Properties

Label 2-5520-1.1-c1-0-87
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 2·7-s + 9-s − 2·11-s − 2·13-s + 15-s − 8·19-s + 2·21-s + 23-s + 25-s + 27-s − 10·29-s − 8·31-s − 2·33-s + 2·35-s + 8·37-s − 2·39-s − 6·41-s − 12·43-s + 45-s − 8·47-s − 3·49-s + 10·53-s − 2·55-s − 8·57-s − 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.258·15-s − 1.83·19-s + 0.436·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 1.43·31-s − 0.348·33-s + 0.338·35-s + 1.31·37-s − 0.320·39-s − 0.937·41-s − 1.82·43-s + 0.149·45-s − 1.16·47-s − 3/7·49-s + 1.37·53-s − 0.269·55-s − 1.05·57-s − 0.520·59-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: $\chi_{5520} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.81417203762081308908148277407, −7.24107948248560444101615145715, −6.41988324886864550344016879934, −5.52932896154542070844987186365, −4.89082478245225597485246334794, −4.11562648032117163784361171754, −3.20071799127292819293597680832, −2.13689586797702783654764076471, −1.75805536690367425869346834365, 0, 1.75805536690367425869346834365, 2.13689586797702783654764076471, 3.20071799127292819293597680832, 4.11562648032117163784361171754, 4.89082478245225597485246334794, 5.52932896154542070844987186365, 6.41988324886864550344016879934, 7.24107948248560444101615145715, 7.81417203762081308908148277407

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