Properties

Label 2-5520-1.1-c1-0-48
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 − 4.86·7-s + 9-s − 4.77·11-s + 2.77·13-s − 15-s − 0.636·17-s + 3.50·19-s + 4.86·21-s + 23-s + 25-s − 27-s + 7.36·29-s + 5.15·31-s + 4.77·33-s − 4.86·35-s + 2.86·37-s − 2.77·39-s − 6.19·41-s + 8.28·43-s + 45-s − 7.50·47-s + 16.6·49-s + 0.636·51-s − 4.91·53-s − 4.77·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.83·7-s + 0.333·9-s − 1.44·11-s + 0.770·13-s − 0.258·15-s − 0.154·17-s + 0.804·19-s + 1.06·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.36·29-s + 0.925·31-s + 0.831·33-s − 0.822·35-s + 0.471·37-s − 0.444·39-s − 0.967·41-s + 1.26·43-s + 0.149·45-s − 1.09·47-s + 2.38·49-s + 0.0891·51-s − 0.675·53-s − 0.644·55-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 \( 1 + 4.86T + 7T^{2} \)
11 \( 1 + 4.77T + 11T^{2} \)
13 \( 1 - 2.77T + 13T^{2} \)
17 \( 1 + 0.636T + 17T^{2} \)
19 \( 1 - 3.50T + 19T^{2} \)
29 \( 1 - 7.36T + 29T^{2} \)
31 \( 1 - 5.15T + 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
41 \( 1 + 6.19T + 41T^{2} \)
43 \( 1 - 8.28T + 43T^{2} \)
47 \( 1 + 7.50T + 47T^{2} \)
53 \( 1 + 4.91T + 53T^{2} \)
59 \( 1 - 8.65T + 59T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 1.45T + 79T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 - 9.29T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−7.76990721637916515834749093198, −6.74686197456505457131150077996, −6.46621790040605593729036306032, −5.68019837925885163838617210115, −5.10492869083676477774891379815, −4.08152357908983427590955622803, −3.05280382271680339413591587885, −2.65124403697881943579303589311, −1.10524444463300926649713717407, 0, 1.10524444463300926649713717407, 2.65124403697881943579303589311, 3.05280382271680339413591587885, 4.08152357908983427590955622803, 5.10492869083676477774891379815, 5.68019837925885163838617210115, 6.46621790040605593729036306032, 6.74686197456505457131150077996, 7.76990721637916515834749093198

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