Properties

Label 2-72-1.1-c11-0-4
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.13e3·5-s − 1.95e4·7-s + 1.96e5·11-s + 3.61e5·13-s + 1.30e5·17-s + 1.85e7·19-s − 2.15e7·23-s + 2.00e6·25-s − 1.91e8·29-s + 2.07e8·31-s − 1.39e8·35-s − 2.00e8·37-s + 1.43e9·41-s + 7.12e8·43-s + 4.96e8·47-s − 1.59e9·49-s + 3.35e9·53-s + 1.39e9·55-s − 4.58e9·59-s + 3.42e9·61-s + 2.57e9·65-s + 1.70e10·67-s + 7.91e9·71-s + 3.15e10·73-s − 3.83e9·77-s + 4.10e10·79-s + 1.99e10·83-s + ⋯
L(s)  = 1  + 1.02·5-s − 0.439·7-s + 0.367·11-s + 0.269·13-s + 0.0223·17-s + 1.71·19-s − 0.698·23-s + 0.0411·25-s − 1.73·29-s + 1.30·31-s − 0.448·35-s − 0.476·37-s + 1.93·41-s + 0.739·43-s + 0.315·47-s − 0.806·49-s + 1.10·53-s + 0.374·55-s − 0.834·59-s + 0.519·61-s + 0.275·65-s + 1.54·67-s + 0.520·71-s + 1.78·73-s − 0.161·77-s + 1.49·79-s + 0.556·83-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: \(0\)
Selberg data: \((2,\ 72,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(2.619083630\)
\(L(\frac12)\) \(\approx\) \(2.619083630\)
\(L(\frac{13}{2})\) 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 \( 1 - 1426 p T + p^{11} T^{2} \)
7 \( 1 + 19536 T + p^{11} T^{2} \)
11 \( 1 - 196148 T + p^{11} T^{2} \)
13 \( 1 - 361414 T + p^{11} T^{2} \)
17 \( 1 - 130942 T + p^{11} T^{2} \)
19 \( 1 - 18516692 T + p^{11} T^{2} \)
23 \( 1 + 21560872 T + p^{11} T^{2} \)
29 \( 1 + 191663742 T + p^{11} T^{2} \)
31 \( 1 - 207933800 T + p^{11} T^{2} \)
37 \( 1 + 200784930 T + p^{11} T^{2} \)
41 \( 1 - 1435256598 T + p^{11} T^{2} \)
43 \( 1 - 712703116 T + p^{11} T^{2} \)
47 \( 1 - 496082400 T + p^{11} T^{2} \)
53 \( 1 - 3350114330 T + p^{11} T^{2} \)
59 \( 1 + 4583222956 T + p^{11} T^{2} \)
61 \( 1 - 3427501702 T + p^{11} T^{2} \)
67 \( 1 - 17079378356 T + p^{11} T^{2} \)
71 \( 1 - 7915078504 T + p^{11} T^{2} \)
73 \( 1 - 31559658778 T + p^{11} T^{2} \)
79 \( 1 - 41023578808 T + p^{11} T^{2} \)
83 \( 1 - 19974672172 T + p^{11} T^{2} \)
89 \( 1 - 10640163606 T + p^{11} T^{2} \)
97 \( 1 - 6441105794 T + p^{11} T^{2} \)
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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

−12.42482173841000269447645922714, −11.22056198637103585537424878672, −9.862282525639540743347731342916, −9.275605329485683540738351471419, −7.68243519957512041751596772721, −6.32070665426465739077781868867, −5.39578347231873285104196580403, −3.69447734581063286435811346582, −2.25985116084076764983613424485, −0.921390856225327980747479711933, 0.921390856225327980747479711933, 2.25985116084076764983613424485, 3.69447734581063286435811346582, 5.39578347231873285104196580403, 6.32070665426465739077781868867, 7.68243519957512041751596772721, 9.275605329485683540738351471419, 9.862282525639540743347731342916, 11.22056198637103585537424878672, 12.42482173841000269447645922714

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