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Properties

Label 2-21780-1.1-c1-0-25
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

−15.81383035953898, −15.18312113323062, −14.64388111641430, −14.31945088689462, −13.61128638554356, −13.29526162848352, −12.42814381657723, −12.13614751513898, −11.21402679826866, −10.82595595552221, −10.64525698449933, −9.729479628737883, −8.978351362901324, −8.515068527308494, −8.189255250872221, −7.381418339727861, −6.700724929810489, −6.109969022337497, −5.518513105908325, −4.707207094665758, −4.319474555614790, −3.638842860903311, −2.330421066132248, −2.073115968824396, −1.270731898248514, 0, 1.270731898248514, 2.073115968824396, 2.330421066132248, 3.638842860903311, 4.319474555614790, 4.707207094665758, 5.518513105908325, 6.109969022337497, 6.700724929810489, 7.381418339727861, 8.189255250872221, 8.515068527308494, 8.978351362901324, 9.729479628737883, 10.64525698449933, 10.82595595552221, 11.21402679826866, 12.13614751513898, 12.42814381657723, 13.29526162848352, 13.61128638554356, 14.31945088689462, 14.64388111641430, 15.18312113323062, 15.81383035953898

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