Properties

Label 2-5520-1.1-c1-0-56
Degree $2$
Conductor $5520$
Sign $-1$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 7-s + 9-s − 2.44·11-s + 4.44·13-s + 15-s − 5.44·17-s − 4.44·19-s − 21-s + 23-s + 25-s − 27-s + 10.3·29-s − 0.101·31-s + 2.44·33-s − 35-s + 3.89·37-s − 4.44·39-s + 5.44·41-s − 2·43-s − 45-s − 8.44·47-s − 6·49-s + 5.44·51-s + 0.550·53-s + 2.44·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s − 0.738·11-s + 1.23·13-s + 0.258·15-s − 1.32·17-s − 1.02·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.92·29-s − 0.0181·31-s + 0.426·33-s − 0.169·35-s + 0.640·37-s − 0.712·39-s + 0.851·41-s − 0.304·43-s − 0.149·45-s − 1.23·47-s − 0.857·49-s + 0.763·51-s + 0.0756·53-s + 0.330·55-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: no
Arithmetic: yes
Character: $\chi_{5520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 + T \)
23 \( 1 - T \)
good7 \( 1 - T + 7T^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
13 \( 1 - 4.44T + 13T^{2} \)
17 \( 1 + 5.44T + 17T^{2} \)
19 \( 1 + 4.44T + 19T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + 0.101T + 31T^{2} \)
37 \( 1 - 3.89T + 37T^{2} \)
41 \( 1 - 5.44T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 8.44T + 47T^{2} \)
53 \( 1 - 0.550T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 0.651T + 61T^{2} \)
67 \( 1 - 7T + 67T^{2} \)
71 \( 1 - 4.34T + 71T^{2} \)
73 \( 1 + 5.34T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 - 3.10T + 97T^{2} \)
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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

−7.998900351165692221970364504029, −6.88614684630696585502663221729, −6.43319724873137140849601587825, −5.71496757545350526494324552262, −4.60763332796836225500383153260, −4.45170956927122151966525697776, −3.29316303795388708183035422721, −2.32756786166233649443248476252, −1.21209491126946288057364593563, 0, 1.21209491126946288057364593563, 2.32756786166233649443248476252, 3.29316303795388708183035422721, 4.45170956927122151966525697776, 4.60763332796836225500383153260, 5.71496757545350526494324552262, 6.43319724873137140849601587825, 6.88614684630696585502663221729, 7.998900351165692221970364504029

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