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Properties

Label 2-72-24.11-c21-0-14
Degree $2$
Conductor $72$
Sign $0.995 - 0.0954i$
Analytic cond. $201.223$
Root an. cond. $14.1853$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.04e3 − 998. i)2-s + (1.03e5 − 2.09e6i)4-s − 2.17e7·5-s + 8.90e8i·7-s + (−1.98e9 − 2.30e9i)8-s + (−2.28e10 + 2.17e10i)10-s − 3.82e10i·11-s − 4.83e11i·13-s + (8.89e11 + 9.34e11i)14-s + (−4.37e12 − 4.35e11i)16-s − 2.21e12i·17-s + 1.16e13·19-s + (−2.26e12 + 4.56e13i)20-s + (−3.81e13 − 4.01e13i)22-s − 8.76e13·23-s + ⋯
L(s)  = 1  + (0.724 − 0.689i)2-s + (0.0495 − 0.998i)4-s − 0.997·5-s + 1.19i·7-s + (−0.652 − 0.757i)8-s + (−0.722 + 0.687i)10-s − 0.444i·11-s − 0.973i·13-s + (0.821 + 0.863i)14-s + (−0.995 − 0.0989i)16-s − 0.266i·17-s + 0.437·19-s + (−0.0494 + 0.996i)20-s + (−0.306 − 0.322i)22-s − 0.441·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0954i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.995 - 0.0954i$
Analytic conductor: \(201.223\)
Root analytic conductor: \(14.1853\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :21/2),\ 0.995 - 0.0954i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.330465682\)
\(L(\frac12)\) \(\approx\) \(1.330465682\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04e3 + 998. i)T \)
3 \( 1 \)
good5 \( 1 + 2.17e7T + 4.76e14T^{2} \)
7 \( 1 - 8.90e8iT - 5.58e17T^{2} \)
11 \( 1 + 3.82e10iT - 7.40e21T^{2} \)
13 \( 1 + 4.83e11iT - 2.47e23T^{2} \)
17 \( 1 + 2.21e12iT - 6.90e25T^{2} \)
19 \( 1 - 1.16e13T + 7.14e26T^{2} \)
23 \( 1 + 8.76e13T + 3.94e28T^{2} \)
29 \( 1 + 4.27e15T + 5.13e30T^{2} \)
31 \( 1 + 2.49e15iT - 2.08e31T^{2} \)
37 \( 1 - 5.91e15iT - 8.55e32T^{2} \)
41 \( 1 + 2.79e16iT - 7.38e33T^{2} \)
43 \( 1 + 5.17e16T + 2.00e34T^{2} \)
47 \( 1 + 2.93e17T + 1.30e35T^{2} \)
53 \( 1 - 7.56e17T + 1.62e36T^{2} \)
59 \( 1 - 4.55e18iT - 1.54e37T^{2} \)
61 \( 1 - 6.83e18iT - 3.10e37T^{2} \)
67 \( 1 - 2.07e18T + 2.22e38T^{2} \)
71 \( 1 - 1.22e19T + 7.52e38T^{2} \)
73 \( 1 - 6.26e19T + 1.34e39T^{2} \)
79 \( 1 + 7.16e19iT - 7.08e39T^{2} \)
83 \( 1 - 1.85e20iT - 1.99e40T^{2} \)
89 \( 1 + 2.85e20iT - 8.65e40T^{2} \)
97 \( 1 - 9.52e20T + 5.27e41T^{2} \)
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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

−11.12847164425121549593053779701, −9.818126523608143507876308586044, −8.692985269398946989126882735249, −7.51736197938798267386626090992, −5.94802382978234315637921591657, −5.24838090673849137030633628542, −3.91475075389051198134250728333, −3.08701296626966249735618284704, −2.06371054265800907205295670309, −0.66986538925932524199823402048, 0.26286436115617450914016361115, 1.86746930431321451097965177986, 3.57238576856829714330004653529, 4.02344260871801965066344322327, 5.05273416823503663327598926736, 6.56064677481700722020687050934, 7.36801206459552562096338630649, 8.072089025533670839070687856370, 9.487724272043478625422285316795, 11.02595348193477954181880455018

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