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Properties

Label 2-72-72.11-c3-0-26
Degree $2$
Conductor $72$
Sign $0.225 + 0.974i$
Analytic cond. $4.24813$
Root an. cond. $2.06110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.42 − 1.45i)2-s + (1.76 − 4.88i)3-s + (3.78 − 7.04i)4-s + (4.95 + 8.58i)5-s + (−2.82 − 14.4i)6-s + (11.4 + 6.61i)7-s + (−1.04 − 22.6i)8-s + (−20.7 − 17.2i)9-s + (24.5 + 13.6i)10-s + (−23.4 − 13.5i)11-s + (−27.7 − 30.9i)12-s + (−41.1 + 23.7i)13-s + (37.3 − 0.577i)14-s + (50.7 − 9.11i)15-s + (−35.3 − 53.3i)16-s + 88.9i·17-s + ⋯
L(s)  = 1  + (0.858 − 0.513i)2-s + (0.338 − 0.940i)3-s + (0.473 − 0.881i)4-s + (0.443 + 0.768i)5-s + (−0.192 − 0.981i)6-s + (0.618 + 0.356i)7-s + (−0.0463 − 0.998i)8-s + (−0.770 − 0.637i)9-s + (0.774 + 0.431i)10-s + (−0.642 − 0.370i)11-s + (−0.668 − 0.743i)12-s + (−0.878 + 0.507i)13-s + (0.713 − 0.0110i)14-s + (0.873 − 0.156i)15-s + (−0.552 − 0.833i)16-s + 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.225 + 0.974i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ 0.225 + 0.974i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.07104 - 1.64656i\)
\(L(\frac12)\) \(\approx\) \(2.07104 - 1.64656i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.42 + 1.45i)T \)
3 \( 1 + (-1.76 + 4.88i)T \)
good5 \( 1 + (-4.95 - 8.58i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-11.4 - 6.61i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (23.4 + 13.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (41.1 - 23.7i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 88.9iT - 4.91e3T^{2} \)
19 \( 1 - 88.1T + 6.85e3T^{2} \)
23 \( 1 + (-89.5 - 155. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-47.7 + 82.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (23.6 - 13.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 117. iT - 5.06e4T^{2} \)
41 \( 1 + (258. - 149. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-151. + 262. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (254. - 440. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 415.T + 1.48e5T^{2} \)
59 \( 1 + (-361. + 208. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (555. + 320. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (210. + 364. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 655.T + 3.57e5T^{2} \)
73 \( 1 - 1.02e3T + 3.89e5T^{2} \)
79 \( 1 + (503. + 290. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (437. + 252. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 407. iT - 7.04e5T^{2} \)
97 \( 1 + (479. - 830. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.91247194444128297858659413744, −12.90987264495768779404400449973, −11.84573556165835793106627463306, −10.91680415587245305741505595949, −9.535476781532531355416343115649, −7.78856744671266819586010299266, −6.53564464127418003177959459776, −5.32647663256936260067794702601, −3.12058678890598239199411571828, −1.83287631348350167287001722023, 2.84776784243613069750441332941, 4.83056697294667807353039183663, 5.14933499592163271503703590476, 7.32569628652441482539556427190, 8.507612595387772481942524930114, 9.800168083618391960895809436879, 11.12831936082991839770580908140, 12.42635470097930364821674624754, 13.54889191088466792645527972223, 14.41182144736593139911086275521

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