Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $0.605 - 0.795i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (1.73 − i)11-s + (−0.5 + 0.866i)16-s + 1.99·22-s + (−1.73 − i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.499i)32-s + (−1 + 1.73i)37-s + (1.73 + 0.999i)44-s + (−0.999 − 1.73i)46-s + 0.999i·50-s − 0.999·64-s − 2i·71-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (1.73 − i)11-s + (−0.5 + 0.866i)16-s + 1.99·22-s + (−1.73 − i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.499i)32-s + (−1 + 1.73i)37-s + (1.73 + 0.999i)44-s + (−0.999 − 1.73i)46-s + 0.999i·50-s − 0.999·64-s − 2i·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.605 - 0.795i$
motivic weight  =  \(0\)
character  :  $\chi_{1764} (667, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :0),\ 0.605 - 0.795i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.981365916\)
\(L(\frac12)\)  \(\approx\)  \(1.981365916\)
\(L(1)\)   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 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.414133468090048691552372361687, −8.604114846485165367377756451998, −8.054065519811579527613742956622, −6.89414551393883860080127986201, −6.39962531471180739800779995331, −5.68903376698362869304282153135, −4.61706481461209074363314089289, −3.82527172320038699897556483198, −3.07882727823879067072400770206, −1.65927051902591981804467826021, 1.45875696382827159344789518132, 2.34089575548995430852041619483, 3.79825535494140145397411078902, 4.10544195877832560964655738112, 5.20454618175549509549389498152, 6.11240860645178291181051337650, 6.78912918638547266293829705291, 7.55310119399822366388242576248, 8.811209351792727814127048820591, 9.587041124781011136057646210977

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