Properties

Label 2-5520-1.1-c1-0-53
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 + 5.02·7-s + 9-s + 3.71·11-s − 6.49·13-s + 15-s + 5.37·17-s + 8.39·19-s + 5.02·21-s + 23-s + 25-s + 27-s + 3.31·29-s − 7.18·31-s + 3.71·33-s + 5.02·35-s − 3.02·37-s − 6.49·39-s + 0.782·41-s + 0.0950·43-s + 45-s − 8.39·47-s + 18.2·49-s + 5.37·51-s + 6.82·53-s + 3.71·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.89·7-s + 0.333·9-s + 1.11·11-s − 1.80·13-s + 0.258·15-s + 1.30·17-s + 1.92·19-s + 1.09·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.615·29-s − 1.28·31-s + 0.645·33-s + 0.848·35-s − 0.496·37-s − 1.03·39-s + 0.122·41-s + 0.0144·43-s + 0.149·45-s − 1.22·47-s + 2.60·49-s + 0.752·51-s + 0.937·53-s + 0.500·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.931597166\)
\(L(\frac12)\) \(\approx\) \(3.931597166\)
\(L(\frac{3}{2})\) 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 \( 1 - 5.02T + 7T^{2} \)
11 \( 1 - 3.71T + 11T^{2} \)
13 \( 1 + 6.49T + 13T^{2} \)
17 \( 1 - 5.37T + 17T^{2} \)
19 \( 1 - 8.39T + 19T^{2} \)
29 \( 1 - 3.31T + 29T^{2} \)
31 \( 1 + 7.18T + 31T^{2} \)
37 \( 1 + 3.02T + 37T^{2} \)
41 \( 1 - 0.782T + 41T^{2} \)
43 \( 1 - 0.0950T + 43T^{2} \)
47 \( 1 + 8.39T + 47T^{2} \)
53 \( 1 - 6.82T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 - 9.71T + 67T^{2} \)
71 \( 1 + 8.73T + 71T^{2} \)
73 \( 1 - 3.64T + 73T^{2} \)
79 \( 1 + 8.59T + 79T^{2} \)
83 \( 1 - 7.37T + 83T^{2} \)
89 \( 1 + 6.29T + 89T^{2} \)
97 \( 1 + 5.32T + 97T^{2} \)
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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

−8.016319642627268180743505909179, −7.47293283587279540368023611043, −7.11711493164003302089578328734, −5.81077495560341634353638201438, −5.07284292329350183543541191941, −4.72590743264831556413512317738, −3.62112123599132451964340525696, −2.75967243989119534039147182264, −1.72928962322335438390710337899, −1.20076151676841946895678406597, 1.20076151676841946895678406597, 1.72928962322335438390710337899, 2.75967243989119534039147182264, 3.62112123599132451964340525696, 4.72590743264831556413512317738, 5.07284292329350183543541191941, 5.81077495560341634353638201438, 7.11711493164003302089578328734, 7.47293283587279540368023611043, 8.016319642627268180743505909179

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