Properties

Label 2-72-24.5-c2-0-1
Degree $2$
Conductor $72$
Sign $0.886 - 0.462i$
Analytic cond. $1.96185$
Root an. cond. $1.40066$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.77 + 0.921i)2-s + (2.30 − 3.27i)4-s + 1.07·5-s + 7.21·7-s + (−1.07 + 7.92i)8-s + (−1.90 + 0.990i)10-s + 16.3·11-s + 21.6i·13-s + (−12.8 + 6.64i)14-s + (−5.39 − 15.0i)16-s − 18.9i·17-s − 17.0i·19-s + (2.47 − 3.51i)20-s + (−29.0 + 15.0i)22-s − 1.11i·23-s + ⋯
L(s)  = 1  + (−0.887 + 0.460i)2-s + (0.575 − 0.817i)4-s + 0.214·5-s + 1.03·7-s + (−0.134 + 0.990i)8-s + (−0.190 + 0.0990i)10-s + 1.48·11-s + 1.66i·13-s + (−0.914 + 0.474i)14-s + (−0.337 − 0.941i)16-s − 1.11i·17-s − 0.897i·19-s + (0.123 − 0.175i)20-s + (−1.31 + 0.684i)22-s − 0.0485i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ 0.886 - 0.462i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.949131 + 0.232620i\)
\(L(\frac12)\) \(\approx\) \(0.949131 + 0.232620i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.77 - 0.921i)T \)
3 \( 1 \)
good5 \( 1 - 1.07T + 25T^{2} \)
7 \( 1 - 7.21T + 49T^{2} \)
11 \( 1 - 16.3T + 121T^{2} \)
13 \( 1 - 21.6iT - 169T^{2} \)
17 \( 1 + 18.9iT - 289T^{2} \)
19 \( 1 + 17.0iT - 361T^{2} \)
23 \( 1 + 1.11iT - 529T^{2} \)
29 \( 1 + 29.4T + 841T^{2} \)
31 \( 1 - 5.63T + 961T^{2} \)
37 \( 1 - 17.0iT - 1.36e3T^{2} \)
41 \( 1 - 27.4iT - 1.68e3T^{2} \)
43 \( 1 + 52.3iT - 1.84e3T^{2} \)
47 \( 1 - 64.5iT - 2.20e3T^{2} \)
53 \( 1 + 35.9T + 2.80e3T^{2} \)
59 \( 1 + 56.8T + 3.48e3T^{2} \)
61 \( 1 + 69.3iT - 3.72e3T^{2} \)
67 \( 1 - 69.3iT - 4.48e3T^{2} \)
71 \( 1 + 98.4iT - 5.04e3T^{2} \)
73 \( 1 - 37.6T + 5.32e3T^{2} \)
79 \( 1 + 127.T + 6.24e3T^{2} \)
83 \( 1 + 7.75T + 6.88e3T^{2} \)
89 \( 1 + 76.1iT - 7.92e3T^{2} \)
97 \( 1 - 4.84T + 9.40e3T^{2} \)
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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

−14.43051492139237569288541420549, −13.94216684839076759412123280051, −11.66866378400644276419247309939, −11.32210784688047764159412499684, −9.541908843228140794866798269016, −8.928669093586210443476644492570, −7.43691227224612107022386888193, −6.37759977511157285116304916054, −4.66953098928050755540552697235, −1.71811595376210578469323126760, 1.59190263552323676275955853787, 3.77811778538907013750361112231, 5.96115217893513013603737199003, 7.66109993310817599311707366898, 8.543200995009783118711686453401, 9.853323764428402412505197572393, 10.88392725304479128027303510678, 11.88788929158869683045588654494, 12.92386923074103123527623277714, 14.45688311605044781899601447711

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