Properties

Degree $2$
Conductor $21780$
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 + 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 + 12·59-s − 2·61-s − 4·65-s − 10·67-s + 12·71-s + 16·73-s − 8·79-s − 6·89-s + 16·91-s − 4·95-s + 14·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-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 + 1.56·59-s − 0.256·61-s − 0.496·65-s − 1.22·67-s + 1.42·71-s + 1.87·73-s − 0.900·79-s − 0.635·89-s + 1.67·91-s − 0.410·95-s + 1.42·97-s + 0.0995·101-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: $1$
Motivic weight: \(1\)
Character: $\chi_{21780} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 21780,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.259308030\)
\(L(\frac12)\) \(\approx\) \(3.259308030\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.45308307669845, −14.96090052463770, −14.62422782501194, −13.86386013392208, −13.49582620058697, −12.89881995127974, −12.15371779579670, −11.46357598145561, −11.32766040516955, −10.83638953430494, −10.06457624922244, −9.396108025973125, −8.666095538413429, −8.295409870550080, −7.761413695996861, −7.168149307936401, −6.506396699801039, −5.666468372420975, −5.111376589382431, −4.577529250261130, −3.838002495694187, −3.200544930289158, −2.288664835623204, −1.371512560714719, −0.8371606478550319, 0.8371606478550319, 1.371512560714719, 2.288664835623204, 3.200544930289158, 3.838002495694187, 4.577529250261130, 5.111376589382431, 5.666468372420975, 6.506396699801039, 7.168149307936401, 7.761413695996861, 8.295409870550080, 8.666095538413429, 9.396108025973125, 10.06457624922244, 10.83638953430494, 11.32766040516955, 11.46357598145561, 12.15371779579670, 12.89881995127974, 13.49582620058697, 13.86386013392208, 14.62422782501194, 14.96090052463770, 15.45308307669845

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