↑

Properties

Label 2-72-24.11-c1-0-1
Degree 22
Conductor 7272
Sign 0.816+0.577i0.816 + 0.577i
Analytic cond. 0.5749220.574922
Root an. cond. 0.7582360.758236
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(72s/2 ΓC(s) L(s)=((0.816+0.577i) Λ‾(2−s)\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}
Λ(s)=(72s/2 ΓC(s+1/2) L(s)=((0.816+0.577i) Λ‾(1−s)\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: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.816+0.577i0.816 + 0.577i
Analytic conductor: 0.5749220.574922
Root analytic conductor: 0.7582360.758236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ72(35,⋅)\chi_{72} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :1/2), 0.816+0.577i)(2,\ 72,\ (\ :1/2),\ 0.816 + 0.577i)

Particular Values

L(1)L(1) ≈\approx 0.694018−0.220585i0.694018 - 0.220585i
L(12)L(\frac12) ≈\approx 0.694018−0.220585i0.694018 - 0.220585i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.22+0.707i)T 1 + (1.22 + 0.707i)T
3 1 1
good5 1−2.44T+5T2 1 - 2.44T + 5T^{2}
7 1+3.46iT−7T2 1 + 3.46iT - 7T^{2}
11 1−2.82iT−11T2 1 - 2.82iT - 11T^{2}
13 1−3.46iT−13T2 1 - 3.46iT - 13T^{2}
17 1+1.41iT−17T2 1 + 1.41iT - 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+4.89T+23T2 1 + 4.89T + 23T^{2}
29 1+2.44T+29T2 1 + 2.44T + 29T^{2}
31 1−3.46iT−31T2 1 - 3.46iT - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1+1.41iT−41T2 1 + 1.41iT - 41T^{2}
43 1−8T+43T2 1 - 8T + 43T^{2}
47 1−4.89T+47T2 1 - 4.89T + 47T^{2}
53 1+7.34T+53T2 1 + 7.34T + 53T^{2}
59 1−11.3iT−59T2 1 - 11.3iT - 59T^{2}
61 1+13.8iT−61T2 1 + 13.8iT - 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1−14.6T+71T2 1 - 14.6T + 71T^{2}
73 1+4T+73T2 1 + 4T + 73T^{2}
79 1+3.46iT−79T2 1 + 3.46iT - 79T^{2}
83 1+14.1iT−83T2 1 + 14.1iT - 83T^{2}
89 1−7.07iT−89T2 1 - 7.07iT - 89T^{2}
97 1−8T+97T2 1 - 8T + 97T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line