Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 $
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-s − 4·7-s + 11-s − 4·13-s − 4·19-s + 6·23-s + 25-s + 6·29-s + 8·31-s + 4·35-s + 2·37-s − 6·41-s + 8·43-s − 6·47-s + 9·49-s + 6·53-s − 55-s + 12·59-s + 2·61-s + 4·65-s − 10·67-s + 12·71-s − 16·73-s − 4·77-s + 8·79-s − 6·89-s + 16·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 0.301·11-s − 1.10·13-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.676·35-s + 0.328·37-s − 0.937·41-s + 1.21·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s − 0.134·55-s + 1.56·59-s + 0.256·61-s + 0.496·65-s − 1.22·67-s + 1.42·71-s − 1.87·73-s − 0.455·77-s + 0.900·79-s − 0.635·89-s + 1.67·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1980\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1980} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1980,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.063789879\)
\(L(\frac12)\)  \(\approx\)  \(1.063789879\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−19.43625676835506, −19.14424745339435, −18.18948916837853, −17.21853474680984, −16.91419690094451, −16.11710695349504, −15.54534930389860, −14.91056800496133, −14.22101601043873, −13.22876970142391, −12.83520568912951, −12.09734767984802, −11.53096413477614, −10.37087020331946, −10.03805365393662, −9.143388368464317, −8.521834734599109, −7.476016433460924, −6.737775491978390, −6.257600674711141, −5.038356278498496, −4.228878003670547, −3.203273562141929, −2.495812936203869, −0.6638036249493917, 0.6638036249493917, 2.495812936203869, 3.203273562141929, 4.228878003670547, 5.038356278498496, 6.257600674711141, 6.737775491978390, 7.476016433460924, 8.521834734599109, 9.143388368464317, 10.03805365393662, 10.37087020331946, 11.53096413477614, 12.09734767984802, 12.83520568912951, 13.22876970142391, 14.22101601043873, 14.91056800496133, 15.54534930389860, 16.11710695349504, 16.91419690094451, 17.21853474680984, 18.18948916837853, 19.14424745339435, 19.43625676835506

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