↑

Properties

Label 2-1200-1.1-c3-0-55
Degree 22
Conductor 12001200
Sign −1-1
Analytic cond. 70.802270.8022
Root an. cond. 8.414408.41440
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 20·7-s + 9·9-s − 16·11-s − 58·13-s − 38·17-s − 4·19-s + 60·21-s − 80·23-s + 27·27-s + 82·29-s + 8·31-s − 48·33-s − 426·37-s − 174·39-s − 246·41-s − 524·43-s − 464·47-s + 57·49-s − 114·51-s + 702·53-s − 12·57-s + 592·59-s + 574·61-s + 180·63-s − 172·67-s − 240·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.07·7-s + 1/3·9-s − 0.438·11-s − 1.23·13-s − 0.542·17-s − 0.0482·19-s + 0.623·21-s − 0.725·23-s + 0.192·27-s + 0.525·29-s + 0.0463·31-s − 0.253·33-s − 1.89·37-s − 0.714·39-s − 0.937·41-s − 1.85·43-s − 1.44·47-s + 0.166·49-s − 0.313·51-s + 1.81·53-s − 0.0278·57-s + 1.30·59-s + 1.20·61-s + 0.359·63-s − 0.313·67-s − 0.418·69-s + ⋯

Functional equation

Λ(s)=(1200s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1200s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12001200    =    24⋅3⋅522^{4} \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 70.802270.8022
Root analytic conductor: 8.414408.41440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1200, ( :3/2), −1)(2,\ 1200,\ (\ :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−pT 1 - p T
5 1 1
good7 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
11 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
13 1+58T+p3T2 1 + 58 T + p^{3} T^{2}
17 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
19 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
23 1+80T+p3T2 1 + 80 T + p^{3} T^{2}
29 1−82T+p3T2 1 - 82 T + p^{3} T^{2}
31 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
37 1+426T+p3T2 1 + 426 T + p^{3} T^{2}
41 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
43 1+524T+p3T2 1 + 524 T + p^{3} T^{2}
47 1+464T+p3T2 1 + 464 T + p^{3} T^{2}
53 1−702T+p3T2 1 - 702 T + p^{3} T^{2}
59 1−592T+p3T2 1 - 592 T + p^{3} T^{2}
61 1−574T+p3T2 1 - 574 T + p^{3} T^{2}
67 1+172T+p3T2 1 + 172 T + p^{3} T^{2}
71 1+768T+p3T2 1 + 768 T + p^{3} T^{2}
73 1−558T+p3T2 1 - 558 T + p^{3} T^{2}
79 1+408T+p3T2 1 + 408 T + p^{3} T^{2}
83 1−164T+p3T2 1 - 164 T + p^{3} T^{2}
89 1+510T+p3T2 1 + 510 T + p^{3} T^{2}
97 1+514T+p3T2 1 + 514 T + p^{3} T^{2}
show more
show less
   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

−8.661866669224269565481907260401, −8.325757662671788554122809905613, −7.38975164334645866079613442341, −6.69880147011743094152562042439, −5.25368741174324151897871037653, −4.78028609485695993390311496027, −3.64134507814836246018594730620, −2.44279926683918990574622020367, −1.66931161354033887472717155188, 0, 1.66931161354033887472717155188, 2.44279926683918990574622020367, 3.64134507814836246018594730620, 4.78028609485695993390311496027, 5.25368741174324151897871037653, 6.69880147011743094152562042439, 7.38975164334645866079613442341, 8.325757662671788554122809905613, 8.661866669224269565481907260401

Graph of the ZZ-function along the critical line