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Properties

Label 2-1800-1.1-c3-0-61
Degree 22
Conductor 18001800
Sign −1-1
Analytic cond. 106.203106.203
Root an. cond. 10.305510.3055
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·7-s + 46·11-s − 34·13-s − 66·17-s + 104·19-s − 164·23-s − 224·29-s − 72·31-s − 22·37-s − 194·41-s + 108·43-s + 480·47-s − 243·49-s − 286·53-s − 426·59-s + 698·61-s + 328·67-s − 188·71-s − 740·73-s + 460·77-s + 1.16e3·79-s − 412·83-s − 1.20e3·89-s − 340·91-s − 1.38e3·97-s + 1.12e3·101-s − 758·103-s + ⋯
L(s)  = 1  + 0.539·7-s + 1.26·11-s − 0.725·13-s − 0.941·17-s + 1.25·19-s − 1.48·23-s − 1.43·29-s − 0.417·31-s − 0.0977·37-s − 0.738·41-s + 0.383·43-s + 1.48·47-s − 0.708·49-s − 0.741·53-s − 0.940·59-s + 1.46·61-s + 0.598·67-s − 0.314·71-s − 1.18·73-s + 0.680·77-s + 1.66·79-s − 0.544·83-s − 1.43·89-s − 0.391·91-s − 1.44·97-s + 1.11·101-s − 0.725·103-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 106.203106.203
Root analytic conductor: 10.305510.3055
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1800, ( :3/2), −1)(2,\ 1800,\ (\ :3/2),\ -1)

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
3 1 1
5 1 1
good7 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
11 1−46T+p3T2 1 - 46 T + p^{3} T^{2}
13 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
17 1+66T+p3T2 1 + 66 T + p^{3} T^{2}
19 1−104T+p3T2 1 - 104 T + p^{3} T^{2}
23 1+164T+p3T2 1 + 164 T + p^{3} T^{2}
29 1+224T+p3T2 1 + 224 T + p^{3} T^{2}
31 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
37 1+22T+p3T2 1 + 22 T + p^{3} T^{2}
41 1+194T+p3T2 1 + 194 T + p^{3} T^{2}
43 1−108T+p3T2 1 - 108 T + p^{3} T^{2}
47 1−480T+p3T2 1 - 480 T + p^{3} T^{2}
53 1+286T+p3T2 1 + 286 T + p^{3} T^{2}
59 1+426T+p3T2 1 + 426 T + p^{3} T^{2}
61 1−698T+p3T2 1 - 698 T + p^{3} T^{2}
67 1−328T+p3T2 1 - 328 T + p^{3} T^{2}
71 1+188T+p3T2 1 + 188 T + p^{3} T^{2}
73 1+740T+p3T2 1 + 740 T + p^{3} T^{2}
79 1−1168T+p3T2 1 - 1168 T + p^{3} T^{2}
83 1+412T+p3T2 1 + 412 T + p^{3} T^{2}
89 1+1206T+p3T2 1 + 1206 T + p^{3} T^{2}
97 1+1384T+p3T2 1 + 1384 T + p^{3} T^{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

−8.574259556317163784698141509614, −7.65002321887676829533358310696, −7.04156272848248849584558376047, −6.09743328249542248670045640864, −5.26480338295921929394123047351, −4.31341305928207759413358955265, −3.58706692950687204880715250725, −2.25952018678081181530397701399, −1.39813602893720354901435763939, 0, 1.39813602893720354901435763939, 2.25952018678081181530397701399, 3.58706692950687204880715250725, 4.31341305928207759413358955265, 5.26480338295921929394123047351, 6.09743328249542248670045640864, 7.04156272848248849584558376047, 7.65002321887676829533358310696, 8.574259556317163784698141509614

Graph of the ZZ-function along the critical line