↑

Properties

Label 2-72-24.11-c1-0-1
Degree $2$
Conductor $72$
Sign $0.816 + 0.577i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + 2.44·5-s − 3.46i·7-s − 2.82i·8-s + (−2.99 − 1.73i)10-s + 2.82i·11-s + 3.46i·13-s + (−2.44 + 4.24i)14-s + (−2.00 + 3.46i)16-s − 1.41i·17-s − 4·19-s + (2.44 + 4.24i)20-s + (2.00 − 3.46i)22-s − 4.89·23-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (0.499 + 0.866i)4-s + 1.09·5-s − 1.30i·7-s − 0.999i·8-s + (−0.948 − 0.547i)10-s + 0.852i·11-s + 0.960i·13-s + (−0.654 + 1.13i)14-s + (−0.500 + 0.866i)16-s − 0.342i·17-s − 0.917·19-s + (0.547 + 0.948i)20-s + (0.426 − 0.738i)22-s − 1.02·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694018 - 0.220585i\)
\(L(\frac12)\) \(\approx\) \(0.694018 - 0.220585i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 \)
good5 \( 1 - 2.44T + 5T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 1.41iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 4.89T + 23T^{2} \)
29 \( 1 + 2.44T + 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 + 7.34T + 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 + 14.1iT - 83T^{2} \)
89 \( 1 - 7.07iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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

−14.29507043209768551684943723357, −13.40338007658979646430857765147, −12.26714419918988143722027725306, −10.86031579940991717493207707704, −10.03287097392496037161348867850, −9.177057214353520543639616944260, −7.56529011520329593222975415410, −6.50039621202296958812085808262, −4.19301040613323135704279591276, −1.94090115405802890267881693810, 2.30323757022301936994344411634, 5.63436225832251089196765755857, 6.12384046882508657970223767193, 8.066099814891969572629206272589, 9.007288407900788220228689770529, 10.01952105990926015691759000204, 11.13584091830949222627430304633, 12.58823283268444817242089857831, 13.90088076020319087180230448325, 14.98288140804421262997902816228

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