↑

Properties

Label 2-2100-105.62-c0-0-5
Degree 22
Conductor 21002100
Sign −0.0299+0.999i-0.0299 + 0.999i
Analytic cond. 1.048031.04803
Root an. cond. 1.023731.02373
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)3-s + (−0.707 + 0.707i)7-s + (−0.866 − 0.499i)9-s − 1.73i·11-s + (−0.707 − 0.707i)13-s + (−1.22 − 1.22i)17-s + (−0.500 − 0.866i)21-s + (0.707 − 0.707i)27-s − 1.73·29-s + (1.67 + 0.448i)33-s + (0.866 − 0.500i)39-s + (1.22 + 1.22i)47-s − 1.00i·49-s + (1.49 − 0.866i)51-s + (0.965 − 0.258i)63-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)3-s + (−0.707 + 0.707i)7-s + (−0.866 − 0.499i)9-s − 1.73i·11-s + (−0.707 − 0.707i)13-s + (−1.22 − 1.22i)17-s + (−0.500 − 0.866i)21-s + (0.707 − 0.707i)27-s − 1.73·29-s + (1.67 + 0.448i)33-s + (0.866 − 0.500i)39-s + (1.22 + 1.22i)47-s − 1.00i·49-s + (1.49 − 0.866i)51-s + (0.965 − 0.258i)63-s + ⋯

Functional equation

Λ(s)=(2100s/2 ΓC(s) L(s)=((−0.0299+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2100s/2 ΓC(s) L(s)=((−0.0299+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21002100    =    22⋅3⋅52⋅72^{2} \cdot 3 \cdot 5^{2} \cdot 7
Sign: −0.0299+0.999i-0.0299 + 0.999i
Analytic conductor: 1.048031.04803
Root analytic conductor: 1.023731.02373
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2100(2057,⋅)\chi_{2100} (2057, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2100, ( :0), −0.0299+0.999i)(2,\ 2100,\ (\ :0),\ -0.0299 + 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.39080622480.3908062248
L(12)L(\frac12) ≈\approx 0.39080622480.3908062248
L(1)L(1) 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+(0.258−0.965i)T 1 + (0.258 - 0.965i)T
5 1 1
7 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good11 1+1.73iT−T2 1 + 1.73iT - T^{2}
13 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
17 1+(1.22+1.22i)T+iT2 1 + (1.22 + 1.22i)T + iT^{2}
19 1+T2 1 + T^{2}
23 1+iT2 1 + iT^{2}
29 1+1.73T+T2 1 + 1.73T + T^{2}
31 1−T2 1 - T^{2}
37 1+iT2 1 + iT^{2}
41 1+T2 1 + T^{2}
43 1−iT2 1 - iT^{2}
47 1+(−1.22−1.22i)T+iT2 1 + (-1.22 - 1.22i)T + iT^{2}
53 1+iT2 1 + iT^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1+iT2 1 + iT^{2}
71 1−T2 1 - T^{2}
73 1+(1.41+1.41i)T+iT2 1 + (1.41 + 1.41i)T + iT^{2}
79 1−iT−T2 1 - iT - T^{2}
83 1−iT2 1 - iT^{2}
89 1−T2 1 - T^{2}
97 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{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

−9.143604353614479549474948401126, −8.659102868117071935943441959170, −7.60296824848240014304343771188, −6.51951082459183972920723039447, −5.73270237183867089668558820625, −5.27921845411633366867182406037, −4.17418227821669131734218705865, −3.17305346256028909147446746792, −2.61652480915699039288634702198, −0.26404630221946519279447254916, 1.71276494134743300240203812512, 2.37329597507435368852888912578, 3.87457871573254250416683507654, 4.58440935401040042313365280920, 5.69626043129586567798778718291, 6.63449778600294310569600708940, 7.11599646064126455582388902905, 7.59779598528324878743823523163, 8.725776025793042739115247657107, 9.473761006526495151673043249086

Graph of the ZZ-function along the critical line