Properties

Label 2-5520-1.1-c1-0-1
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 − 4.46·7-s + 9-s − 3.39·11-s − 4.70·13-s − 15-s − 6.54·17-s + 6.22·19-s − 4.46·21-s − 23-s + 25-s + 27-s + 0.448·29-s + 5.15·31-s − 3.39·33-s + 4.46·35-s − 5.98·37-s − 4.70·39-s + 3.07·41-s + 4.47·43-s − 45-s − 6.22·47-s + 12.9·49-s − 6.54·51-s − 2.59·53-s + 3.39·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.68·7-s + 0.333·9-s − 1.02·11-s − 1.30·13-s − 0.258·15-s − 1.58·17-s + 1.42·19-s − 0.974·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.0832·29-s + 0.925·31-s − 0.590·33-s + 0.754·35-s − 0.984·37-s − 0.753·39-s + 0.479·41-s + 0.683·43-s − 0.149·45-s − 0.908·47-s + 1.84·49-s − 0.916·51-s − 0.355·53-s + 0.457·55-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

\(L(1)\) \(\approx\) \(0.8654461534\)
\(L(\frac12)\) \(\approx\) \(0.8654461534\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.46T + 7T^{2} \)
11 \( 1 + 3.39T + 11T^{2} \)
13 \( 1 + 4.70T + 13T^{2} \)
17 \( 1 + 6.54T + 17T^{2} \)
19 \( 1 - 6.22T + 19T^{2} \)
29 \( 1 - 0.448T + 29T^{2} \)
31 \( 1 - 5.15T + 31T^{2} \)
37 \( 1 + 5.98T + 37T^{2} \)
41 \( 1 - 3.07T + 41T^{2} \)
43 \( 1 - 4.47T + 43T^{2} \)
47 \( 1 + 6.22T + 47T^{2} \)
53 \( 1 + 2.59T + 53T^{2} \)
59 \( 1 - 1.55T + 59T^{2} \)
61 \( 1 + 4.91T + 61T^{2} \)
67 \( 1 + 6.08T + 67T^{2} \)
71 \( 1 + 1.07T + 71T^{2} \)
73 \( 1 + 8.70T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 - 4.30T + 97T^{2} \)
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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

−8.005510937914596088162700017536, −7.46799117326591776416891949987, −6.84640733730721831575239808709, −6.17676926113442828921726172439, −5.12704350304752739576578327657, −4.47986372596935319762172090626, −3.44149420932361365039407454792, −2.89194634994068628803332016180, −2.21802730382417712860980062184, −0.44719517368359605850515577495, 0.44719517368359605850515577495, 2.21802730382417712860980062184, 2.89194634994068628803332016180, 3.44149420932361365039407454792, 4.47986372596935319762172090626, 5.12704350304752739576578327657, 6.17676926113442828921726172439, 6.84640733730721831575239808709, 7.46799117326591776416891949987, 8.005510937914596088162700017536

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