Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $0.999 - 0.0131i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.658 − 1.25i)2-s + (−1.13 + 1.64i)4-s + 2.08i·5-s + (2.80 + 0.333i)8-s + (2.60 − 1.37i)10-s + 4.26·11-s + 4.80·13-s + (−1.43 − 3.73i)16-s − 3.20i·17-s + 2.81i·19-s + (−3.42 − 2.35i)20-s + (−2.80 − 5.34i)22-s − 4.66·23-s + 0.669·25-s + (−3.16 − 6.01i)26-s + ⋯
L(s)  = 1  + (−0.465 − 0.885i)2-s + (−0.566 + 0.823i)4-s + 0.930i·5-s + (0.993 + 0.117i)8-s + (0.823 − 0.433i)10-s + 1.28·11-s + 1.33·13-s + (−0.357 − 0.933i)16-s − 0.777i·17-s + 0.646i·19-s + (−0.766 − 0.527i)20-s + (−0.599 − 1.13i)22-s − 0.971·23-s + 0.133·25-s + (−0.620 − 1.17i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.999 - 0.0131i$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (1079, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :1/2),\ 0.999 - 0.0131i)\)
\(L(1)\)  \(\approx\)  \(1.442491901\)
\(L(\frac12)\)  \(\approx\)  \(1.442491901\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) 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 \( 1 + (0.658 + 1.25i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2.08iT - 5T^{2} \)
11 \( 1 - 4.26T + 11T^{2} \)
13 \( 1 - 4.80T + 13T^{2} \)
17 \( 1 + 3.20iT - 17T^{2} \)
19 \( 1 - 2.81iT - 19T^{2} \)
23 \( 1 + 4.66T + 23T^{2} \)
29 \( 1 + 3.87iT - 29T^{2} \)
31 \( 1 - 10.2iT - 31T^{2} \)
37 \( 1 - 0.273T + 37T^{2} \)
41 \( 1 - 0.387iT - 41T^{2} \)
43 \( 1 - 0.907iT - 43T^{2} \)
47 \( 1 + 7.85T + 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 - 3.70T + 59T^{2} \)
61 \( 1 - 8.02T + 61T^{2} \)
67 \( 1 - 1.40iT - 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 0.826iT - 79T^{2} \)
83 \( 1 - 5.69T + 83T^{2} \)
89 \( 1 - 3.02iT - 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−9.449516969358691707569193555534, −8.594336045662419286028025090604, −7.986147124074007232437852509297, −6.84145564691885684219403261718, −6.39361571474529576020113956621, −5.05263600934335585527283047274, −3.77375155844775727937718106614, −3.46577722634917244555473040079, −2.20412504242820188849183128493, −1.11156910859636149200449404898, 0.823827823122380737496298896036, 1.75354352526080615316113641475, 3.79238434535590850600755862825, 4.37026328311227158344762220097, 5.44964377144282694118501409771, 6.20070846985992552843202700615, 6.77704936836468084724831252641, 7.961902147309033693725303096662, 8.473384541241876488692925418635, 9.153203811988284553305716520479

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