↑

Properties

Label 2-253-11.10-c2-0-18
Degree 22
Conductor 253253
Sign 0.833−0.552i0.833 - 0.552i
Analytic cond. 6.893756.89375
Root an. cond. 2.625592.62559
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  + 3.71i·2-s − 2.82·3-s − 9.78·4-s + 2.21·5-s − 10.4i·6-s − 5.12i·7-s − 21.4i·8-s − 1.04·9-s + 8.23i·10-s + (9.16 − 6.08i)11-s + 27.5·12-s + 2.92i·13-s + 19.0·14-s − 6.25·15-s + 40.5·16-s − 12.0i·17-s + ⋯
L(s)  = 1  + 1.85i·2-s − 0.940·3-s − 2.44·4-s + 0.443·5-s − 1.74i·6-s − 0.731i·7-s − 2.68i·8-s − 0.116·9-s + 0.823i·10-s + (0.833 − 0.552i)11-s + 2.29·12-s + 0.224i·13-s + 1.35·14-s − 0.417·15-s + 2.53·16-s − 0.708i·17-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((0.833−0.552i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((0.833−0.552i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: 0.833−0.552i0.833 - 0.552i
Analytic conductor: 6.893756.89375
Root analytic conductor: 2.625592.62559
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ253(208,⋅)\chi_{253} (208, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 253, ( :1), 0.833−0.552i)(2,\ 253,\ (\ :1),\ 0.833 - 0.552i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.728182+0.219612i0.728182 + 0.219612i
L(12)L(\frac12) ≈\approx 0.728182+0.219612i0.728182 + 0.219612i
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)
bad11 1+(−9.16+6.08i)T 1 + (-9.16 + 6.08i)T
23 1−4.79T 1 - 4.79T
good2 1−3.71iT−4T2 1 - 3.71iT - 4T^{2}
3 1+2.82T+9T2 1 + 2.82T + 9T^{2}
5 1−2.21T+25T2 1 - 2.21T + 25T^{2}
7 1+5.12iT−49T2 1 + 5.12iT - 49T^{2}
13 1−2.92iT−169T2 1 - 2.92iT - 169T^{2}
17 1+12.0iT−289T2 1 + 12.0iT - 289T^{2}
19 1+1.76iT−361T2 1 + 1.76iT - 361T^{2}
29 1+7.43iT−841T2 1 + 7.43iT - 841T^{2}
31 1+15.3T+961T2 1 + 15.3T + 961T^{2}
37 1−53.4T+1.36e3T2 1 - 53.4T + 1.36e3T^{2}
41 1+58.9iT−1.68e3T2 1 + 58.9iT - 1.68e3T^{2}
43 1+45.8iT−1.84e3T2 1 + 45.8iT - 1.84e3T^{2}
47 1−5.73T+2.20e3T2 1 - 5.73T + 2.20e3T^{2}
53 1+90.1T+2.80e3T2 1 + 90.1T + 2.80e3T^{2}
59 1−71.4T+3.48e3T2 1 - 71.4T + 3.48e3T^{2}
61 1+26.1iT−3.72e3T2 1 + 26.1iT - 3.72e3T^{2}
67 1−20.7T+4.48e3T2 1 - 20.7T + 4.48e3T^{2}
71 1−20.2T+5.04e3T2 1 - 20.2T + 5.04e3T^{2}
73 1+111.iT−5.32e3T2 1 + 111. iT - 5.32e3T^{2}
79 1+13.9iT−6.24e3T2 1 + 13.9iT - 6.24e3T^{2}
83 1+86.4iT−6.88e3T2 1 + 86.4iT - 6.88e3T^{2}
89 1+29.8T+7.92e3T2 1 + 29.8T + 7.92e3T^{2}
97 1+79.0T+9.40e3T2 1 + 79.0T + 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

−11.96321346127232001150304740696, −10.88917940532138367055390119261, −9.639037637462278258577804394840, −8.833078479661988056831250284462, −7.62585763703528833947305867014, −6.70277060956268660832530640327, −5.98853632248213511284610531111, −5.14872496818217818719591101682, −3.97596178774092732395001198902, −0.50462492750501281215532622621, 1.35177480376511975819286678930, 2.67508913921440626306147032429, 4.14401827217144789117721412147, 5.29647434536082133590784330809, 6.25905593412270191987115921409, 8.287724653510086834304429259561, 9.368023867878179122570752503227, 9.977299358383131329194031286735, 11.12507649102234221088493935611, 11.53326798508993595030767083158

Graph of the ZZ-function along the critical line