Properties

Label 2-5520-1.1-c1-0-46
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 + 1.56·7-s + 9-s + 3.12·11-s + 2·13-s + 15-s + 3.56·17-s + 2·19-s + 1.56·21-s − 23-s + 25-s + 27-s + 6.68·29-s − 4.68·31-s + 3.12·33-s + 1.56·35-s + 2.43·37-s + 2·39-s − 2.68·41-s + 45-s − 4·47-s − 4.56·49-s + 3.56·51-s + 7.56·53-s + 3.12·55-s + 2·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.590·7-s + 0.333·9-s + 0.941·11-s + 0.554·13-s + 0.258·15-s + 0.863·17-s + 0.458·19-s + 0.340·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 1.24·29-s − 0.841·31-s + 0.543·33-s + 0.263·35-s + 0.400·37-s + 0.320·39-s − 0.419·41-s + 0.149·45-s − 0.583·47-s − 0.651·49-s + 0.498·51-s + 1.03·53-s + 0.421·55-s + 0.264·57-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.544853237\)
\(L(\frac12)\) \(\approx\) \(3.544853237\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.56T + 7T^{2} \)
11 \( 1 - 3.12T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 3.56T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + 4.68T + 31T^{2} \)
37 \( 1 - 2.43T + 37T^{2} \)
41 \( 1 + 2.68T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 7.56T + 53T^{2} \)
59 \( 1 + 3.56T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 + 8.68T + 67T^{2} \)
71 \( 1 + 0.438T + 71T^{2} \)
73 \( 1 + 4.24T + 73T^{2} \)
79 \( 1 - 2.87T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + 5.12T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−8.277266617868684416563278175221, −7.49798166321625942112823659404, −6.77541428932317051535252519573, −6.02289684934293703856018097975, −5.27182153989776344005139817517, −4.41651864173653427063270264343, −3.63389199898900878820170045706, −2.86322783719428630909232653588, −1.76050307300593083884306030466, −1.09899276835850695300664812720, 1.09899276835850695300664812720, 1.76050307300593083884306030466, 2.86322783719428630909232653588, 3.63389199898900878820170045706, 4.41651864173653427063270264343, 5.27182153989776344005139817517, 6.02289684934293703856018097975, 6.77541428932317051535252519573, 7.49798166321625942112823659404, 8.277266617868684416563278175221

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