Properties

Label 2-5520-1.1-c1-0-29
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 + 3·7-s + 9-s − 0.732·11-s − 4.19·13-s + 15-s − 5.73·17-s + 6.73·19-s + 3·21-s − 23-s + 25-s + 27-s − 0.267·29-s − 3.92·31-s − 0.732·33-s + 3·35-s + 11.9·37-s − 4.19·39-s − 0.267·41-s + 4.53·43-s + 45-s + 9.66·47-s + 2·49-s − 5.73·51-s − 2.26·53-s − 0.732·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.13·7-s + 0.333·9-s − 0.220·11-s − 1.16·13-s + 0.258·15-s − 1.39·17-s + 1.54·19-s + 0.654·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s − 0.0497·29-s − 0.705·31-s − 0.127·33-s + 0.507·35-s + 1.96·37-s − 0.671·39-s − 0.0418·41-s + 0.691·43-s + 0.149·45-s + 1.40·47-s + 0.285·49-s − 0.802·51-s − 0.311·53-s − 0.0987·55-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: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.023453361\)
\(L(\frac12)\) \(\approx\) \(3.023453361\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T + 7T^{2} \)
11 \( 1 + 0.732T + 11T^{2} \)
13 \( 1 + 4.19T + 13T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 - 6.73T + 19T^{2} \)
29 \( 1 + 0.267T + 29T^{2} \)
31 \( 1 + 3.92T + 31T^{2} \)
37 \( 1 - 11.9T + 37T^{2} \)
41 \( 1 + 0.267T + 41T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
47 \( 1 - 9.66T + 47T^{2} \)
53 \( 1 + 2.26T + 53T^{2} \)
59 \( 1 - 3.19T + 59T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 + 0.464T + 67T^{2} \)
71 \( 1 - 0.267T + 71T^{2} \)
73 \( 1 - 9.66T + 73T^{2} \)
79 \( 1 - 6.92T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 - 9.46T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−8.019663219313853471754741685138, −7.57057733184152510942458826939, −6.94156256406943812965732465440, −5.92628983817436065660355130575, −5.09536655742963787837109325148, −4.62726337760161498988752723831, −3.70387052193895692338936175229, −2.48825745595824878734006289163, −2.15800128589694955721755914532, −0.922397691644222319169880871688, 0.922397691644222319169880871688, 2.15800128589694955721755914532, 2.48825745595824878734006289163, 3.70387052193895692338936175229, 4.62726337760161498988752723831, 5.09536655742963787837109325148, 5.92628983817436065660355130575, 6.94156256406943812965732465440, 7.57057733184152510942458826939, 8.019663219313853471754741685138

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