Properties

Label 2-5520-1.1-c1-0-26
Degree $2$
Conductor $5520$
Sign $1$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 3·7-s + 9-s + 4·11-s + 15-s − 3·17-s + 8·19-s − 3·21-s − 23-s + 25-s − 27-s + 9·29-s + 5·31-s − 4·33-s − 3·35-s − 9·37-s + 7·41-s − 4·43-s − 45-s + 2·47-s + 2·49-s + 3·51-s + 13·53-s − 4·55-s − 8·57-s + 3·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.20·11-s + 0.258·15-s − 0.727·17-s + 1.83·19-s − 0.654·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 0.898·31-s − 0.696·33-s − 0.507·35-s − 1.47·37-s + 1.09·41-s − 0.609·43-s − 0.149·45-s + 0.291·47-s + 2/7·49-s + 0.420·51-s + 1.78·53-s − 0.539·55-s − 1.05·57-s + 0.390·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(44.0774\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.028325750\)
\(L(\frac12)\) \(\approx\) \(2.028325750\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + T \)
23 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.154748981038880777868297478790, −7.35431694975267318577548707562, −6.80466454616981725602709625544, −6.01223528851390510636283506656, −5.12893599196902185809428477339, −4.57855600420135458094374913034, −3.88136271853996912281498524537, −2.85499157171565939866611329878, −1.59030470424870116207137233576, −0.867145429131301457442003022812, 0.867145429131301457442003022812, 1.59030470424870116207137233576, 2.85499157171565939866611329878, 3.88136271853996912281498524537, 4.57855600420135458094374913034, 5.12893599196902185809428477339, 6.01223528851390510636283506656, 6.80466454616981725602709625544, 7.35431694975267318577548707562, 8.154748981038880777868297478790

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