Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·13-s − 19-s − 5·25-s + 11·31-s + 11·37-s − 13·43-s + 14·61-s + 5·67-s + 17·73-s + 17·79-s + 14·97-s − 13·103-s − 19·109-s + ⋯
L(s)  = 1  + 1.38·13-s − 0.229·19-s − 25-s + 1.97·31-s + 1.80·37-s − 1.98·43-s + 1.79·61-s + 0.610·67-s + 1.98·73-s + 1.91·79-s + 1.42·97-s − 1.28·103-s − 1.81·109-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.824793164\)
\(L(\frac12)\)  \(\approx\)  \(1.824793164\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−9.371369340475474030335733727421, −8.228115246436835029120734828381, −8.087898060104447287606612070089, −6.68988040532618663572401167961, −6.23769545148704938178746476280, −5.25343964239659451061264663366, −4.23748187348637361897475637498, −3.43898861300086424137501327609, −2.26111515808195169942016527355, −0.965089271400007404454750534601, 0.965089271400007404454750534601, 2.26111515808195169942016527355, 3.43898861300086424137501327609, 4.23748187348637361897475637498, 5.25343964239659451061264663366, 6.23769545148704938178746476280, 6.68988040532618663572401167961, 8.087898060104447287606612070089, 8.228115246436835029120734828381, 9.371369340475474030335733727421

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