Properties

Label 2-528-1.1-c1-0-5
Degree $2$
Conductor $528$
Sign $1$
Analytic cond. $4.21610$
Root an. cond. $2.05331$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 9-s − 11-s + 2·13-s + 2·15-s + 6·17-s − 4·23-s − 25-s + 27-s + 2·29-s − 33-s − 10·37-s + 2·39-s + 6·41-s + 8·43-s + 2·45-s + 4·47-s − 7·49-s + 6·51-s − 6·53-s − 2·55-s + 12·59-s + 2·61-s + 4·65-s − 4·67-s − 4·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.174·33-s − 1.64·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.298·45-s + 0.583·47-s − 49-s + 0.840·51-s − 0.824·53-s − 0.269·55-s + 1.56·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s − 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $1$
Analytic conductor: \(4.21610\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.048195850\)
\(L(\frac12)\) \(\approx\) \(2.048195850\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−10.54048685043449941200110881885, −9.981987073174720407794705697877, −9.140836382652818830334248334151, −8.212378907948040729561958605852, −7.36587637640732761296700605801, −6.12991681965878756173970153874, −5.39498548145393328063580392099, −3.98296680233403436899260776008, −2.82255351943050054606485500714, −1.54699235007435022323925928448, 1.54699235007435022323925928448, 2.82255351943050054606485500714, 3.98296680233403436899260776008, 5.39498548145393328063580392099, 6.12991681965878756173970153874, 7.36587637640732761296700605801, 8.212378907948040729561958605852, 9.140836382652818830334248334151, 9.981987073174720407794705697877, 10.54048685043449941200110881885

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