Properties

Label 2-21780-1.1-c1-0-26
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 + 3·7-s − 2·13-s − 2·17-s + 19-s + 2·23-s + 25-s + 8·29-s + 5·31-s + 3·35-s − 11·37-s − 6·41-s − 8·43-s − 8·47-s + 2·49-s − 2·53-s − 4·59-s − 3·61-s − 2·65-s − 9·67-s + 6·71-s − 11·73-s − 9·79-s − 8·83-s − 2·85-s − 6·89-s − 6·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s − 0.554·13-s − 0.485·17-s + 0.229·19-s + 0.417·23-s + 1/5·25-s + 1.48·29-s + 0.898·31-s + 0.507·35-s − 1.80·37-s − 0.937·41-s − 1.21·43-s − 1.16·47-s + 2/7·49-s − 0.274·53-s − 0.520·59-s − 0.384·61-s − 0.248·65-s − 1.09·67-s + 0.712·71-s − 1.28·73-s − 1.01·79-s − 0.878·83-s − 0.216·85-s − 0.635·89-s − 0.628·91-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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−15.65501169184784, −15.31634279127026, −14.66480588156006, −14.17190387319874, −13.76543086956919, −13.21115907756830, −12.52145876344017, −11.83624452054918, −11.61821372144492, −10.84780206420789, −10.22023454649564, −9.950896977648980, −9.034121284691917, −8.510434558877831, −8.137922086599499, −7.311338900362441, −6.772997988795221, −6.197613308025778, −5.279821296279866, −4.858347345046910, −4.449122923761532, −3.324261824178293, −2.734763769981017, −1.792918180125112, −1.324120071984594, 0, 1.324120071984594, 1.792918180125112, 2.734763769981017, 3.324261824178293, 4.449122923761532, 4.858347345046910, 5.279821296279866, 6.197613308025778, 6.772997988795221, 7.311338900362441, 8.137922086599499, 8.510434558877831, 9.034121284691917, 9.950896977648980, 10.22023454649564, 10.84780206420789, 11.61821372144492, 11.83624452054918, 12.52145876344017, 13.21115907756830, 13.76543086956919, 14.17190387319874, 14.66480588156006, 15.31634279127026, 15.65501169184784

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