↑

Properties

Label 2-60-20.19-c2-0-7
Degree 22
Conductor 6060
Sign 0.866−0.5i0.866 - 0.5i
Analytic cond. 1.634881.63488
Root an. cond. 1.278621.27862
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 + i)2-s + 1.73·3-s + (1.99 + 3.46i)4-s − 5i·5-s + (2.99 + 1.73i)6-s − 10.3·7-s + 7.99i·8-s + 2.99·9-s + (5 − 8.66i)10-s + 10.3i·11-s + (3.46 + 5.99i)12-s − 18i·13-s + (−18 − 10.3i)14-s − 8.66i·15-s + (−8 + 13.8i)16-s + 10i·17-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + 0.577·3-s + (0.499 + 0.866i)4-s − i·5-s + (0.499 + 0.288i)6-s − 1.48·7-s + 0.999i·8-s + 0.333·9-s + (0.5 − 0.866i)10-s + 0.944i·11-s + (0.288 + 0.499i)12-s − 1.38i·13-s + (−1.28 − 0.742i)14-s − 0.577i·15-s + (−0.5 + 0.866i)16-s + 0.588i·17-s + ⋯

Functional equation

Λ(s)=(60s/2 ΓC(s) L(s)=((0.866−0.5i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(60s/2 ΓC(s+1) L(s)=((0.866−0.5i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6060    =    22⋅3⋅52^{2} \cdot 3 \cdot 5
Sign: 0.866−0.5i0.866 - 0.5i
Analytic conductor: 1.634881.63488
Root analytic conductor: 1.278621.27862
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ60(19,⋅)\chi_{60} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 60, ( :1), 0.866−0.5i)(2,\ 60,\ (\ :1),\ 0.866 - 0.5i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.85513+0.497081i1.85513 + 0.497081i
L(12)L(\frac12) ≈\approx 1.85513+0.497081i1.85513 + 0.497081i
L(2)L(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+(−1.73−i)T 1 + (-1.73 - i)T
3 1−1.73T 1 - 1.73T
5 1+5iT 1 + 5iT
good7 1+10.3T+49T2 1 + 10.3T + 49T^{2}
11 1−10.3iT−121T2 1 - 10.3iT - 121T^{2}
13 1+18iT−169T2 1 + 18iT - 169T^{2}
17 1−10iT−289T2 1 - 10iT - 289T^{2}
19 1+13.8iT−361T2 1 + 13.8iT - 361T^{2}
23 1−6.92T+529T2 1 - 6.92T + 529T^{2}
29 1−36T+841T2 1 - 36T + 841T^{2}
31 1+6.92iT−961T2 1 + 6.92iT - 961T^{2}
37 1−54iT−1.36e3T2 1 - 54iT - 1.36e3T^{2}
41 1−18T+1.68e3T2 1 - 18T + 1.68e3T^{2}
43 1−20.7T+1.84e3T2 1 - 20.7T + 1.84e3T^{2}
47 1+2.20e3T2 1 + 2.20e3T^{2}
53 1−26iT−2.80e3T2 1 - 26iT - 2.80e3T^{2}
59 1−31.1iT−3.48e3T2 1 - 31.1iT - 3.48e3T^{2}
61 1+74T+3.72e3T2 1 + 74T + 3.72e3T^{2}
67 1+41.5T+4.48e3T2 1 + 41.5T + 4.48e3T^{2}
71 1+103.iT−5.04e3T2 1 + 103. iT - 5.04e3T^{2}
73 1+36iT−5.32e3T2 1 + 36iT - 5.32e3T^{2}
79 1+90.0iT−6.24e3T2 1 + 90.0iT - 6.24e3T^{2}
83 1+90.0T+6.88e3T2 1 + 90.0T + 6.88e3T^{2}
89 1−18T+7.92e3T2 1 - 18T + 7.92e3T^{2}
97 1+72iT−9.40e3T2 1 + 72iT - 9.40e3T^{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

−15.19727982633406335003594988742, −13.56578440478616414484173606680, −12.88113925322346961057012887202, −12.26720916770949534475969984552, −10.17171427084914204247436510682, −8.878908673860368783240290797367, −7.60897122852066003840228852115, −6.18785581928845022646909641453, −4.63458114106254751109070966319, −3.04697893126555157947514251848, 2.73947392994343412870070891116, 3.82152555811029791515926039426, 6.12348328898362290817991877767, 7.02873970708748645772864447080, 9.244690731137069458376173257719, 10.24941218672404963073890935219, 11.44004042588776171888541551729, 12.66395745557934144883157750148, 13.88261097351605199089322512916, 14.27689735493798379471524957053

Graph of the ZZ-function along the critical line