Properties

Label 2-5520-1.1-c1-0-58
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 1.89·7-s + 9-s + 2.38·11-s − 4.38·13-s − 15-s + 0.761·17-s − 0.864·19-s + 1.89·21-s + 23-s + 25-s − 27-s + 8.76·29-s − 9.35·31-s − 2.38·33-s − 1.89·35-s − 0.103·37-s + 4.38·39-s + 9.53·41-s − 3.25·43-s + 45-s − 3.13·47-s − 3.40·49-s − 0.761·51-s + 8.01·53-s + 2.38·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.716·7-s + 0.333·9-s + 0.719·11-s − 1.21·13-s − 0.258·15-s + 0.184·17-s − 0.198·19-s + 0.413·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.62·29-s − 1.68·31-s − 0.415·33-s − 0.320·35-s − 0.0169·37-s + 0.702·39-s + 1.48·41-s − 0.495·43-s + 0.149·45-s − 0.457·47-s − 0.485·49-s − 0.106·51-s + 1.10·53-s + 0.321·55-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: \(1\)
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 \( 1 + 1.89T + 7T^{2} \)
11 \( 1 - 2.38T + 11T^{2} \)
13 \( 1 + 4.38T + 13T^{2} \)
17 \( 1 - 0.761T + 17T^{2} \)
19 \( 1 + 0.864T + 19T^{2} \)
29 \( 1 - 8.76T + 29T^{2} \)
31 \( 1 + 9.35T + 31T^{2} \)
37 \( 1 + 0.103T + 37T^{2} \)
41 \( 1 - 9.53T + 41T^{2} \)
43 \( 1 + 3.25T + 43T^{2} \)
47 \( 1 + 3.13T + 47T^{2} \)
53 \( 1 - 8.01T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 3.64T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 - 6.55T + 71T^{2} \)
73 \( 1 + 4.92T + 73T^{2} \)
79 \( 1 - 7.04T + 79T^{2} \)
83 \( 1 + 5.23T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + 0.0645T + 97T^{2} \)
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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

−7.59379925235903249898669233788, −6.96735843547707639723555199673, −6.37888235372392597281462585962, −5.70983806368819028613914951523, −4.93530801048408249226542803945, −4.21205268443420562031152096122, −3.22222279573673447142068945571, −2.35964621995168022875338465977, −1.25584017134138386024805223605, 0, 1.25584017134138386024805223605, 2.35964621995168022875338465977, 3.22222279573673447142068945571, 4.21205268443420562031152096122, 4.93530801048408249226542803945, 5.70983806368819028613914951523, 6.37888235372392597281462585962, 6.96735843547707639723555199673, 7.59379925235903249898669233788

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