Properties

Label 2-72-1.1-c11-0-2
Degree $2$
Conductor $72$
Sign $1$
Analytic cond. $55.3207$
Root an. cond. $7.43778$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.87e3·5-s − 7.23e4·7-s − 1.47e5·11-s − 1.56e6·13-s + 1.45e5·17-s + 1.09e6·19-s + 6.00e7·23-s − 4.53e7·25-s + 1.96e7·29-s − 2.39e8·31-s + 1.35e8·35-s + 4.88e8·37-s − 4.70e7·41-s + 4.28e8·43-s − 4.50e8·47-s + 3.25e9·49-s − 4.33e9·53-s + 2.76e8·55-s + 8.93e9·59-s + 4.67e9·61-s + 2.92e9·65-s + 7.49e9·67-s + 2.70e10·71-s + 1.16e10·73-s + 1.06e10·77-s + 2.47e9·79-s − 4.27e10·83-s + ⋯
L(s)  = 1  − 0.267·5-s − 1.62·7-s − 0.276·11-s − 1.16·13-s + 0.0249·17-s + 0.101·19-s + 1.94·23-s − 0.928·25-s + 0.177·29-s − 1.50·31-s + 0.435·35-s + 1.15·37-s − 0.0634·41-s + 0.444·43-s − 0.286·47-s + 1.64·49-s − 1.42·53-s + 0.0741·55-s + 1.62·59-s + 0.708·61-s + 0.312·65-s + 0.678·67-s + 1.77·71-s + 0.659·73-s + 0.450·77-s + 0.0906·79-s − 1.19·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(55.3207\)
Root analytic conductor: \(7.43778\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: $\chi_{72} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(1.001055725\)
\(L(\frac12)\) \(\approx\) \(1.001055725\)
\(L(\frac{13}{2})\) 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 \( 1 \)
good5 \( 1 + 374 p T + p^{11} T^{2} \)
7 \( 1 + 72312 T + p^{11} T^{2} \)
11 \( 1 + 147940 T + p^{11} T^{2} \)
13 \( 1 + 1562858 T + p^{11} T^{2} \)
17 \( 1 - 145774 T + p^{11} T^{2} \)
19 \( 1 - 1096796 T + p^{11} T^{2} \)
23 \( 1 - 60014264 T + p^{11} T^{2} \)
29 \( 1 - 19626954 T + p^{11} T^{2} \)
31 \( 1 + 239950480 T + p^{11} T^{2} \)
37 \( 1 - 488238078 T + p^{11} T^{2} \)
41 \( 1 + 47066010 T + p^{11} T^{2} \)
43 \( 1 - 428866948 T + p^{11} T^{2} \)
47 \( 1 + 450903216 T + p^{11} T^{2} \)
53 \( 1 + 4336685950 T + p^{11} T^{2} \)
59 \( 1 - 8937556460 T + p^{11} T^{2} \)
61 \( 1 - 4673884486 T + p^{11} T^{2} \)
67 \( 1 - 7498937612 T + p^{11} T^{2} \)
71 \( 1 - 27032101480 T + p^{11} T^{2} \)
73 \( 1 - 159947146 p T + p^{11} T^{2} \)
79 \( 1 - 2478876544 T + p^{11} T^{2} \)
83 \( 1 + 42745596956 T + p^{11} T^{2} \)
89 \( 1 - 93270772662 T + p^{11} T^{2} \)
97 \( 1 - 118032786914 T + p^{11} T^{2} \)
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

−12.59678177238233129412971064921, −11.24855588633908657044424047591, −9.930823907344200284468404485601, −9.197823501338072973264715401734, −7.56330240470941563855340378736, −6.59485612218428395584999094508, −5.20702218138044875656733107474, −3.62327435260340793407216199774, −2.53394080892337475836381326680, −0.52280468260721427830718497975, 0.52280468260721427830718497975, 2.53394080892337475836381326680, 3.62327435260340793407216199774, 5.20702218138044875656733107474, 6.59485612218428395584999094508, 7.56330240470941563855340378736, 9.197823501338072973264715401734, 9.930823907344200284468404485601, 11.24855588633908657044424047591, 12.59678177238233129412971064921

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