Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $0.895 + 0.444i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·13-s + (1.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + (1.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + 43-s + (0.5 + 0.866i)67-s + (−1.5 + 0.866i)73-s + (0.5 − 0.866i)79-s + (−1.5 − 0.866i)103-s + (−0.5 − 0.866i)109-s + ⋯
L(s)  = 1  − 1.73i·13-s + (1.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + (1.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + 43-s + (0.5 + 0.866i)67-s + (−1.5 + 0.866i)73-s + (0.5 − 0.866i)79-s + (−1.5 − 0.866i)103-s + (−0.5 − 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.895 + 0.444i$
motivic weight  =  \(0\)
character  :  $\chi_{1764} (901, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :0),\ 0.895 + 0.444i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.159088361\)
\(L(\frac12)\)  \(\approx\)  \(1.159088361\)
\(L(1)\)   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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.687184356593749704493411338288, −8.403023092783124412015173049755, −7.972400335505062075757378460648, −7.18860862345448173196603323893, −6.02399397173807095499728119634, −5.53021260668522422814805144838, −4.49610482149762231237602058076, −3.41057191562855977404957052690, −2.60865273860038604852418654443, −1.04076825266703248020012614521, 1.39681928976638470793928278990, 2.61058847721582603051547490613, 3.70144622381404672599981968592, 4.62941671262291896795223466798, 5.42507612909214849578848355285, 6.49898595070475263277650220619, 7.11930821621898335104347898863, 7.891062340729175464212660120786, 9.081666323979300436048649978424, 9.275247870096338604079306453823

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