Properties

Label 2-21780-1.1-c1-0-28
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 + 4·13-s − 6·17-s − 2·19-s + 25-s − 4·31-s + 4·35-s − 10·37-s + 4·43-s − 12·47-s + 9·49-s − 6·53-s − 12·59-s + 10·61-s + 4·65-s − 4·67-s − 8·73-s + 10·79-s − 6·83-s − 6·85-s + 6·89-s + 16·91-s − 2·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 1.10·13-s − 1.45·17-s − 0.458·19-s + 1/5·25-s − 0.718·31-s + 0.676·35-s − 1.64·37-s + 0.609·43-s − 1.75·47-s + 9/7·49-s − 0.824·53-s − 1.56·59-s + 1.28·61-s + 0.496·65-s − 0.488·67-s − 0.936·73-s + 1.12·79-s − 0.658·83-s − 0.650·85-s + 0.635·89-s + 1.67·91-s − 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.80412187349826, −15.18231953504560, −14.77624847466948, −14.10589569608724, −13.76449429925498, −13.17142827294494, −12.65799384368991, −11.89461712370386, −11.29412005929072, −10.88194521429956, −10.60998918430196, −9.689622160503713, −8.973603002852378, −8.596742858824746, −8.120072588016054, −7.403333680531554, −6.658703689231612, −6.197749556890042, −5.374424794742147, −4.880559067993424, −4.251653625691743, −3.570407754172462, −2.570179849855544, −1.749405339285517, −1.420129610281989, 0, 1.420129610281989, 1.749405339285517, 2.570179849855544, 3.570407754172462, 4.251653625691743, 4.880559067993424, 5.374424794742147, 6.197749556890042, 6.658703689231612, 7.403333680531554, 8.120072588016054, 8.596742858824746, 8.973603002852378, 9.689622160503713, 10.60998918430196, 10.88194521429956, 11.29412005929072, 11.89461712370386, 12.65799384368991, 13.17142827294494, 13.76449429925498, 14.10589569608724, 14.77624847466948, 15.18231953504560, 15.80412187349826

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