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Properties

Label 2-1800-5.4-c3-0-66
Degree 22
Conductor 18001800
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 106.203106.203
Root an. cond. 10.305510.3055
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 34i·7-s − 16·11-s − 58i·13-s + 70i·17-s − 4·19-s − 134i·23-s − 242·29-s + 100·31-s − 438i·37-s + 138·41-s − 178i·43-s − 22i·47-s − 813·49-s + 162i·53-s − 268·59-s + ⋯
L(s)  = 1  − 1.83i·7-s − 0.438·11-s − 1.23i·13-s + 0.998i·17-s − 0.0482·19-s − 1.21i·23-s − 1.54·29-s + 0.579·31-s − 1.94i·37-s + 0.525·41-s − 0.631i·43-s − 0.0682i·47-s − 2.37·49-s + 0.419i·53-s − 0.591·59-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s+3/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 106.203106.203
Root analytic conductor: 10.305510.3055
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1800(649,⋅)\chi_{1800} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1800, ( :3/2), −0.894−0.447i)(2,\ 1800,\ (\ :3/2),\ -0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 0.80234367080.8023436708
L(12)L(\frac12) ≈\approx 0.80234367080.8023436708
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
3 1 1
5 1 1
good7 1+34iT−343T2 1 + 34iT - 343T^{2}
11 1+16T+1.33e3T2 1 + 16T + 1.33e3T^{2}
13 1+58iT−2.19e3T2 1 + 58iT - 2.19e3T^{2}
17 1−70iT−4.91e3T2 1 - 70iT - 4.91e3T^{2}
19 1+4T+6.85e3T2 1 + 4T + 6.85e3T^{2}
23 1+134iT−1.21e4T2 1 + 134iT - 1.21e4T^{2}
29 1+242T+2.43e4T2 1 + 242T + 2.43e4T^{2}
31 1−100T+2.97e4T2 1 - 100T + 2.97e4T^{2}
37 1+438iT−5.06e4T2 1 + 438iT - 5.06e4T^{2}
41 1−138T+6.89e4T2 1 - 138T + 6.89e4T^{2}
43 1+178iT−7.95e4T2 1 + 178iT - 7.95e4T^{2}
47 1+22iT−1.03e5T2 1 + 22iT - 1.03e5T^{2}
53 1−162iT−1.48e5T2 1 - 162iT - 1.48e5T^{2}
59 1+268T+2.05e5T2 1 + 268T + 2.05e5T^{2}
61 1−250T+2.26e5T2 1 - 250T + 2.26e5T^{2}
67 1−422iT−3.00e5T2 1 - 422iT - 3.00e5T^{2}
71 1−852T+3.57e5T2 1 - 852T + 3.57e5T^{2}
73 1+306iT−3.89e5T2 1 + 306iT - 3.89e5T^{2}
79 1−456T+4.93e5T2 1 - 456T + 4.93e5T^{2}
83 1−434iT−5.71e5T2 1 - 434iT - 5.71e5T^{2}
89 1+726T+7.04e5T2 1 + 726T + 7.04e5T^{2}
97 1−1.37e3iT−9.12e5T2 1 - 1.37e3iT - 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

−8.216653917690370161414310019503, −7.70824885489226415366625975491, −7.01702403633827120924261003847, −6.07891544147464450692316539687, −5.18062215942209140810155891597, −4.11095301543375202653994383317, −3.60834072190163192861077403549, −2.33241406423316551513256639311, −0.986169811466296005768752808220, −0.18587571312521825345759844268, 1.60949101075104289735804464423, 2.45697838985817334358055238736, 3.32107028601643397210609178351, 4.64639228900083435063532685245, 5.34619753767099391865582756977, 6.08439767770628909758181768335, 6.93885057789568231693330367550, 7.904493480950619413955930623874, 8.655594404750326920676172206107, 9.497600651252117366081922348247

Graph of the ZZ-function along the critical line