Properties

Label 2-10780-1.1-c1-0-7
Degree $2$
Conductor $10780$
Sign $-1$
Analytic cond. $86.0787$
Root an. cond. $9.27786$
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  − 3-s − 5-s − 2·9-s − 11-s + 4·13-s + 15-s − 6·17-s − 2·19-s − 9·23-s + 25-s + 5·27-s + 9·29-s + 4·31-s + 33-s + 2·37-s − 4·39-s − 3·41-s + 11·43-s + 2·45-s + 12·47-s + 6·51-s + 55-s + 2·57-s + 7·61-s − 4·65-s + 5·67-s + 9·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.301·11-s + 1.10·13-s + 0.258·15-s − 1.45·17-s − 0.458·19-s − 1.87·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s + 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.640·39-s − 0.468·41-s + 1.67·43-s + 0.298·45-s + 1.75·47-s + 0.840·51-s + 0.134·55-s + 0.264·57-s + 0.896·61-s − 0.496·65-s + 0.610·67-s + 1.08·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10780\)    =    \(2^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(86.0787\)
Root analytic conductor: \(9.27786\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10780} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10780,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−16.90577569627787, −15.91994022999472, −15.86383408531792, −15.43884502959586, −14.28067856592035, −14.10812260169424, −13.37988290600961, −12.73525563828917, −12.07856306973311, −11.58898867727913, −11.09539850234481, −10.48791950692025, −10.04961982796493, −8.857317481326777, −8.629071153085343, −8.034754176834296, −7.196999701150778, −6.258146874829633, −6.176027443709012, −5.295808297087206, −4.301033127137961, −4.087379318388491, −2.878172435199233, −2.256504095732686, −0.9646738707022310, 0, 0.9646738707022310, 2.256504095732686, 2.878172435199233, 4.087379318388491, 4.301033127137961, 5.295808297087206, 6.176027443709012, 6.258146874829633, 7.196999701150778, 8.034754176834296, 8.629071153085343, 8.857317481326777, 10.04961982796493, 10.48791950692025, 11.09539850234481, 11.58898867727913, 12.07856306973311, 12.73525563828917, 13.37988290600961, 14.10812260169424, 14.28067856592035, 15.43884502959586, 15.86383408531792, 15.91994022999472, 16.90577569627787

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