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Properties

Label 2-72-8.3-c6-0-23
Degree $2$
Conductor $72$
Sign $-0.970 + 0.241i$
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  + (−7.97 + 0.648i)2-s + (63.1 − 10.3i)4-s − 232. i·5-s − 483. i·7-s + (−496. + 123. i)8-s + (150. + 1.85e3i)10-s + 538.·11-s − 764. i·13-s + (313. + 3.85e3i)14-s + (3.88e3 − 1.30e3i)16-s + 4.27e3·17-s − 5.00e3·19-s + (−2.40e3 − 1.46e4i)20-s + (−4.29e3 + 349. i)22-s + 1.26e4i·23-s + ⋯
L(s)  = 1  + (−0.996 + 0.0811i)2-s + (0.986 − 0.161i)4-s − 1.85i·5-s − 1.41i·7-s + (−0.970 + 0.241i)8-s + (0.150 + 1.85i)10-s + 0.404·11-s − 0.347i·13-s + (0.114 + 1.40i)14-s + (0.947 − 0.319i)16-s + 0.869·17-s − 0.729·19-s + (−0.300 − 1.83i)20-s + (−0.403 + 0.0328i)22-s + 1.03i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.970 + 0.241i$
Analytic conductor: \(16.5638\)
Root analytic conductor: \(4.06987\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3),\ -0.970 + 0.241i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.107687 - 0.879704i\)
\(L(\frac12)\) \(\approx\) \(0.107687 - 0.879704i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.97 - 0.648i)T \)
3 \( 1 \)
good5 \( 1 + 232. iT - 1.56e4T^{2} \)
7 \( 1 + 483. iT - 1.17e5T^{2} \)
11 \( 1 - 538.T + 1.77e6T^{2} \)
13 \( 1 + 764. iT - 4.82e6T^{2} \)
17 \( 1 - 4.27e3T + 2.41e7T^{2} \)
19 \( 1 + 5.00e3T + 4.70e7T^{2} \)
23 \( 1 - 1.26e4iT - 1.48e8T^{2} \)
29 \( 1 + 2.61e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.63e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.60e4iT - 2.56e9T^{2} \)
41 \( 1 + 5.31e4T + 4.75e9T^{2} \)
43 \( 1 - 8.43e4T + 6.32e9T^{2} \)
47 \( 1 + 1.15e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.58e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.20e3T + 4.21e10T^{2} \)
61 \( 1 - 3.04e5iT - 5.15e10T^{2} \)
67 \( 1 - 1.44e5T + 9.04e10T^{2} \)
71 \( 1 - 1.59e4iT - 1.28e11T^{2} \)
73 \( 1 + 2.61e5T + 1.51e11T^{2} \)
79 \( 1 - 3.37e5iT - 2.43e11T^{2} \)
83 \( 1 - 4.61e5T + 3.26e11T^{2} \)
89 \( 1 + 1.14e6T + 4.96e11T^{2} \)
97 \( 1 + 2.88e5T + 8.32e11T^{2} \)
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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

−12.76870194633266625178069175306, −11.74650070809727552388509472583, −10.36568627098105793830741294463, −9.395297092676980439942762250967, −8.321815810853824247186375399688, −7.36350360092429538456347523907, −5.63650017597487428397291421402, −4.03590958820244738903771088148, −1.40817051733691794487627142079, −0.49505992837715682039693944878, 2.13642083354024229068561681618, 3.16737707720546097418188611720, 6.04460088481896150577848054435, 6.86142086350364594148814950138, 8.219898550471365512500174178843, 9.466747476257107874607198012568, 10.53670539345197609931279183318, 11.44478399558087286828875693106, 12.37207495446262684632891269742, 14.45836227162048886975773777442

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