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Properties

Label 2-72-8.3-c6-0-22
Degree $2$
Conductor $72$
Sign $0.683 + 0.729i$
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 + (−56.1 + 30.7i)4-s + 106. i·5-s − 614. i·7-s + (−350. − 373. i)8-s + (−826. + 212. i)10-s − 1.31e3·11-s − 626. i·13-s + (4.75e3 − 1.22e3i)14-s + (2.19e3 − 3.45e3i)16-s + 8.17e3·17-s − 9.29e3·19-s + (−3.28e3 − 5.98e3i)20-s + (−2.62e3 − 1.02e4i)22-s − 1.42e4i·23-s + ⋯
L(s)  = 1  + (0.248 + 0.968i)2-s + (−0.876 + 0.481i)4-s + 0.853i·5-s − 1.79i·7-s + (−0.683 − 0.729i)8-s + (−0.826 + 0.212i)10-s − 0.991·11-s − 0.285i·13-s + (1.73 − 0.444i)14-s + (0.536 − 0.843i)16-s + 1.66·17-s − 1.35·19-s + (−0.410 − 0.748i)20-s + (−0.246 − 0.959i)22-s − 1.17i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.683 + 0.729i$
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: \((2,\ 72,\ (\ :3),\ 0.683 + 0.729i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.899565 - 0.389745i\)
\(L(\frac12)\) \(\approx\) \(0.899565 - 0.389745i\)
\(L(4)\) 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 \( 1 - 106. iT - 1.56e4T^{2} \)
7 \( 1 + 614. iT - 1.17e5T^{2} \)
11 \( 1 + 1.31e3T + 1.77e6T^{2} \)
13 \( 1 + 626. iT - 4.82e6T^{2} \)
17 \( 1 - 8.17e3T + 2.41e7T^{2} \)
19 \( 1 + 9.29e3T + 4.70e7T^{2} \)
23 \( 1 + 1.42e4iT - 1.48e8T^{2} \)
29 \( 1 + 2.01e4iT - 5.94e8T^{2} \)
31 \( 1 + 1.01e4iT - 8.87e8T^{2} \)
37 \( 1 + 1.00e4iT - 2.56e9T^{2} \)
41 \( 1 + 4.21e4T + 4.75e9T^{2} \)
43 \( 1 + 6.85e4T + 6.32e9T^{2} \)
47 \( 1 + 3.55e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.87e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.58e5T + 4.21e10T^{2} \)
61 \( 1 + 3.63e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.87e5T + 9.04e10T^{2} \)
71 \( 1 + 3.38e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.91e5T + 1.51e11T^{2} \)
79 \( 1 + 6.30e4iT - 2.43e11T^{2} \)
83 \( 1 - 4.12e5T + 3.26e11T^{2} \)
89 \( 1 + 1.13e6T + 4.96e11T^{2} \)
97 \( 1 - 8.67e5T + 8.32e11T^{2} \)
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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

−13.57923563376084152456237361559, −12.58112446903374898980590444452, −10.67741414060259499733185098750, −10.08631919078478002425448927264, −8.107811842162301142278893254871, −7.33905464241681736869129732533, −6.25356597621350409712378425547, −4.59395864381768245028735787851, −3.27775711165982530606069456175, −0.35215990018469500552317535755, 1.64599289113910062491265343604, 3.04682129907278956131228989713, 4.97668258924745603329430314161, 5.71859132912116986082903038049, 8.281459275740121794041661618841, 9.054840892470346866549489777336, 10.20913492285009235298922462264, 11.65496961646705485635856906799, 12.43824170418375129096982602348, 13.08436131584986910665379330580

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