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Properties

Label 2-72-9.4-c1-0-1
Degree $2$
Conductor $72$
Sign $0.972 - 0.234i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.68 + 0.396i)3-s + (−1.18 + 2.05i)5-s + (−2.18 − 3.78i)7-s + (2.68 + 1.33i)9-s + (−0.5 − 0.866i)11-s + (0.186 − 0.322i)13-s + (−2.81 + 2.99i)15-s − 5.37·17-s + 0.627·19-s + (−2.18 − 7.25i)21-s + (0.186 − 0.322i)23-s + (−0.313 − 0.543i)25-s + (4 + 3.31i)27-s + (2.18 + 3.78i)29-s + (−3.18 + 5.51i)31-s + ⋯
L(s)  = 1  + (0.973 + 0.228i)3-s + (−0.530 + 0.918i)5-s + (−0.826 − 1.43i)7-s + (0.895 + 0.445i)9-s + (−0.150 − 0.261i)11-s + (0.0516 − 0.0894i)13-s + (−0.726 + 0.773i)15-s − 1.30·17-s + 0.144·19-s + (−0.477 − 1.58i)21-s + (0.0388 − 0.0672i)23-s + (−0.0627 − 0.108i)25-s + (0.769 + 0.638i)27-s + (0.405 + 0.703i)29-s + (−0.572 + 0.991i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.972 - 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05148 + 0.124984i\)
\(L(\frac12)\) \(\approx\) \(1.05148 + 0.124984i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.396i)T \)
good5 \( 1 + (1.18 - 2.05i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.18 + 3.78i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.186 + 0.322i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.37T + 17T^{2} \)
19 \( 1 - 0.627T + 19T^{2} \)
23 \( 1 + (-0.186 + 0.322i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.18 - 3.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.18 - 5.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.74T + 37T^{2} \)
41 \( 1 + (-5.87 + 10.1i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.872 - 1.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.18 + 3.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.744T + 53T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.18 + 2.05i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.87 + 3.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + (-3.18 - 5.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.81 + 8.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (0.872 + 1.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.59885884120698211995189632813, −13.72820938222386480611499348133, −12.87688605500792286663976548578, −10.97484239703290360874038635021, −10.33316642614616915853722407192, −9.009342894345427883741841716162, −7.52659070916985522092147891192, −6.78796326320592240347188043600, −4.17387174021514465059944413867, −3.08366219427620883466983368697, 2.57422165224831863614353373906, 4.40959661098213397712541338428, 6.25523529252752935890943165888, 7.909028627599230400162437591922, 8.919556668285640819836691952917, 9.585166706487300286107305189198, 11.62070730074264037553971097131, 12.70651803559246465556139321599, 13.20579877834166898002105661499, 14.81438599945287301475911967199

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