949T + 47T^{2} \)
53 \( 1 - 4.51T + 53T^{2} \)
59 \( 1 + 8.02T + 59T^{2} \)
61 \( 1 + 5.43T + 61T^{2} \)
67 \( 1 + 7.76T + 67T^{2} \)
71 \( 1 - 3.53T + 71T^{2} \)
73 \( 1 + 16.1T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 9.81T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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

−7.32243093458217316445378590205, −6.63081503260843206897252596232, −5.76843724492194050768891434964, −5.60751186879212159484689771214, −4.38149237771943922601639589606, −4.27802983598255640921736603340, −3.18184861399603950120041373680, −2.58977444438676861200116652286, −1.75885500882358783873392697265, 0, 1.75885500882358783873392697265, 2.58977444438676861200116652286, 3.18184861399603950120041373680, 4.27802983598255640921736603340, 4.38149237771943922601639589606, 5.60751186879212159484689771214, 5.76843724492194050768891434964, 6.63081503260843206897252596232, 7.32243093458217316445378590205

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