↑

Properties

Label 2-1224-8.5-c1-0-22
Degree 22
Conductor 12241224
Sign 0.676+0.736i0.676 + 0.736i
Analytic cond. 9.773689.77368
Root an. cond. 3.126293.12629
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  + (−0.346 − 1.37i)2-s + (−1.75 + 0.951i)4-s − 1.24i·5-s − 3.59·7-s + (1.91 + 2.08i)8-s + (−1.71 + 0.432i)10-s + 1.76i·11-s + 1.45i·13-s + (1.24 + 4.93i)14-s + (2.19 − 3.34i)16-s + 17-s + 3.75i·19-s + (1.18 + 2.19i)20-s + (2.42 − 0.612i)22-s + 4.84·23-s + ⋯
L(s)  = 1  + (−0.245 − 0.969i)2-s + (−0.879 + 0.475i)4-s − 0.558i·5-s − 1.35·7-s + (0.676 + 0.736i)8-s + (−0.541 + 0.136i)10-s + 0.532i·11-s + 0.403i·13-s + (0.333 + 1.31i)14-s + (0.547 − 0.836i)16-s + 0.242·17-s + 0.860i·19-s + (0.265 + 0.490i)20-s + (0.516 − 0.130i)22-s + 1.01·23-s + ⋯

Functional equation

Λ(s)=(1224s/2 ΓC(s) L(s)=((0.676+0.736i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1224s/2 ΓC(s+1/2) L(s)=((0.676+0.736i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12241224    =    23⋅32⋅172^{3} \cdot 3^{2} \cdot 17
Sign: 0.676+0.736i0.676 + 0.736i
Analytic conductor: 9.773689.77368
Root analytic conductor: 3.126293.12629
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1224(613,⋅)\chi_{1224} (613, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1224, ( :1/2), 0.676+0.736i)(2,\ 1224,\ (\ :1/2),\ 0.676 + 0.736i)

Particular Values

L(1)L(1) ≈\approx 1.0244172581.024417258
L(12)L(\frac12) ≈\approx 1.0244172581.024417258
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+(0.346+1.37i)T 1 + (0.346 + 1.37i)T
3 1 1
17 1−T 1 - T
good5 1+1.24iT−5T2 1 + 1.24iT - 5T^{2}
7 1+3.59T+7T2 1 + 3.59T + 7T^{2}
11 1−1.76iT−11T2 1 - 1.76iT - 11T^{2}
13 1−1.45iT−13T2 1 - 1.45iT - 13T^{2}
19 1−3.75iT−19T2 1 - 3.75iT - 19T^{2}
23 1−4.84T+23T2 1 - 4.84T + 23T^{2}
29 1+7.54iT−29T2 1 + 7.54iT - 29T^{2}
31 1+1.06T+31T2 1 + 1.06T + 31T^{2}
37 1−4.97iT−37T2 1 - 4.97iT - 37T^{2}
41 1−6.51T+41T2 1 - 6.51T + 41T^{2}
43 1+0.946iT−43T2 1 + 0.946iT - 43T^{2}
47 1−3.46T+47T2 1 - 3.46T + 47T^{2}
53 1+6.03iT−53T2 1 + 6.03iT - 53T^{2}
59 1−6.30iT−59T2 1 - 6.30iT - 59T^{2}
61 1+4.77iT−61T2 1 + 4.77iT - 61T^{2}
67 1−11.6iT−67T2 1 - 11.6iT - 67T^{2}
71 1−8.69T+71T2 1 - 8.69T + 71T^{2}
73 1−14.6T+73T2 1 - 14.6T + 73T^{2}
79 1−10.8T+79T2 1 - 10.8T + 79T^{2}
83 1+11.5iT−83T2 1 + 11.5iT - 83T^{2}
89 1+7.56T+89T2 1 + 7.56T + 89T^{2}
97 1−7.32T+97T2 1 - 7.32T + 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

−9.668540619133072335007267951394, −9.105729073626433502560746546686, −8.241528549053139495493264405490, −7.26877890734085025962345087424, −6.26486924029069905424570984084, −5.16229873515260996451149424190, −4.19698434610901721258132592209, −3.33830703556564641968264915314, −2.28848111587801542122097619031, −0.868948022525394123319433110421, 0.70197278102131051777393659550, 2.87002166344696793564199683185, 3.65124242472671573375678382816, 4.98852151870066186116440428126, 5.82903993301798597417925153886, 6.72070933491219345674870079468, 7.08281337070766631774232550687, 8.130520178469692360814958262730, 9.186769942018635118304705061685, 9.429148616796200465252141979655

Graph of the ZZ-function along the critical line