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  :  $\chi_{1764} (1243, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :0),\ 0.605 + 0.795i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5869705535\)
\(L(\frac12)\)  \(\approx\)  \(0.5869705535\)
\(L(1)\)   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

−9.100777793763634865762488981956, −8.613029783060397991315509707163, −7.87451798494159881769975370341, −7.15292978823360098662071039641, −6.27756029136800055621069338263, −5.42966471062023955239344779615, −4.78997946875933757265319941348, −3.13222330033251035775996215260, −2.29251402830539862297653218458, −0.60097015769081178863827817118, 1.44193440244662016617745613553, 2.61889142530479702165042925502, 3.34078547643171601110707041047, 4.71714491947273489038642057406, 5.42616194856615565403755394025, 6.87157246585498388548340881002, 7.32152872094821419581859032643, 8.105007153093755029020923519337, 8.877359005296056440187324528902, 9.706769129691241505604179192285

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