↑

Properties

Label 2-6120-17.16-c1-0-45
Degree 22
Conductor 61206120
Sign 0.846+0.532i0.846 + 0.532i
Analytic cond. 48.868448.8684
Root an. cond. 6.990596.99059
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  − i·5-s − 0.489i·7-s − 0.489i·11-s + 0.292·13-s + (−2.19 + 3.48i)17-s − 7.17·19-s + 2.29i·23-s − 25-s + 1.51i·29-s − 4.68i·31-s − 0.489·35-s + 1.51i·37-s + 7.86i·41-s + 9.95·43-s + 13.5·47-s + ⋯
L(s)  = 1  − 0.447i·5-s − 0.184i·7-s − 0.147i·11-s + 0.0811·13-s + (−0.532 + 0.846i)17-s − 1.64·19-s + 0.478i·23-s − 0.200·25-s + 0.280i·29-s − 0.841i·31-s − 0.0827·35-s + 0.248i·37-s + 1.22i·41-s + 1.51·43-s + 1.97·47-s + ⋯

Functional equation

Λ(s)=(6120s/2 ΓC(s) L(s)=((0.846+0.532i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(6120s/2 ΓC(s+1/2) L(s)=((0.846+0.532i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 61206120    =    23⋅32⋅5⋅172^{3} \cdot 3^{2} \cdot 5 \cdot 17
Sign: 0.846+0.532i0.846 + 0.532i
Analytic conductor: 48.868448.8684
Root analytic conductor: 6.990596.99059
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ6120(1801,⋅)\chi_{6120} (1801, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 6120, ( :1/2), 0.846+0.532i)(2,\ 6120,\ (\ :1/2),\ 0.846 + 0.532i)

Particular Values

L(1)L(1) ≈\approx 1.6583003991.658300399
L(12)L(\frac12) ≈\approx 1.6583003991.658300399
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
3 1 1
5 1+iT 1 + iT
17 1+(2.19−3.48i)T 1 + (2.19 - 3.48i)T
good7 1+0.489iT−7T2 1 + 0.489iT - 7T^{2}
11 1+0.489iT−11T2 1 + 0.489iT - 11T^{2}
13 1−0.292T+13T2 1 - 0.292T + 13T^{2}
19 1+7.17T+19T2 1 + 7.17T + 19T^{2}
23 1−2.29iT−23T2 1 - 2.29iT - 23T^{2}
29 1−1.51iT−29T2 1 - 1.51iT - 29T^{2}
31 1+4.68iT−31T2 1 + 4.68iT - 31T^{2}
37 1−1.51iT−37T2 1 - 1.51iT - 37T^{2}
41 1−7.86iT−41T2 1 - 7.86iT - 41T^{2}
43 1−9.95T+43T2 1 - 9.95T + 43T^{2}
47 1−13.5T+47T2 1 - 13.5T + 47T^{2}
53 1+1.80T+53T2 1 + 1.80T + 53T^{2}
59 1−7.07T+59T2 1 - 7.07T + 59T^{2}
61 1+5.70iT−61T2 1 + 5.70iT - 61T^{2}
67 1−7.27T+67T2 1 - 7.27T + 67T^{2}
71 1−0.585iT−71T2 1 - 0.585iT - 71T^{2}
73 1+16.8iT−73T2 1 + 16.8iT - 73T^{2}
79 1+15.9iT−79T2 1 + 15.9iT - 79T^{2}
83 1−10.3T+83T2 1 - 10.3T + 83T^{2}
89 1−4.68T+89T2 1 - 4.68T + 89T^{2}
97 1−3.95iT−97T2 1 - 3.95iT - 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

−8.007064335075448980749951120372, −7.41501938213716902610375169277, −6.38398946142241896259946840457, −6.04632002546085230151611608308, −5.08445865238494763986500161557, −4.23835399643771627919248962002, −3.81367570666025512299658884843, −2.56705030079380637250679071922, −1.78630415111860286542859030890, −0.59778565897287705433973027982, 0.73542659849945289134755050936, 2.26750431824503419211150423726, 2.56169827577738180335151126682, 3.88174848661140195149499417311, 4.32980926577032009770173838860, 5.37142278222587794695119173880, 5.98511578217747247752329019223, 6.92977140801263009584416800744, 7.14523516236785133112943586421, 8.236615322691790416040992617494

Graph of the ZZ-function along the critical line