Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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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

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{1764} (883, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(2.470754300\)
\(L(\frac12)\)  \(\approx\)  \(2.470754300\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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 \( 1 - 1.41T + T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−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