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Properties

Label 2-72-72.59-c1-0-5
Degree $2$
Conductor $72$
Sign $0.744 - 0.667i$
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  + (0.533 + 1.30i)2-s + (0.925 − 1.46i)3-s + (−1.43 + 1.39i)4-s + (−0.895 + 1.55i)5-s + (2.41 + 0.431i)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 + (0.721 + 3.38i)12-s + (−5.35 − 3.09i)13-s + (2.69 + 2.09i)14-s + (1.44 + 2.74i)15-s + (0.0944 − 3.99i)16-s + 3.69i·17-s + ⋯
L(s)  = 1  + (0.377 + 0.926i)2-s + (0.534 − 0.845i)3-s + (−0.715 + 0.698i)4-s + (−0.400 + 0.693i)5-s + (0.984 + 0.176i)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 + (0.208 + 0.978i)12-s + (−1.48 − 0.857i)13-s + (0.719 + 0.558i)14-s + (0.372 + 0.709i)15-s + (0.0236 − 0.999i)16-s + 0.897i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.744 - 0.667i$
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: \((2,\ 72,\ (\ :1/2),\ 0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04584 + 0.400197i\)
\(L(\frac12)\) \(\approx\) \(1.04584 + 0.400197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.533 - 1.30i)T \)
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

−14.78485085842360707182257363971, −13.96116810120142225973402370246, −12.82326970881920507172996458741, −11.89663371890698023864312286290, −10.20835609643184969609770170189, −8.431183285862719658741090699993, −7.59236814073773039267490019293, −6.81654025325012612689010375168, −5.02600643192640580898232083022, −3.12429892296544354208659903568, 2.59902006919522766563605002911, 4.48390000971309823116307073552, 5.16637401116567275598190997053, 7.966935798586887028346955693476, 9.127084900552053979621382297883, 9.959902827096572999638842151323, 11.43609410781781962833545215933, 12.04018112215033660531766578266, 13.54125094780331610123203966842, 14.41587201802865461071811790578

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