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Properties

Label 2-72-8.3-c6-0-9
Degree $2$
Conductor $72$
Sign $0.104 - 0.994i$
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  + (−0.278 + 7.99i)2-s + (−63.8 − 4.44i)4-s − 111. i·5-s + 106. i·7-s + (53.3 − 509. i)8-s + (890. + 30.9i)10-s + 948.·11-s + 2.79e3i·13-s + (−854. − 29.7i)14-s + (4.05e3 + 567. i)16-s + 8.80e3·17-s − 5.26e3·19-s + (−495. + 7.11e3i)20-s + (−263. + 7.58e3i)22-s + 1.41e4i·23-s + ⋯
L(s)  = 1  + (−0.0347 + 0.999i)2-s + (−0.997 − 0.0695i)4-s − 0.891i·5-s + 0.311i·7-s + (0.104 − 0.994i)8-s + (0.890 + 0.0309i)10-s + 0.712·11-s + 1.27i·13-s + (−0.311 − 0.0108i)14-s + (0.990 + 0.138i)16-s + 1.79·17-s − 0.767·19-s + (−0.0619 + 0.889i)20-s + (−0.0247 + 0.712i)22-s + 1.16i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.104 - 0.994i$
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: \((2,\ 72,\ (\ :3),\ 0.104 - 0.994i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.21849 + 1.09756i\)
\(L(\frac12)\) \(\approx\) \(1.21849 + 1.09756i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.278 - 7.99i)T \)
3 \( 1 \)
good5 \( 1 + 111. iT - 1.56e4T^{2} \)
7 \( 1 - 106. iT - 1.17e5T^{2} \)
11 \( 1 - 948.T + 1.77e6T^{2} \)
13 \( 1 - 2.79e3iT - 4.82e6T^{2} \)
17 \( 1 - 8.80e3T + 2.41e7T^{2} \)
19 \( 1 + 5.26e3T + 4.70e7T^{2} \)
23 \( 1 - 1.41e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.61e4iT - 5.94e8T^{2} \)
31 \( 1 + 893. iT - 8.87e8T^{2} \)
37 \( 1 - 7.63e4iT - 2.56e9T^{2} \)
41 \( 1 - 8.44e3T + 4.75e9T^{2} \)
43 \( 1 - 9.37e4T + 6.32e9T^{2} \)
47 \( 1 - 1.28e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.70e5iT - 2.21e10T^{2} \)
59 \( 1 - 3.65e5T + 4.21e10T^{2} \)
61 \( 1 + 3.64e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.74e3T + 9.04e10T^{2} \)
71 \( 1 + 3.69e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.05e5T + 1.51e11T^{2} \)
79 \( 1 - 9.28e5iT - 2.43e11T^{2} \)
83 \( 1 + 2.28e5T + 3.26e11T^{2} \)
89 \( 1 - 3.05e5T + 4.96e11T^{2} \)
97 \( 1 - 5.74e5T + 8.32e11T^{2} \)
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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

−13.91077145195391061328835920747, −12.71198241445300141700463918552, −11.75483504333809099739810123647, −9.773346774757214853026259978427, −8.978015370176023813919603688547, −7.87786963320678476979375800191, −6.47264398350360014097266583842, −5.24436129541186614314242451788, −3.97464334115430182262314786938, −1.18137276169043004175064636434, 0.832356190441951942615864303010, 2.71705104945267512417186135934, 3.89292516121728435083122443632, 5.67132486329863458740360103321, 7.39294418087887520298322559711, 8.745025386813847825159724709975, 10.25490805087088310865391496861, 10.67174799580206025139601901806, 12.05281032218095607070469385209, 12.91253836606072854487188659063

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