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Properties

Label 2-1800-1.1-c3-0-13
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  − 26·7-s + 59·11-s − 28·13-s + 5·17-s + 109·19-s − 194·23-s + 32·29-s + 10·31-s + 198·37-s − 117·41-s − 388·43-s − 68·47-s + 333·49-s − 18·53-s − 392·59-s − 710·61-s + 253·67-s + 612·71-s + 549·73-s − 1.53e3·77-s + 414·79-s − 121·83-s + 81·89-s + 728·91-s + 1.50e3·97-s + 234·101-s + 1.17e3·103-s + ⋯
L(s)  = 1  − 1.40·7-s + 1.61·11-s − 0.597·13-s + 0.0713·17-s + 1.31·19-s − 1.75·23-s + 0.204·29-s + 0.0579·31-s + 0.879·37-s − 0.445·41-s − 1.37·43-s − 0.211·47-s + 0.970·49-s − 0.0466·53-s − 0.864·59-s − 1.49·61-s + 0.461·67-s + 1.02·71-s + 0.880·73-s − 2.27·77-s + 0.589·79-s − 0.160·83-s + 0.0964·89-s + 0.838·91-s + 1.57·97-s + 0.230·101-s + 1.12·103-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 1.6231821141.623182114
L(12)L(\frac12) ≈\approx 1.6231821141.623182114
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+26T+p3T2 1 + 26 T + p^{3} T^{2}
11 1−59T+p3T2 1 - 59 T + p^{3} T^{2}
13 1+28T+p3T2 1 + 28 T + p^{3} T^{2}
17 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
19 1−109T+p3T2 1 - 109 T + p^{3} T^{2}
23 1+194T+p3T2 1 + 194 T + p^{3} T^{2}
29 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
31 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
37 1−198T+p3T2 1 - 198 T + p^{3} T^{2}
41 1+117T+p3T2 1 + 117 T + p^{3} T^{2}
43 1+388T+p3T2 1 + 388 T + p^{3} T^{2}
47 1+68T+p3T2 1 + 68 T + p^{3} T^{2}
53 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
59 1+392T+p3T2 1 + 392 T + p^{3} T^{2}
61 1+710T+p3T2 1 + 710 T + p^{3} T^{2}
67 1−253T+p3T2 1 - 253 T + p^{3} T^{2}
71 1−612T+p3T2 1 - 612 T + p^{3} T^{2}
73 1−549T+p3T2 1 - 549 T + p^{3} T^{2}
79 1−414T+p3T2 1 - 414 T + p^{3} T^{2}
83 1+121T+p3T2 1 + 121 T + p^{3} T^{2}
89 1−81T+p3T2 1 - 81 T + p^{3} T^{2}
97 1−1502T+p3T2 1 - 1502 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

−9.170474303194529578552590647186, −8.111302615699579225715583640660, −7.21921641606883845754540527517, −6.44418372106866791886892183739, −5.96641922761082019807857810914, −4.75251869095824657444713976859, −3.73323162285852791753673394707, −3.14786917416481041115075918618, −1.84442289015236427275908077816, −0.59519584277076253382780006874, 0.59519584277076253382780006874, 1.84442289015236427275908077816, 3.14786917416481041115075918618, 3.73323162285852791753673394707, 4.75251869095824657444713976859, 5.96641922761082019807857810914, 6.44418372106866791886892183739, 7.21921641606883845754540527517, 8.111302615699579225715583640660, 9.170474303194529578552590647186

Graph of the ZZ-function along the critical line