Properties

Label 2-72-1.1-c11-0-9
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.19e3·5-s + 1.84e4·7-s − 1.35e5·11-s − 8.48e5·13-s + 7.12e6·17-s − 5.04e6·19-s + 1.48e7·23-s − 4.74e7·25-s + 1.15e8·29-s − 1.63e8·31-s − 2.19e7·35-s − 2.23e8·37-s − 1.05e8·41-s + 1.41e9·43-s − 2.46e9·47-s − 1.63e9·49-s + 4.83e8·53-s + 1.61e8·55-s − 6.15e9·59-s − 7.53e9·61-s + 1.00e9·65-s − 8.76e9·67-s + 1.04e10·71-s − 3.17e10·73-s − 2.51e9·77-s − 3.98e10·79-s − 1.35e10·83-s + ⋯
L(s)  = 1  − 0.170·5-s + 0.415·7-s − 0.254·11-s − 0.633·13-s + 1.21·17-s − 0.467·19-s + 0.482·23-s − 0.970·25-s + 1.04·29-s − 1.02·31-s − 0.0707·35-s − 0.530·37-s − 0.142·41-s + 1.47·43-s − 1.57·47-s − 0.827·49-s + 0.158·53-s + 0.0433·55-s − 1.12·59-s − 1.14·61-s + 0.107·65-s − 0.793·67-s + 0.684·71-s − 1.79·73-s − 0.105·77-s − 1.45·79-s − 0.376·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 72,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 + 238 p T + p^{11} T^{2} \)
7 \( 1 - 2640 p T + p^{11} T^{2} \)
11 \( 1 + 135884 T + p^{11} T^{2} \)
13 \( 1 + 848186 T + p^{11} T^{2} \)
17 \( 1 - 7124606 T + p^{11} T^{2} \)
19 \( 1 + 5046316 T + p^{11} T^{2} \)
23 \( 1 - 14891224 T + p^{11} T^{2} \)
29 \( 1 - 115001346 T + p^{11} T^{2} \)
31 \( 1 + 163990552 T + p^{11} T^{2} \)
37 \( 1 + 223622178 T + p^{11} T^{2} \)
41 \( 1 + 105358314 T + p^{11} T^{2} \)
43 \( 1 - 1419475852 T + p^{11} T^{2} \)
47 \( 1 + 2469276960 T + p^{11} T^{2} \)
53 \( 1 - 483704986 T + p^{11} T^{2} \)
59 \( 1 + 6151842476 T + p^{11} T^{2} \)
61 \( 1 + 7532732282 T + p^{11} T^{2} \)
67 \( 1 + 8764949068 T + p^{11} T^{2} \)
71 \( 1 - 10401627752 T + p^{11} T^{2} \)
73 \( 1 + 31738391270 T + p^{11} T^{2} \)
79 \( 1 + 39880016072 T + p^{11} T^{2} \)
83 \( 1 + 13513323988 T + p^{11} T^{2} \)
89 \( 1 + 81514517226 T + p^{11} T^{2} \)
97 \( 1 - 30783027074 T + p^{11} T^{2} \)
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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

−11.84812196191024582740870245321, −10.68192945597756813832942610858, −9.592505180442608584531936503029, −8.235491195221083297853121371203, −7.27043119287031754576630458666, −5.74419682716315104183926686585, −4.53572009400874848739329284462, −3.05064459256594690459725398210, −1.55300842706871748756266367622, 0, 1.55300842706871748756266367622, 3.05064459256594690459725398210, 4.53572009400874848739329284462, 5.74419682716315104183926686585, 7.27043119287031754576630458666, 8.235491195221083297853121371203, 9.592505180442608584531936503029, 10.68192945597756813832942610858, 11.84812196191024582740870245321

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