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Properties

Label 2-72-8.3-c6-0-12
Degree $2$
Conductor $72$
Sign $0.684 + 0.728i$
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  + (−1.98 − 7.74i)2-s + (−56.0 + 30.8i)4-s − 82.3i·5-s + 351. i·7-s + (350. + 373. i)8-s + (−637. + 163. i)10-s + 704.·11-s + 713. i·13-s + (2.72e3 − 699. i)14-s + (2.19e3 − 3.45e3i)16-s − 2.68e3·17-s + 1.31e4·19-s + (2.53e3 + 4.61e3i)20-s + (−1.40e3 − 5.45e3i)22-s + 6.56e3i·23-s + ⋯
L(s)  = 1  + (−0.248 − 0.968i)2-s + (−0.876 + 0.481i)4-s − 0.658i·5-s + 1.02i·7-s + (0.684 + 0.728i)8-s + (−0.637 + 0.163i)10-s + 0.528·11-s + 0.324i·13-s + (0.992 − 0.254i)14-s + (0.535 − 0.844i)16-s − 0.546·17-s + 1.91·19-s + (0.317 + 0.576i)20-s + (−0.131 − 0.512i)22-s + 0.539i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.684 + 0.728i$
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: \((2,\ 72,\ (\ :3),\ 0.684 + 0.728i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.38741 - 0.600396i\)
\(L(\frac12)\) \(\approx\) \(1.38741 - 0.600396i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.98 + 7.74i)T \)
3 \( 1 \)
good5 \( 1 + 82.3iT - 1.56e4T^{2} \)
7 \( 1 - 351. iT - 1.17e5T^{2} \)
11 \( 1 - 704.T + 1.77e6T^{2} \)
13 \( 1 - 713. iT - 4.82e6T^{2} \)
17 \( 1 + 2.68e3T + 2.41e7T^{2} \)
19 \( 1 - 1.31e4T + 4.70e7T^{2} \)
23 \( 1 - 6.56e3iT - 1.48e8T^{2} \)
29 \( 1 + 3.24e4iT - 5.94e8T^{2} \)
31 \( 1 + 5.25e4iT - 8.87e8T^{2} \)
37 \( 1 - 5.00e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.22e5T + 4.75e9T^{2} \)
43 \( 1 - 2.72e3T + 6.32e9T^{2} \)
47 \( 1 - 1.41e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.62e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.53e5T + 4.21e10T^{2} \)
61 \( 1 - 3.16e5iT - 5.15e10T^{2} \)
67 \( 1 - 2.55e5T + 9.04e10T^{2} \)
71 \( 1 + 2.49e5iT - 1.28e11T^{2} \)
73 \( 1 + 8.98e3T + 1.51e11T^{2} \)
79 \( 1 - 2.68e5iT - 2.43e11T^{2} \)
83 \( 1 - 4.10e4T + 3.26e11T^{2} \)
89 \( 1 + 6.40e5T + 4.96e11T^{2} \)
97 \( 1 - 1.70e5T + 8.32e11T^{2} \)
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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

−13.03837674058404058248799311649, −11.95132309232764150899077017159, −11.38130250889480080726089447874, −9.611588492662242972709786736920, −9.084126117402874216564295867029, −7.79249176600878097028219917112, −5.68149300039124079902516214163, −4.30695394595562218380301362035, −2.60913283151080297600609507179, −1.04168838339747863639587709650, 0.902723895598727458956116191037, 3.57866122902268270573757087884, 5.13761382180475097850527011400, 6.74702092942556664081228873228, 7.37992088874243244888480396949, 8.845686339138529650208568870487, 10.08691201305755011373210901408, 11.01706597205714827298481697159, 12.75520358698760087497773967162, 14.08283001407804724973122236772

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