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Properties

Label 2-1800-5.4-c3-0-21
Degree 22
Conductor 18001800
Sign 0.447−0.894i0.447 - 0.894i
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  + 19i·7-s − 22·11-s + i·13-s − 58i·17-s + 53·19-s − 58i·23-s + 22·29-s − 35·31-s + 270i·37-s + 468·41-s − 431i·43-s − 230i·47-s − 18·49-s + 446·59-s + 127·61-s + ⋯
L(s)  = 1  + 1.02i·7-s − 0.603·11-s + 0.0213i·13-s − 0.827i·17-s + 0.639·19-s − 0.525i·23-s + 0.140·29-s − 0.202·31-s + 1.19i·37-s + 1.78·41-s − 1.52i·43-s − 0.713i·47-s − 0.0524·49-s + 0.984·59-s + 0.266·61-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s+3/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{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: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: 0.447−0.894i0.447 - 0.894i
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), 0.447−0.894i)(2,\ 1800,\ (\ :3/2),\ 0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.8119798511.811979851
L(12)L(\frac12) ≈\approx 1.8119798511.811979851
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 1−19iT−343T2 1 - 19iT - 343T^{2}
11 1+22T+1.33e3T2 1 + 22T + 1.33e3T^{2}
13 1−iT−2.19e3T2 1 - iT - 2.19e3T^{2}
17 1+58iT−4.91e3T2 1 + 58iT - 4.91e3T^{2}
19 1−53T+6.85e3T2 1 - 53T + 6.85e3T^{2}
23 1+58iT−1.21e4T2 1 + 58iT - 1.21e4T^{2}
29 1−22T+2.43e4T2 1 - 22T + 2.43e4T^{2}
31 1+35T+2.97e4T2 1 + 35T + 2.97e4T^{2}
37 1−270iT−5.06e4T2 1 - 270iT - 5.06e4T^{2}
41 1−468T+6.89e4T2 1 - 468T + 6.89e4T^{2}
43 1+431iT−7.95e4T2 1 + 431iT - 7.95e4T^{2}
47 1+230iT−1.03e5T2 1 + 230iT - 1.03e5T^{2}
53 1−1.48e5T2 1 - 1.48e5T^{2}
59 1−446T+2.05e5T2 1 - 446T + 2.05e5T^{2}
61 1−127T+2.26e5T2 1 - 127T + 2.26e5T^{2}
67 1−811iT−3.00e5T2 1 - 811iT - 3.00e5T^{2}
71 1+36T+3.57e5T2 1 + 36T + 3.57e5T^{2}
73 1−522iT−3.89e5T2 1 - 522iT - 3.89e5T^{2}
79 1+1.36e3T+4.93e5T2 1 + 1.36e3T + 4.93e5T^{2}
83 1−1.13e3iT−5.71e5T2 1 - 1.13e3iT - 5.71e5T^{2}
89 1−144T+7.04e5T2 1 - 144T + 7.04e5T^{2}
97 1−1.07e3iT−9.12e5T2 1 - 1.07e3iT - 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

−8.978953864563239822186012059711, −8.386410561804469857769750618171, −7.47625665065262454068042498570, −6.71394656566363103196436034771, −5.62838586415054428676283680000, −5.21112062651992409561914800821, −4.10577686294779830313008645946, −2.88938623916626015743178619688, −2.28651331479010322288551122849, −0.840988129822077940071024306653, 0.49358286890047522587309732690, 1.56349116405121224044442575562, 2.82882624961970486927670791969, 3.80897897010173835143947235725, 4.56214095566332762806990356334, 5.59283556204901695296822275329, 6.34742313617586669267331525610, 7.54008219943278856057609697814, 7.64580490535863809511218551226, 8.797085239517514994844994507050

Graph of the ZZ-function along the critical line