Properties

Label 2-5520-1.1-c1-0-61
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 + 7-s + 9-s + 2.44·11-s − 0.449·13-s + 15-s − 0.550·17-s + 0.449·19-s − 21-s + 23-s + 25-s − 27-s − 4.34·29-s − 9.89·31-s − 2.44·33-s − 35-s − 5.89·37-s + 0.449·39-s + 0.550·41-s − 2·43-s − 45-s − 3.55·47-s − 6·49-s + 0.550·51-s + 5.44·53-s − 2.44·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.738·11-s − 0.124·13-s + 0.258·15-s − 0.133·17-s + 0.103·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.807·29-s − 1.77·31-s − 0.426·33-s − 0.169·35-s − 0.969·37-s + 0.0719·39-s + 0.0859·41-s − 0.304·43-s − 0.149·45-s − 0.517·47-s − 0.857·49-s + 0.0770·51-s + 0.748·53-s − 0.330·55-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: $\chi_{5520} (1, \cdot )$
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 \( 1 - T + 7T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + 0.449T + 13T^{2} \)
17 \( 1 + 0.550T + 17T^{2} \)
19 \( 1 - 0.449T + 19T^{2} \)
29 \( 1 + 4.34T + 29T^{2} \)
31 \( 1 + 9.89T + 31T^{2} \)
37 \( 1 + 5.89T + 37T^{2} \)
41 \( 1 - 0.550T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 3.55T + 47T^{2} \)
53 \( 1 - 5.44T + 53T^{2} \)
59 \( 1 - 4.34T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 - 7T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 9.34T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 9.24T + 83T^{2} \)
89 \( 1 + 7.10T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−7.70186646102301411667562043034, −7.02690533959664022719426071644, −6.49851412208641600319588913561, −5.45613047221678302940544463000, −5.05722448009057045034167075535, −3.99294084488089109264212353206, −3.55336477302658470044256243508, −2.19968150372779865790424260110, −1.27416998067059293327661325630, 0, 1.27416998067059293327661325630, 2.19968150372779865790424260110, 3.55336477302658470044256243508, 3.99294084488089109264212353206, 5.05722448009057045034167075535, 5.45613047221678302940544463000, 6.49851412208641600319588913561, 7.02690533959664022719426071644, 7.70186646102301411667562043034

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