Properties

Label 2-5520-1.1-c1-0-51
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 − 4.48·7-s + 9-s + 1.14·11-s + 0.853·13-s − 15-s + 1.34·17-s + 3.83·19-s + 4.48·21-s − 23-s + 25-s − 27-s − 8.02·29-s − 2.19·31-s − 1.14·33-s − 4.48·35-s − 2.48·37-s − 0.853·39-s + 11.3·41-s − 10.6·43-s + 45-s − 1.53·47-s + 13.1·49-s − 1.34·51-s + 4.02·53-s + 1.14·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.69·7-s + 0.333·9-s + 0.345·11-s + 0.236·13-s − 0.258·15-s + 0.325·17-s + 0.879·19-s + 0.979·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s − 1.49·29-s − 0.394·31-s − 0.199·33-s − 0.758·35-s − 0.409·37-s − 0.136·39-s + 1.76·41-s − 1.62·43-s + 0.149·45-s − 0.224·47-s + 1.87·49-s − 0.188·51-s + 0.553·53-s + 0.154·55-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: \(1\)
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 \( 1 + 4.48T + 7T^{2} \)
11 \( 1 - 1.14T + 11T^{2} \)
13 \( 1 - 0.853T + 13T^{2} \)
17 \( 1 - 1.34T + 17T^{2} \)
19 \( 1 - 3.83T + 19T^{2} \)
29 \( 1 + 8.02T + 29T^{2} \)
31 \( 1 + 2.19T + 31T^{2} \)
37 \( 1 + 2.48T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 1.53T + 47T^{2} \)
53 \( 1 - 4.02T + 53T^{2} \)
59 \( 1 - 15.0T + 59T^{2} \)
61 \( 1 + 5.83T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 4.32T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 + 0.585T + 79T^{2} \)
83 \( 1 + 5.63T + 83T^{2} \)
89 \( 1 + 0.100T + 89T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
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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

−7.51753910948058578219960947674, −7.01820475239984164932649222759, −6.24594280271300422061256946029, −5.79559474047823067209460175795, −5.10690573756739483658473182178, −3.88283372747718658164252006795, −3.41516017393529501970121449245, −2.39218842680616095565867475805, −1.18854558734192236960784281190, 0, 1.18854558734192236960784281190, 2.39218842680616095565867475805, 3.41516017393529501970121449245, 3.88283372747718658164252006795, 5.10690573756739483658473182178, 5.79559474047823067209460175795, 6.24594280271300422061256946029, 7.01820475239984164932649222759, 7.51753910948058578219960947674

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