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Properties

Label 2-1800-1.1-c3-0-8
Degree 22
Conductor 18001800
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·7-s − 36·11-s + 42·13-s − 110·17-s − 116·19-s + 16·23-s − 198·29-s + 240·31-s + 258·37-s − 442·41-s + 292·43-s + 392·47-s − 87·49-s + 142·53-s + 348·59-s − 570·61-s − 692·67-s − 168·71-s + 134·73-s + 576·77-s + 784·79-s + 564·83-s − 1.03e3·89-s − 672·91-s + 382·97-s + 674·101-s + 992·103-s + ⋯
L(s)  = 1  − 0.863·7-s − 0.986·11-s + 0.896·13-s − 1.56·17-s − 1.40·19-s + 0.145·23-s − 1.26·29-s + 1.39·31-s + 1.14·37-s − 1.68·41-s + 1.03·43-s + 1.21·47-s − 0.253·49-s + 0.368·53-s + 0.767·59-s − 1.19·61-s − 1.26·67-s − 0.280·71-s + 0.214·73-s + 0.852·77-s + 1.11·79-s + 0.745·83-s − 1.23·89-s − 0.774·91-s + 0.399·97-s + 0.664·101-s + 0.948·103-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: 11
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: 00
Selberg data: (2, 1800, ( :3/2), 1)(2,\ 1800,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.0619076101.061907610
L(12)L(\frac12) ≈\approx 1.0619076101.061907610
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+16T+p3T2 1 + 16 T + p^{3} T^{2}
11 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
13 1−42T+p3T2 1 - 42 T + p^{3} T^{2}
17 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
19 1+116T+p3T2 1 + 116 T + p^{3} T^{2}
23 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
29 1+198T+p3T2 1 + 198 T + p^{3} T^{2}
31 1−240T+p3T2 1 - 240 T + p^{3} T^{2}
37 1−258T+p3T2 1 - 258 T + p^{3} T^{2}
41 1+442T+p3T2 1 + 442 T + p^{3} T^{2}
43 1−292T+p3T2 1 - 292 T + p^{3} T^{2}
47 1−392T+p3T2 1 - 392 T + p^{3} T^{2}
53 1−142T+p3T2 1 - 142 T + p^{3} T^{2}
59 1−348T+p3T2 1 - 348 T + p^{3} T^{2}
61 1+570T+p3T2 1 + 570 T + p^{3} T^{2}
67 1+692T+p3T2 1 + 692 T + p^{3} T^{2}
71 1+168T+p3T2 1 + 168 T + p^{3} T^{2}
73 1−134T+p3T2 1 - 134 T + p^{3} T^{2}
79 1−784T+p3T2 1 - 784 T + p^{3} T^{2}
83 1−564T+p3T2 1 - 564 T + p^{3} T^{2}
89 1+1034T+p3T2 1 + 1034 T + p^{3} T^{2}
97 1−382T+p3T2 1 - 382 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

−8.851753834284614091392857368247, −8.276090193346409480076597597649, −7.27047022555279141619102725140, −6.40697515062134833477875001115, −5.92580119689374399210033020739, −4.71396372682087169102626217987, −3.95307759304663441299461447116, −2.87091582790039404897616209390, −2.02558001185043872110530288433, −0.45968537237450876577772116559, 0.45968537237450876577772116559, 2.02558001185043872110530288433, 2.87091582790039404897616209390, 3.95307759304663441299461447116, 4.71396372682087169102626217987, 5.92580119689374399210033020739, 6.40697515062134833477875001115, 7.27047022555279141619102725140, 8.276090193346409480076597597649, 8.851753834284614091392857368247

Graph of the ZZ-function along the critical line