↑

Properties

Label 2-72-24.11-c3-0-7
Degree $2$
Conductor $72$
Sign $0.927 - 0.373i$
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  + (2.77 + 0.536i)2-s + (7.42 + 2.97i)4-s + 5.96·5-s + 4.64i·7-s + (19.0 + 12.2i)8-s + (16.5 + 3.20i)10-s − 40.5i·11-s + 63.6i·13-s + (−2.49 + 12.9i)14-s + (46.2 + 44.2i)16-s − 93.9i·17-s − 73.6·19-s + (44.3 + 17.7i)20-s + (21.7 − 112. i)22-s − 149.·23-s + ⋯
L(s)  = 1  + (0.981 + 0.189i)2-s + (0.928 + 0.372i)4-s + 0.533·5-s + 0.250i·7-s + (0.840 + 0.541i)8-s + (0.524 + 0.101i)10-s − 1.11i·11-s + 1.35i·13-s + (−0.0475 + 0.246i)14-s + (0.722 + 0.691i)16-s − 1.34i·17-s − 0.888·19-s + (0.495 + 0.198i)20-s + (0.210 − 1.09i)22-s − 1.35·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.64880 + 0.513459i\)
\(L(\frac12)\) \(\approx\) \(2.64880 + 0.513459i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.77 - 0.536i)T \)
3 \( 1 \)
good5 \( 1 - 5.96T + 125T^{2} \)
7 \( 1 - 4.64iT - 343T^{2} \)
11 \( 1 + 40.5iT - 1.33e3T^{2} \)
13 \( 1 - 63.6iT - 2.19e3T^{2} \)
17 \( 1 + 93.9iT - 4.91e3T^{2} \)
19 \( 1 + 73.6T + 6.85e3T^{2} \)
23 \( 1 + 149.T + 1.21e4T^{2} \)
29 \( 1 - 67.3T + 2.43e4T^{2} \)
31 \( 1 + 323. iT - 2.97e4T^{2} \)
37 \( 1 - 269. iT - 5.06e4T^{2} \)
41 \( 1 - 33.8iT - 6.89e4T^{2} \)
43 \( 1 + 112.T + 7.95e4T^{2} \)
47 \( 1 + 301.T + 1.03e5T^{2} \)
53 \( 1 - 520.T + 1.48e5T^{2} \)
59 \( 1 - 257. iT - 2.05e5T^{2} \)
61 \( 1 - 445. iT - 2.26e5T^{2} \)
67 \( 1 - 643.T + 3.00e5T^{2} \)
71 \( 1 - 588.T + 3.57e5T^{2} \)
73 \( 1 - 503.T + 3.89e5T^{2} \)
79 \( 1 + 81.5iT - 4.93e5T^{2} \)
83 \( 1 - 703. iT - 5.71e5T^{2} \)
89 \( 1 + 97.3iT - 7.04e5T^{2} \)
97 \( 1 + 1.26e3T + 9.12e5T^{2} \)
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

−13.85195867268863445567729334613, −13.58026325727101792064713274473, −11.98547240639316053601460722353, −11.30184518447782179687931529142, −9.742624600188181257779485062431, −8.259691625089320640587925919654, −6.67640345966703351881359400994, −5.69118805047022592528472648414, −4.14866915001683313506875045100, −2.33238487802225518771285189923, 2.00944508351848309228086172592, 3.89096313985330228406558036417, 5.39129522609201311165828088019, 6.58176688684486357496139784472, 8.037079543897166995962962122254, 10.04990305785358238234953802148, 10.63484437289615309672499909569, 12.30857785394589154252116214451, 12.86619952422440591807816961819, 14.03924312536466172536148877079

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