↑

Properties

Label 2-72-24.11-c21-0-0
Degree $2$
Conductor $72$
Sign $-0.384 - 0.923i$
Analytic cond. $201.223$
Root an. cond. $14.1853$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36e3 + 478. i)2-s + (1.63e6 − 1.30e6i)4-s − 1.38e7·5-s − 1.09e9i·7-s + (−1.61e9 + 2.57e9i)8-s + (1.88e10 − 6.60e9i)10-s − 4.42e10i·11-s + 9.88e10i·13-s + (5.22e11 + 1.49e12i)14-s + (9.78e11 − 4.28e12i)16-s + 4.30e12i·17-s − 6.19e12·19-s + (−2.26e13 + 1.80e13i)20-s + (2.11e13 + 6.05e13i)22-s + 1.44e14·23-s + ⋯
L(s)  = 1  + (−0.943 + 0.330i)2-s + (0.781 − 0.623i)4-s − 0.632·5-s − 1.46i·7-s + (−0.532 + 0.846i)8-s + (0.597 − 0.208i)10-s − 0.514i·11-s + 0.198i·13-s + (0.482 + 1.37i)14-s + (0.222 − 0.974i)16-s + 0.518i·17-s − 0.231·19-s + (−0.494 + 0.394i)20-s + (0.169 + 0.485i)22-s + 0.727·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.384 - 0.923i$
Analytic conductor: \(201.223\)
Root analytic conductor: \(14.1853\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :21/2),\ -0.384 - 0.923i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.02056406791\)
\(L(\frac12)\) \(\approx\) \(0.02056406791\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36e3 - 478. i)T \)
3 \( 1 \)
good5 \( 1 + 1.38e7T + 4.76e14T^{2} \)
7 \( 1 + 1.09e9iT - 5.58e17T^{2} \)
11 \( 1 + 4.42e10iT - 7.40e21T^{2} \)
13 \( 1 - 9.88e10iT - 2.47e23T^{2} \)
17 \( 1 - 4.30e12iT - 6.90e25T^{2} \)
19 \( 1 + 6.19e12T + 7.14e26T^{2} \)
23 \( 1 - 1.44e14T + 3.94e28T^{2} \)
29 \( 1 + 1.75e14T + 5.13e30T^{2} \)
31 \( 1 + 3.72e15iT - 2.08e31T^{2} \)
37 \( 1 + 3.46e16iT - 8.55e32T^{2} \)
41 \( 1 + 1.63e16iT - 7.38e33T^{2} \)
43 \( 1 + 1.19e17T + 2.00e34T^{2} \)
47 \( 1 + 2.37e17T + 1.30e35T^{2} \)
53 \( 1 - 3.01e17T + 1.62e36T^{2} \)
59 \( 1 + 1.83e18iT - 1.54e37T^{2} \)
61 \( 1 - 1.78e18iT - 3.10e37T^{2} \)
67 \( 1 + 1.98e19T + 2.22e38T^{2} \)
71 \( 1 + 4.06e19T + 7.52e38T^{2} \)
73 \( 1 + 8.70e18T + 1.34e39T^{2} \)
79 \( 1 - 3.94e19iT - 7.08e39T^{2} \)
83 \( 1 + 5.89e19iT - 1.99e40T^{2} \)
89 \( 1 + 3.06e20iT - 8.65e40T^{2} \)
97 \( 1 - 8.97e20T + 5.27e41T^{2} \)
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

−10.86132388847769170892640132869, −10.02160760526458356722103318806, −8.777456231070154193547203639811, −7.75532249524154713911521915005, −7.08192430928741164890066569922, −5.93528957776740292308350167251, −4.39667926460702973533418718652, −3.32290696029129225600824502739, −1.75909580150700093699538524152, −0.68076613142950523384761081668, 0.00746980967300814516617844286, 1.39458874428113589378849582360, 2.46397985181050862727732812147, 3.34257488040877385429899859169, 4.89261115620107809617030952227, 6.26164745358750877098285918276, 7.41595308184708066923703852518, 8.431531706700713644663151347628, 9.192245091056178340081448671465, 10.26403454062706405550453944943

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