Properties

Label 2-5520-1.1-c1-0-0
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 4.20·7-s + 9-s − 2.75·11-s − 0.753·13-s + 15-s − 4.96·17-s − 4.75·19-s + 4.20·21-s − 23-s + 25-s − 27-s + 4.96·29-s + 0.209·31-s + 2.75·33-s + 4.20·35-s − 5.71·37-s + 0.753·39-s − 9.38·41-s − 12.4·43-s − 45-s − 7.17·47-s + 10.7·49-s + 4.96·51-s − 9.38·53-s + 2.75·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.59·7-s + 0.333·9-s − 0.830·11-s − 0.208·13-s + 0.258·15-s − 1.20·17-s − 1.09·19-s + 0.918·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.921·29-s + 0.0375·31-s + 0.479·33-s + 0.711·35-s − 0.939·37-s + 0.120·39-s − 1.46·41-s − 1.89·43-s − 0.149·45-s − 1.04·47-s + 1.53·49-s + 0.694·51-s − 1.28·53-s + 0.371·55-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: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1298868796\)
\(L(\frac12)\) \(\approx\) \(0.1298868796\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.20T + 7T^{2} \)
11 \( 1 + 2.75T + 11T^{2} \)
13 \( 1 + 0.753T + 13T^{2} \)
17 \( 1 + 4.96T + 17T^{2} \)
19 \( 1 + 4.75T + 19T^{2} \)
29 \( 1 - 4.96T + 29T^{2} \)
31 \( 1 - 0.209T + 31T^{2} \)
37 \( 1 + 5.71T + 37T^{2} \)
41 \( 1 + 9.38T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + 7.17T + 47T^{2} \)
53 \( 1 + 9.38T + 53T^{2} \)
59 \( 1 - 4.96T + 59T^{2} \)
61 \( 1 + 5.17T + 61T^{2} \)
67 \( 1 - 0.209T + 67T^{2} \)
71 \( 1 + 9.38T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 - 2.41T + 79T^{2} \)
83 \( 1 + 5.45T + 83T^{2} \)
89 \( 1 - 4.91T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.249168274403541843662902930521, −7.23447779617899550283364496424, −6.44200395169925744241350885666, −6.42757920877965860724614371320, −5.16592220236755315666087666599, −4.60486045149467756356698393001, −3.61996602786703754091005141598, −2.94862976774238059131393036388, −1.91550436238486790551727118414, −0.18496868059260596846442361952, 0.18496868059260596846442361952, 1.91550436238486790551727118414, 2.94862976774238059131393036388, 3.61996602786703754091005141598, 4.60486045149467756356698393001, 5.16592220236755315666087666599, 6.42757920877965860724614371320, 6.44200395169925744241350885666, 7.23447779617899550283364496424, 8.249168274403541843662902930521

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