Properties

Label 2-72-8.3-c6-0-8
Degree $2$
Conductor $72$
Sign $0.377 - 0.926i$
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  + (−7.38 + 3.07i)2-s + (45.0 − 45.4i)4-s + 70.4i·5-s − 87.7i·7-s + (−193. + 474. i)8-s + (−216. − 520. i)10-s + 407.·11-s − 1.89e3i·13-s + (270. + 648. i)14-s + (−33.3 − 4.09e3i)16-s + 2.51e3·17-s + 437.·19-s + (3.20e3 + 3.17e3i)20-s + (−3.00e3 + 1.25e3i)22-s + 1.40e4i·23-s + ⋯
L(s)  = 1  + (−0.923 + 0.384i)2-s + (0.704 − 0.709i)4-s + 0.563i·5-s − 0.255i·7-s + (−0.377 + 0.926i)8-s + (−0.216 − 0.520i)10-s + 0.305·11-s − 0.861i·13-s + (0.0984 + 0.236i)14-s + (−0.00813 − 0.999i)16-s + 0.511·17-s + 0.0638·19-s + (0.400 + 0.396i)20-s + (−0.282 + 0.117i)22-s + 1.15i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.377 - 0.926i$
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: \((2,\ 72,\ (\ :3),\ 0.377 - 0.926i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.944437 + 0.635226i\)
\(L(\frac12)\) \(\approx\) \(0.944437 + 0.635226i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.38 - 3.07i)T \)
3 \( 1 \)
good5 \( 1 - 70.4iT - 1.56e4T^{2} \)
7 \( 1 + 87.7iT - 1.17e5T^{2} \)
11 \( 1 - 407.T + 1.77e6T^{2} \)
13 \( 1 + 1.89e3iT - 4.82e6T^{2} \)
17 \( 1 - 2.51e3T + 2.41e7T^{2} \)
19 \( 1 - 437.T + 4.70e7T^{2} \)
23 \( 1 - 1.40e4iT - 1.48e8T^{2} \)
29 \( 1 - 3.15e4iT - 5.94e8T^{2} \)
31 \( 1 - 3.84e4iT - 8.87e8T^{2} \)
37 \( 1 - 5.50e4iT - 2.56e9T^{2} \)
41 \( 1 - 4.57e4T + 4.75e9T^{2} \)
43 \( 1 - 8.99e4T + 6.32e9T^{2} \)
47 \( 1 + 3.02e4iT - 1.07e10T^{2} \)
53 \( 1 + 4.40e4iT - 2.21e10T^{2} \)
59 \( 1 + 2.74e5T + 4.21e10T^{2} \)
61 \( 1 - 2.33e5iT - 5.15e10T^{2} \)
67 \( 1 - 2.75e5T + 9.04e10T^{2} \)
71 \( 1 + 4.81e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.80e5T + 1.51e11T^{2} \)
79 \( 1 - 7.01e5iT - 2.43e11T^{2} \)
83 \( 1 + 7.40e5T + 3.26e11T^{2} \)
89 \( 1 - 1.17e6T + 4.96e11T^{2} \)
97 \( 1 + 1.05e5T + 8.32e11T^{2} \)
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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

−13.95043431187818275283107188561, −12.33467438901469405126694899188, −11.00875448888083673107011171161, −10.24497195534568820046848272682, −9.030749517405967394049439380940, −7.74582052298044206490325418228, −6.76867949808081292864723882454, −5.39408778891404339972163834213, −3.08865901187157745589391218820, −1.16315793031807131012515106333, 0.71911722945033265403846838973, 2.32069476467525288971736118632, 4.18972321896254706485498246703, 6.17634532478694084901913365089, 7.60629540363273769526037238695, 8.808980060983517084225112332111, 9.607500666365840392136378196717, 10.94177555665323085599023988103, 11.99549722306521157778612561732, 12.82946209071871718038244906314

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