Properties

Label 2-5520-1.1-c1-0-11
Degree $2$
Conductor $5520$
Sign $1$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 3·7-s + 9-s + 6·11-s − 2·13-s + 15-s + 3·17-s + 6·19-s + 3·21-s + 23-s + 25-s − 27-s − 9·29-s + 3·31-s − 6·33-s + 3·35-s + 3·37-s + 2·39-s − 3·41-s − 45-s − 4·47-s + 2·49-s − 3·51-s − 9·53-s − 6·55-s − 6·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.258·15-s + 0.727·17-s + 1.37·19-s + 0.654·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.67·29-s + 0.538·31-s − 1.04·33-s + 0.507·35-s + 0.493·37-s + 0.320·39-s − 0.468·41-s − 0.149·45-s − 0.583·47-s + 2/7·49-s − 0.420·51-s − 1.23·53-s − 0.809·55-s − 0.794·57-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.267219657\)
\(L(\frac12)\) \(\approx\) \(1.267219657\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−7.989586710103824428067856864786, −7.29384453127105234509533939558, −6.70902466654960435696911857755, −6.10594516285092160704014512345, −5.34477195253805987643204761809, −4.45076781536427426432080930506, −3.58206158092981118469442472445, −3.15396175658548843645329551116, −1.64627640654696770580664686070, −0.64378858879566636241022575623, 0.64378858879566636241022575623, 1.64627640654696770580664686070, 3.15396175658548843645329551116, 3.58206158092981118469442472445, 4.45076781536427426432080930506, 5.34477195253805987643204761809, 6.10594516285092160704014512345, 6.70902466654960435696911857755, 7.29384453127105234509533939558, 7.989586710103824428067856864786

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