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Properties

Label 2-200-5.4-c3-0-10
Degree 22
Conductor 200200
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 11.800311.8003
Root an. cond. 3.435163.43516
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9i·3-s − 26i·7-s − 54·9-s − 59·11-s + 28i·13-s − 5i·17-s − 109·19-s + 234·21-s − 194i·23-s − 243i·27-s + 32·29-s + 10·31-s − 531i·33-s + 198i·37-s − 252·39-s + ⋯
L(s)  = 1  + 1.73i·3-s − 1.40i·7-s − 2·9-s − 1.61·11-s + 0.597i·13-s − 0.0713i·17-s − 1.31·19-s + 2.43·21-s − 1.75i·23-s − 1.73i·27-s + 0.204·29-s + 0.0579·31-s − 2.80i·33-s + 0.879i·37-s − 1.03·39-s + ⋯

Functional equation

Λ(s)=(200s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(200s/2 ΓC(s+3/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23⋅522^{3} \cdot 5^{2}
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 11.800311.8003
Root analytic conductor: 3.435163.43516
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ200(49,⋅)\chi_{200} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 200, ( :3/2), −0.447+0.894i)(2,\ 200,\ (\ :3/2),\ -0.447 + 0.894i)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
5 1 1
good3 1−9iT−27T2 1 - 9iT - 27T^{2}
7 1+26iT−343T2 1 + 26iT - 343T^{2}
11 1+59T+1.33e3T2 1 + 59T + 1.33e3T^{2}
13 1−28iT−2.19e3T2 1 - 28iT - 2.19e3T^{2}
17 1+5iT−4.91e3T2 1 + 5iT - 4.91e3T^{2}
19 1+109T+6.85e3T2 1 + 109T + 6.85e3T^{2}
23 1+194iT−1.21e4T2 1 + 194iT - 1.21e4T^{2}
29 1−32T+2.43e4T2 1 - 32T + 2.43e4T^{2}
31 1−10T+2.97e4T2 1 - 10T + 2.97e4T^{2}
37 1−198iT−5.06e4T2 1 - 198iT - 5.06e4T^{2}
41 1−117T+6.89e4T2 1 - 117T + 6.89e4T^{2}
43 1−388iT−7.95e4T2 1 - 388iT - 7.95e4T^{2}
47 1−68iT−1.03e5T2 1 - 68iT - 1.03e5T^{2}
53 1+18iT−1.48e5T2 1 + 18iT - 1.48e5T^{2}
59 1+392T+2.05e5T2 1 + 392T + 2.05e5T^{2}
61 1+710T+2.26e5T2 1 + 710T + 2.26e5T^{2}
67 1−253iT−3.00e5T2 1 - 253iT - 3.00e5T^{2}
71 1+612T+3.57e5T2 1 + 612T + 3.57e5T^{2}
73 1+549iT−3.89e5T2 1 + 549iT - 3.89e5T^{2}
79 1+414T+4.93e5T2 1 + 414T + 4.93e5T^{2}
83 1+121iT−5.71e5T2 1 + 121iT - 5.71e5T^{2}
89 1−81T+7.04e5T2 1 - 81T + 7.04e5T^{2}
97 1−1.50e3iT−9.12e5T2 1 - 1.50e3iT - 9.12e5T^{2}
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   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

−11.15393938352522419346794942557, −10.49402007663718025376895868752, −10.11515541157881633164754281975, −8.845914893732051975166137237585, −7.82334988108034490830774446384, −6.31954877183554949997475755517, −4.74456189187046857897397817155, −4.30213702937028984728472696706, −2.86651536518036680132163365638, 0, 1.93969814527912622473200520647, 2.83505152557841791525048598942, 5.40348380460050031519800036264, 6.01862108603600452737511220259, 7.38160533140963653868149082931, 8.104698253017225594076831966140, 8.979985282582842188120706929513, 10.57688960264807055982568372232, 11.67348721665217986390393719824

Graph of the ZZ-function along the critical line