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Properties

Label 2-72-72.59-c1-0-8
Degree $2$
Conductor $72$
Sign $-0.0190 + 0.999i$
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  + (−0.867 − 1.11i)2-s + (0.925 − 1.46i)3-s + (−0.494 + 1.93i)4-s + (0.895 − 1.55i)5-s + (−2.43 + 0.236i)6-s + (−2.08 + 1.20i)7-s + (2.59 − 1.12i)8-s + (−1.28 − 2.71i)9-s + (−2.50 + 0.345i)10-s + (−1.36 + 0.790i)11-s + (2.37 + 2.51i)12-s + (5.35 + 3.09i)13-s + (3.15 + 1.28i)14-s + (−1.44 − 2.74i)15-s + (−3.51 − 1.91i)16-s + 3.69i·17-s + ⋯
L(s)  = 1  + (−0.613 − 0.789i)2-s + (0.534 − 0.845i)3-s + (−0.247 + 0.968i)4-s + (0.400 − 0.693i)5-s + (−0.995 + 0.0964i)6-s + (−0.789 + 0.455i)7-s + (0.916 − 0.398i)8-s + (−0.428 − 0.903i)9-s + (−0.793 + 0.109i)10-s + (−0.412 + 0.238i)11-s + (0.686 + 0.726i)12-s + (1.48 + 0.857i)13-s + (0.843 + 0.343i)14-s + (−0.372 − 0.709i)15-s + (−0.877 − 0.479i)16-s + 0.897i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0190 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0190 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.0190 + 0.999i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ -0.0190 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557861 - 0.568592i\)
\(L(\frac12)\) \(\approx\) \(0.557861 - 0.568592i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.867 + 1.11i)T \)
3 \( 1 + (-0.925 + 1.46i)T \)
good5 \( 1 + (-0.895 + 1.55i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.08 - 1.20i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.36 - 0.790i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.35 - 3.09i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.69iT - 17T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
23 \( 1 + (1.36 - 2.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.55 + 4.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.95 + 3.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.24iT - 37T^{2} \)
41 \( 1 + (5.32 + 3.07i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.452 + 0.783i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.88 - 8.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.05T + 53T^{2} \)
59 \( 1 + (6.10 + 3.52i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.05 - 1.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.03 + 1.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.31T + 71T^{2} \)
73 \( 1 - 0.631T + 73T^{2} \)
79 \( 1 + (-7.82 + 4.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-13.5 + 7.82i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.16iT - 89T^{2} \)
97 \( 1 + (6.72 + 11.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.81561802632836683664132372672, −13.07494794110286469792504139554, −12.37888741818295180983508300333, −11.14609220950177076944027272382, −9.469863797576338160079024456249, −8.913116502771241013430763378463, −7.67866593927411241083481450206, −6.08306461681509906852253753695, −3.59258993065301058805722835022, −1.76701152001641069368753915369, 3.29798548591285374836194236134, 5.35831864526167550140914835563, 6.70275537341565672925403474175, 8.096560497510472378891376742566, 9.264034393539575668465836313275, 10.29881406296236357935278118390, 10.89008229767147210836481426816, 13.38822450543277078190925204356, 14.00469507057460900047444685226, 15.13586488262535530659374223330

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