Properties

Label 2-72-9.2-c6-0-0
Degree $2$
Conductor $72$
Sign $-0.819 + 0.573i$
Analytic cond. $16.5638$
Root an. cond. $4.06987$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16.4 + 21.4i)3-s + (−124. + 71.9i)5-s + (53.2 − 92.3i)7-s + (−188. + 704. i)9-s + (−1.73e3 − 1.00e3i)11-s + (48.1 + 83.3i)13-s + (−3.59e3 − 1.48e3i)15-s − 4.37e3i·17-s + 2.32e3·19-s + (2.85e3 − 375. i)21-s + (−1.35e3 + 782. i)23-s + (2.55e3 − 4.41e3i)25-s + (−1.81e4 + 7.53e3i)27-s + (−2.88e4 − 1.66e4i)29-s + (−1.52e4 − 2.63e4i)31-s + ⋯
L(s)  = 1  + (0.608 + 0.793i)3-s + (−0.997 + 0.575i)5-s + (0.155 − 0.269i)7-s + (−0.258 + 0.965i)9-s + (−1.30 − 0.754i)11-s + (0.0219 + 0.0379i)13-s + (−1.06 − 0.440i)15-s − 0.889i·17-s + 0.339·19-s + (0.308 − 0.0405i)21-s + (−0.111 + 0.0642i)23-s + (0.163 − 0.282i)25-s + (−0.923 + 0.382i)27-s + (−1.18 − 0.682i)29-s + (−0.510 − 0.884i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.819 + 0.573i$
Analytic conductor: \(16.5638\)
Root analytic conductor: \(4.06987\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3),\ -0.819 + 0.573i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0555834 - 0.176275i\)
\(L(\frac12)\) \(\approx\) \(0.0555834 - 0.176275i\)
\(L(4)\) 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 \( 1 + (-16.4 - 21.4i)T \)
good5 \( 1 + (124. - 71.9i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-53.2 + 92.3i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (1.73e3 + 1.00e3i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (-48.1 - 83.3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + 4.37e3iT - 2.41e7T^{2} \)
19 \( 1 - 2.32e3T + 4.70e7T^{2} \)
23 \( 1 + (1.35e3 - 782. i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (2.88e4 + 1.66e4i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (1.52e4 + 2.63e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + 4.13e4T + 2.56e9T^{2} \)
41 \( 1 + (8.07e4 - 4.66e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (3.38e4 - 5.86e4i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-1.41e5 - 8.14e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + 1.56e5iT - 2.21e10T^{2} \)
59 \( 1 + (3.20e5 - 1.85e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-1.34e5 + 2.32e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-2.66e5 - 4.61e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 2.79e5iT - 1.28e11T^{2} \)
73 \( 1 - 5.06e5T + 1.51e11T^{2} \)
79 \( 1 + (1.81e5 - 3.14e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (1.98e5 + 1.14e5i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 - 8.22e5iT - 4.96e11T^{2} \)
97 \( 1 + (-3.70e5 + 6.40e5i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−14.16557307101801923725465791044, −13.24322066984620607524265663817, −11.48088607596406073531975157650, −10.81902316042233297527149688454, −9.609956758204010835890705860720, −8.173464396020368553502417839667, −7.43351499487820528957593771013, −5.33248597158587673576797774494, −3.87674214501107492751650464609, −2.76476574420295976340079606864, 0.06257260786511931969484462089, 1.90734522063049814626275283416, 3.61306593474821219211299659664, 5.28063640685597008783217732252, 7.15079445342071158132400255691, 8.028697594134739239932948919690, 8.921589146838439265806326829636, 10.55896215093510665218599324135, 12.10635565057810858525674994552, 12.59679987195757259326584480531

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