Properties

Degree $2$
Conductor $1764$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 1.41·5-s + 8-s − 1.41·10-s + 1.41·13-s + 16-s + 1.41·17-s − 1.41·20-s + 1.00·25-s + 1.41·26-s − 2·29-s + 32-s + 1.41·34-s − 1.41·40-s + 1.41·41-s + 1.00·50-s + 1.41·52-s − 2·58-s − 1.41·61-s + 64-s − 2.00·65-s + 1.41·68-s − 1.41·73-s − 1.41·80-s + 1.41·82-s − 2.00·85-s + ⋯
L(s)  = 1  + 2-s + 4-s − 1.41·5-s + 8-s − 1.41·10-s + 1.41·13-s + 16-s + 1.41·17-s − 1.41·20-s + 1.00·25-s + 1.41·26-s − 2·29-s + 32-s + 1.41·34-s − 1.41·40-s + 1.41·41-s + 1.00·50-s + 1.41·52-s − 2·58-s − 1.41·61-s + 64-s − 2.00·65-s + 1.41·68-s − 1.41·73-s − 1.41·80-s + 1.41·82-s − 2.00·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{1764} (883, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.859520300\)
\(L(\frac12)\) \(\approx\) \(1.859520300\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.41T + T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.507238998594422910190845815944, −8.409820620381303945906461342835, −7.71028414437749712951653289369, −7.22004980049350719028832459650, −6.07407141785245440799488904403, −5.46804349956058881904074425448, −4.25645906281573239712506438275, −3.74845511281811893891380091675, −3.01196616588470107968316419511, −1.39157937932907991579790493048, 1.39157937932907991579790493048, 3.01196616588470107968316419511, 3.74845511281811893891380091675, 4.25645906281573239712506438275, 5.46804349956058881904074425448, 6.07407141785245440799488904403, 7.22004980049350719028832459650, 7.71028414437749712951653289369, 8.409820620381303945906461342835, 9.507238998594422910190845815944

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