Properties

Label 2-5520-1.1-c1-0-21
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 1.39·7-s + 9-s + 6.64·13-s − 15-s + 3.39·17-s + 1.39·21-s − 23-s + 25-s − 27-s + 0.601·29-s + 6.04·31-s − 1.39·35-s − 8.69·37-s − 6.64·39-s + 5.24·41-s + 7.44·43-s + 45-s − 2.79·47-s − 5.04·49-s − 3.39·51-s + 9.24·53-s + 3.24·59-s − 9.44·61-s − 1.39·63-s + 6.64·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.528·7-s + 0.333·9-s + 1.84·13-s − 0.258·15-s + 0.824·17-s + 0.305·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.111·29-s + 1.08·31-s − 0.236·35-s − 1.42·37-s − 1.06·39-s + 0.819·41-s + 1.13·43-s + 0.149·45-s − 0.407·47-s − 0.720·49-s − 0.475·51-s + 1.27·53-s + 0.422·59-s − 1.20·61-s − 0.176·63-s + 0.824·65-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.883923257\)
\(L(\frac12)\) \(\approx\) \(1.883923257\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.39T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.64T + 13T^{2} \)
17 \( 1 - 3.39T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 - 0.601T + 29T^{2} \)
31 \( 1 - 6.04T + 31T^{2} \)
37 \( 1 + 8.69T + 37T^{2} \)
41 \( 1 - 5.24T + 41T^{2} \)
43 \( 1 - 7.44T + 43T^{2} \)
47 \( 1 + 2.79T + 47T^{2} \)
53 \( 1 - 9.24T + 53T^{2} \)
59 \( 1 - 3.24T + 59T^{2} \)
61 \( 1 + 9.44T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 7.89T + 71T^{2} \)
73 \( 1 - 4.79T + 73T^{2} \)
79 \( 1 + 5.85T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 6.09T + 89T^{2} \)
97 \( 1 - 9.44T + 97T^{2} \)
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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

−8.229954626853797993918535022307, −7.32842386384325546406610955580, −6.53318293216438492153201046754, −5.99099502085968091416421990083, −5.52226090038736556058170429469, −4.49132583112241414593627236035, −3.68022516963048021982753018020, −2.93135205548507371469263614531, −1.66426877941703981416363646784, −0.802828088709736895396299559422, 0.802828088709736895396299559422, 1.66426877941703981416363646784, 2.93135205548507371469263614531, 3.68022516963048021982753018020, 4.49132583112241414593627236035, 5.52226090038736556058170429469, 5.99099502085968091416421990083, 6.53318293216438492153201046754, 7.32842386384325546406610955580, 8.229954626853797993918535022307

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