Properties

Label 2-21780-1.1-c1-0-13
Degree $2$
Conductor $21780$
Sign $-1$
Analytic cond. $173.914$
Root an. cond. $13.1876$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s − 2·13-s + 5·17-s + 8·19-s − 5·23-s + 25-s − 6·29-s − 9·31-s − 4·35-s − 4·37-s − 6·41-s + 6·43-s + 13·47-s + 9·49-s − 9·53-s + 10·59-s + 11·61-s − 2·65-s + 12·67-s − 8·71-s − 4·73-s + 5·79-s − 8·83-s + 5·85-s + 8·89-s + 8·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 0.554·13-s + 1.21·17-s + 1.83·19-s − 1.04·23-s + 1/5·25-s − 1.11·29-s − 1.61·31-s − 0.676·35-s − 0.657·37-s − 0.937·41-s + 0.914·43-s + 1.89·47-s + 9/7·49-s − 1.23·53-s + 1.30·59-s + 1.40·61-s − 0.248·65-s + 1.46·67-s − 0.949·71-s − 0.468·73-s + 0.562·79-s − 0.878·83-s + 0.542·85-s + 0.847·89-s + 0.838·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21780\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(173.914\)
Root analytic conductor: \(13.1876\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 21780,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.95771094092056, −15.44829297258835, −14.51870476841951, −14.22797054606080, −13.71481685582306, −12.92329724666440, −12.74255842187274, −12.01917547105925, −11.63280536096701, −10.75670012631581, −9.986299048561770, −9.886576907648280, −9.295138499316067, −8.795847049479891, −7.712828104276213, −7.392745777811692, −6.825997789406254, −5.983204365382665, −5.541676250719310, −5.138171305726312, −3.753227570641452, −3.622045528813126, −2.790299405676194, −2.027154230084821, −1.007998711877648, 0, 1.007998711877648, 2.027154230084821, 2.790299405676194, 3.622045528813126, 3.753227570641452, 5.138171305726312, 5.541676250719310, 5.983204365382665, 6.825997789406254, 7.392745777811692, 7.712828104276213, 8.795847049479891, 9.295138499316067, 9.886576907648280, 9.986299048561770, 10.75670012631581, 11.63280536096701, 12.01917547105925, 12.74255842187274, 12.92329724666440, 13.71481685582306, 14.22797054606080, 14.51870476841951, 15.44829297258835, 15.95771094092056

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