↑

Properties

Label 2-72-72.11-c3-0-1
Degree $2$
Conductor $72$
Sign $-0.861 + 0.507i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.250 + 2.81i)2-s + (−3.80 + 3.53i)3-s + (−7.87 − 1.41i)4-s + (10.8 + 18.8i)5-s + (−9.00 − 11.6i)6-s + (−10.5 − 6.09i)7-s + (5.94 − 21.8i)8-s + (1.99 − 26.9i)9-s + (−55.7 + 25.8i)10-s + (−22.8 − 13.1i)11-s + (34.9 − 22.4i)12-s + (−24.0 + 13.8i)13-s + (19.8 − 28.2i)14-s + (−107. − 33.2i)15-s + (60.0 + 22.2i)16-s + 56.3i·17-s + ⋯
L(s)  = 1  + (−0.0885 + 0.996i)2-s + (−0.732 + 0.680i)3-s + (−0.984 − 0.176i)4-s + (0.971 + 1.68i)5-s + (−0.612 − 0.790i)6-s + (−0.570 − 0.329i)7-s + (0.262 − 0.964i)8-s + (0.0739 − 0.997i)9-s + (−1.76 + 0.819i)10-s + (−0.625 − 0.360i)11-s + (0.841 − 0.540i)12-s + (−0.512 + 0.295i)13-s + (0.378 − 0.538i)14-s + (−1.85 − 0.572i)15-s + (0.937 + 0.347i)16-s + 0.804i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.861 + 0.507i$
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: \((2,\ 72,\ (\ :3/2),\ -0.861 + 0.507i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.210360 - 0.770911i\)
\(L(\frac12)\) \(\approx\) \(0.210360 - 0.770911i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.250 - 2.81i)T \)
3 \( 1 + (3.80 - 3.53i)T \)
good5 \( 1 + (-10.8 - 18.8i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (10.5 + 6.09i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (22.8 + 13.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (24.0 - 13.8i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 56.3iT - 4.91e3T^{2} \)
19 \( 1 - 80.9T + 6.85e3T^{2} \)
23 \( 1 + (4.32 + 7.49i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (43.8 - 75.8i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (174. - 100. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 53.3iT - 5.06e4T^{2} \)
41 \( 1 + (-398. + 229. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (105. - 181. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (168. - 292. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 84.2T + 1.48e5T^{2} \)
59 \( 1 + (86.9 - 50.2i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-342. - 197. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-68.5 - 118. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 766.T + 3.57e5T^{2} \)
73 \( 1 - 464.T + 3.89e5T^{2} \)
79 \( 1 + (906. + 523. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-1.13e3 - 654. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 403. iT - 7.04e5T^{2} \)
97 \( 1 + (29.9 - 51.9i)T + (-4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.79150291439974619905043387110, −14.11253224151836345373487404946, −12.89826294614310209684176852068, −10.99509344397618257737659609293, −10.17502575361191576278567203865, −9.415210421050602709405628343449, −7.30064237414090203654681257455, −6.37039881749866421680548495470, −5.43873623785836356670440934761, −3.45703226123120746340417404636, 0.58498524360196863225773825356, 2.17399078352381050768041401999, 4.91299536642407423504543157094, 5.65083235934047549341546772076, 7.83150958437451100260480394218, 9.272535661579276706896907433138, 9.978995065653751104933496912568, 11.57865035751800894546295563496, 12.59478911738940310739243288515, 12.98598387135374826671233218014

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