Properties

Degree $2$
Conductor $5520$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s − 4·11-s − 2·13-s − 15-s − 6·17-s − 4·19-s + 23-s + 25-s − 27-s − 2·29-s + 4·33-s − 2·37-s + 2·39-s + 10·41-s + 4·43-s + 45-s − 7·49-s + 6·51-s + 6·53-s − 4·55-s + 4·57-s + 4·59-s − 10·61-s − 2·65-s + 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.328·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 49-s + 0.840·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s + 0.520·59-s − 1.28·61-s − 0.248·65-s + 1.46·67-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: $1$
Motivic weight: \(1\)
Character: $\chi_{5520} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.047652508\)
\(L(\frac12)\) \(\approx\) \(1.047652508\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−17.43528493253976, −17.21553274503261, −16.33916718326422, −15.83744803533281, −15.23983197030649, −14.67172137947804, −13.83027330388902, −13.20494316163544, −12.76650358161383, −12.23341670654482, −11.17490486179053, −10.90454802557263, −10.28980248782943, −9.522596671327744, −8.926390869841280, −8.072859314969052, −7.405599079202819, −6.605496922306835, −6.068076199051600, −5.167983194327930, −4.729267181755777, −3.807341904158315, −2.527992435531672, −2.070816735168106, −0.5348308113447385, 0.5348308113447385, 2.070816735168106, 2.527992435531672, 3.807341904158315, 4.729267181755777, 5.167983194327930, 6.068076199051600, 6.605496922306835, 7.405599079202819, 8.072859314969052, 8.926390869841280, 9.522596671327744, 10.28980248782943, 10.90454802557263, 11.17490486179053, 12.23341670654482, 12.76650358161383, 13.20494316163544, 13.83027330388902, 14.67172137947804, 15.23983197030649, 15.83744803533281, 16.33916718326422, 17.21553274503261, 17.43528493253976

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