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Properties

Label 2-10780-1.1-c1-0-12
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  + 2·3-s − 5-s + 9-s − 11-s + 4·13-s − 2·15-s + 4·19-s − 6·23-s + 25-s − 4·27-s − 6·29-s − 8·31-s − 2·33-s + 2·37-s + 8·39-s − 6·41-s + 8·43-s − 45-s − 6·47-s − 6·53-s + 55-s + 8·57-s + 12·59-s − 2·61-s − 4·65-s − 10·67-s − 12·69-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.516·15-s + 0.917·19-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 1.43·31-s − 0.348·33-s + 0.328·37-s + 1.28·39-s − 0.937·41-s + 1.21·43-s − 0.149·45-s − 0.875·47-s − 0.824·53-s + 0.134·55-s + 1.05·57-s + 1.56·59-s − 0.256·61-s − 0.496·65-s − 1.22·67-s − 1.44·69-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: Trivial
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 \( 1 - 2 T + p T^{2} \)
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

−16.44855808911975, −16.28815806473185, −15.58131130058686, −15.08395993382699, −14.49322820731591, −14.02349025293589, −13.42343655874213, −13.02303350727700, −12.26178045268364, −11.53070340622507, −11.09153394487086, −10.38228147123479, −9.543726873016603, −9.176779934743572, −8.479695873924181, −7.911237713411769, −7.568010288019511, −6.723619120703306, −5.823912108693032, −5.319019048540907, −4.158655597319819, −3.671639880575456, −3.122884008619565, −2.202845762764706, −1.419917945051841, 0, 1.419917945051841, 2.202845762764706, 3.122884008619565, 3.671639880575456, 4.158655597319819, 5.319019048540907, 5.823912108693032, 6.723619120703306, 7.568010288019511, 7.911237713411769, 8.479695873924181, 9.176779934743572, 9.543726873016603, 10.38228147123479, 11.09153394487086, 11.53070340622507, 12.26178045268364, 13.02303350727700, 13.42343655874213, 14.02349025293589, 14.49322820731591, 15.08395993382699, 15.58131130058686, 16.28815806473185, 16.44855808911975

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