Properties

Label 2-5520-1.1-c1-0-16
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 − 2.56·7-s + 9-s − 5.12·11-s + 2·13-s + 15-s − 0.561·17-s + 2·19-s − 2.56·21-s − 23-s + 25-s + 27-s − 5.68·29-s + 7.68·31-s − 5.12·33-s − 2.56·35-s + 6.56·37-s + 2·39-s + 9.68·41-s + 45-s − 4·47-s − 0.438·49-s − 0.561·51-s + 3.43·53-s − 5.12·55-s + 2·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.968·7-s + 0.333·9-s − 1.54·11-s + 0.554·13-s + 0.258·15-s − 0.136·17-s + 0.458·19-s − 0.558·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s − 1.05·29-s + 1.38·31-s − 0.891·33-s − 0.432·35-s + 1.07·37-s + 0.320·39-s + 1.51·41-s + 0.149·45-s − 0.583·47-s − 0.0626·49-s − 0.0786·51-s + 0.472·53-s − 0.690·55-s + 0.264·57-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

\(L(1)\) \(\approx\) \(2.087594527\)
\(L(\frac12)\) \(\approx\) \(2.087594527\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.56T + 7T^{2} \)
11 \( 1 + 5.12T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 0.561T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 - 7.68T + 31T^{2} \)
37 \( 1 - 6.56T + 37T^{2} \)
41 \( 1 - 9.68T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 3.43T + 53T^{2} \)
59 \( 1 - 0.561T + 59T^{2} \)
61 \( 1 - 0.876T + 61T^{2} \)
67 \( 1 - 3.68T + 67T^{2} \)
71 \( 1 + 4.56T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 0.315T + 83T^{2} \)
89 \( 1 - 3.12T + 89T^{2} \)
97 \( 1 - 2.87T + 97T^{2} \)
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

−8.024489923883684005752256161734, −7.64639972560922797713387958247, −6.67894870060593962394100505544, −6.04079091317727681557359135647, −5.34711075332276694848618076125, −4.44719790658571881666587106621, −3.48836734729989045598260061026, −2.79920083663139171802560760543, −2.13340463348737489454252899177, −0.73011891525904154094942911784, 0.73011891525904154094942911784, 2.13340463348737489454252899177, 2.79920083663139171802560760543, 3.48836734729989045598260061026, 4.44719790658571881666587106621, 5.34711075332276694848618076125, 6.04079091317727681557359135647, 6.67894870060593962394100505544, 7.64639972560922797713387958247, 8.024489923883684005752256161734

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