↑

Properties

Label 2-42e2-4.3-c0-0-6
Degree $2$
Conductor $1764$
Sign $1$
Analytic cond. $0.880350$
Root an. cond. $0.938270$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

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: $1$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (883, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.470754300\)
\(L(\frac12)\) \(\approx\) \(2.470754300\)
\(L(1)\) 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

−9.659383482076321123103911949970, −8.845207762284210828230873090792, −7.60420758735182749836449887121, −6.86504228984014016252219503405, −6.18460956923895546934513110706, −5.31451908067224196334220458268, −4.79943829362728001219307872881, −3.63332251567808974695681433885, −2.36291571661188020919983003193, −1.94540061496984898281872094842, 1.94540061496984898281872094842, 2.36291571661188020919983003193, 3.63332251567808974695681433885, 4.79943829362728001219307872881, 5.31451908067224196334220458268, 6.18460956923895546934513110706, 6.86504228984014016252219503405, 7.60420758735182749836449887121, 8.845207762284210828230873090792, 9.659383482076321123103911949970

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