↑

Properties

Label 2-72-24.11-c21-0-10
Degree $2$
Conductor $72$
Sign $-0.892 + 0.451i$
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  + (−917. + 1.12e3i)2-s + (−4.14e5 − 2.05e6i)4-s − 3.53e7·5-s − 2.13e8i·7-s + (2.68e9 + 1.42e9i)8-s + (3.24e10 − 3.96e10i)10-s + 6.53e10i·11-s + 8.18e11i·13-s + (2.39e11 + 1.95e11i)14-s + (−4.05e12 + 1.70e12i)16-s − 8.70e12i·17-s − 6.82e12·19-s + (1.46e13 + 7.26e13i)20-s + (−7.32e13 − 5.99e13i)22-s − 3.72e14·23-s + ⋯
L(s)  = 1  + (−0.633 + 0.773i)2-s + (−0.197 − 0.980i)4-s − 1.61·5-s − 0.285i·7-s + (0.883 + 0.467i)8-s + (1.02 − 1.25i)10-s + 0.760i·11-s + 1.64i·13-s + (0.221 + 0.180i)14-s + (−0.921 + 0.387i)16-s − 1.04i·17-s − 0.255·19-s + (0.319 + 1.58i)20-s + (−0.588 − 0.481i)22-s − 1.87·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.892 + 0.451i$
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: \((2,\ 72,\ (\ :21/2),\ -0.892 + 0.451i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.3944407957\)
\(L(\frac12)\) \(\approx\) \(0.3944407957\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (917. - 1.12e3i)T \)
3 \( 1 \)
good5 \( 1 + 3.53e7T + 4.76e14T^{2} \)
7 \( 1 + 2.13e8iT - 5.58e17T^{2} \)
11 \( 1 - 6.53e10iT - 7.40e21T^{2} \)
13 \( 1 - 8.18e11iT - 2.47e23T^{2} \)
17 \( 1 + 8.70e12iT - 6.90e25T^{2} \)
19 \( 1 + 6.82e12T + 7.14e26T^{2} \)
23 \( 1 + 3.72e14T + 3.94e28T^{2} \)
29 \( 1 + 2.96e14T + 5.13e30T^{2} \)
31 \( 1 + 7.68e15iT - 2.08e31T^{2} \)
37 \( 1 - 5.18e16iT - 8.55e32T^{2} \)
41 \( 1 - 1.54e17iT - 7.38e33T^{2} \)
43 \( 1 - 2.16e17T + 2.00e34T^{2} \)
47 \( 1 - 4.92e17T + 1.30e35T^{2} \)
53 \( 1 - 1.02e17T + 1.62e36T^{2} \)
59 \( 1 - 2.54e18iT - 1.54e37T^{2} \)
61 \( 1 - 2.77e18iT - 3.10e37T^{2} \)
67 \( 1 - 8.26e18T + 2.22e38T^{2} \)
71 \( 1 + 2.50e19T + 7.52e38T^{2} \)
73 \( 1 - 4.37e19T + 1.34e39T^{2} \)
79 \( 1 + 7.98e19iT - 7.08e39T^{2} \)
83 \( 1 + 3.14e19iT - 1.99e40T^{2} \)
89 \( 1 + 1.80e20iT - 8.65e40T^{2} \)
97 \( 1 - 1.12e21T + 5.27e41T^{2} \)
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

−11.40550496025723700381294050747, −10.01775905900320972745334301356, −9.032440927469484207472593345323, −7.84532513761910871948042921301, −7.31553879637522227126695651086, −6.26650593612124771485484408580, −4.52333971091695750006304215844, −4.13729758884869308450970666053, −2.22272835456917784944747984392, −0.864431503708121457774057043192, 0.16212132049370155010803442693, 0.76268871877369297018874323976, 2.30732519128522210015645840234, 3.52771047996915662819020627270, 3.99220167371002400497762114837, 5.66201300141816486670947040132, 7.40060929064944840118898341655, 8.142013889207015416055846309049, 8.792901214657515857609547241947, 10.49144987147309275015412760937

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