Properties

Label 2-72-1.1-c11-0-10
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  + 3.49e3·5-s − 5.54e4·7-s + 5.97e5·11-s + 1.37e6·13-s − 1.01e7·17-s − 7.29e6·19-s + 3.20e7·23-s − 3.66e7·25-s + 1.36e7·29-s + 2.33e8·31-s − 1.93e8·35-s − 2.57e8·37-s + 2.21e8·41-s − 1.69e9·43-s − 5.27e8·47-s + 1.09e9·49-s − 3.27e9·53-s + 2.08e9·55-s + 3.00e9·59-s − 1.16e10·61-s + 4.79e9·65-s − 1.71e10·67-s − 2.61e10·71-s − 7.03e9·73-s − 3.31e10·77-s − 4.19e9·79-s + 3.97e10·83-s + ⋯
L(s)  = 1  + 0.499·5-s − 1.24·7-s + 1.11·11-s + 1.02·13-s − 1.73·17-s − 0.676·19-s + 1.03·23-s − 0.750·25-s + 0.123·29-s + 1.46·31-s − 0.622·35-s − 0.611·37-s + 0.298·41-s − 1.76·43-s − 0.335·47-s + 0.555·49-s − 1.07·53-s + 0.558·55-s + 0.546·59-s − 1.76·61-s + 0.512·65-s − 1.55·67-s − 1.72·71-s − 0.397·73-s − 1.39·77-s − 0.153·79-s + 1.10·83-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 \( 1 - 698 p T + p^{11} T^{2} \)
7 \( 1 + 55464 T + p^{11} T^{2} \)
11 \( 1 - 597004 T + p^{11} T^{2} \)
13 \( 1 - 1373878 T + p^{11} T^{2} \)
17 \( 1 + 10140850 T + p^{11} T^{2} \)
19 \( 1 + 7297396 T + p^{11} T^{2} \)
23 \( 1 - 32057464 T + p^{11} T^{2} \)
29 \( 1 - 13605402 T + p^{11} T^{2} \)
31 \( 1 - 233160800 T + p^{11} T^{2} \)
37 \( 1 + 6967194 p T + p^{11} T^{2} \)
41 \( 1 - 221438598 T + p^{11} T^{2} \)
43 \( 1 + 1697758892 T + p^{11} T^{2} \)
47 \( 1 + 527509392 T + p^{11} T^{2} \)
53 \( 1 + 3277379822 T + p^{11} T^{2} \)
59 \( 1 - 3001908988 T + p^{11} T^{2} \)
61 \( 1 + 11630023610 T + p^{11} T^{2} \)
67 \( 1 + 17189000548 T + p^{11} T^{2} \)
71 \( 1 + 26169539608 T + p^{11} T^{2} \)
73 \( 1 + 7039021094 T + p^{11} T^{2} \)
79 \( 1 + 4199910416 T + p^{11} T^{2} \)
83 \( 1 - 39739936436 T + p^{11} T^{2} \)
89 \( 1 + 10565331594 T + p^{11} T^{2} \)
97 \( 1 + 69851645662 T + p^{11} T^{2} \)
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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

−11.79862511465648457274650231111, −10.62128131277090014794656483418, −9.430583013471741892744439912246, −8.655754776372209917409313847386, −6.66853640703438522848840159542, −6.23126239224997448463342937829, −4.36486952022787405286851382575, −3.08826482265313388913654896999, −1.56271838474469756894641894194, 0, 1.56271838474469756894641894194, 3.08826482265313388913654896999, 4.36486952022787405286851382575, 6.23126239224997448463342937829, 6.66853640703438522848840159542, 8.655754776372209917409313847386, 9.430583013471741892744439912246, 10.62128131277090014794656483418, 11.79862511465648457274650231111

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