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Properties

Label 2-1200-1.1-c3-0-43
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 32·7-s + 9·9-s − 36·11-s + 10·13-s + 78·17-s − 140·19-s − 96·21-s − 192·23-s − 27·27-s + 6·29-s + 16·31-s + 108·33-s + 34·37-s − 30·39-s − 390·41-s − 52·43-s + 408·47-s + 681·49-s − 234·51-s + 114·53-s + 420·57-s − 516·59-s − 58·61-s + 288·63-s − 892·67-s + 576·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.72·7-s + 1/3·9-s − 0.986·11-s + 0.213·13-s + 1.11·17-s − 1.69·19-s − 0.997·21-s − 1.74·23-s − 0.192·27-s + 0.0384·29-s + 0.0926·31-s + 0.569·33-s + 0.151·37-s − 0.123·39-s − 1.48·41-s − 0.184·43-s + 1.26·47-s + 1.98·49-s − 0.642·51-s + 0.295·53-s + 0.975·57-s − 1.13·59-s − 0.121·61-s + 0.575·63-s − 1.62·67-s + 1.00·69-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 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
11 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
13 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
17 1−78T+p3T2 1 - 78 T + p^{3} T^{2}
19 1+140T+p3T2 1 + 140 T + p^{3} T^{2}
23 1+192T+p3T2 1 + 192 T + p^{3} T^{2}
29 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
31 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
37 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
41 1+390T+p3T2 1 + 390 T + p^{3} T^{2}
43 1+52T+p3T2 1 + 52 T + p^{3} T^{2}
47 1−408T+p3T2 1 - 408 T + p^{3} T^{2}
53 1−114T+p3T2 1 - 114 T + p^{3} T^{2}
59 1+516T+p3T2 1 + 516 T + p^{3} T^{2}
61 1+58T+p3T2 1 + 58 T + p^{3} T^{2}
67 1+892T+p3T2 1 + 892 T + p^{3} T^{2}
71 1−120T+p3T2 1 - 120 T + p^{3} T^{2}
73 1−646T+p3T2 1 - 646 T + p^{3} T^{2}
79 1−1168T+p3T2 1 - 1168 T + p^{3} T^{2}
83 1+732T+p3T2 1 + 732 T + p^{3} T^{2}
89 1+1590T+p3T2 1 + 1590 T + p^{3} T^{2}
97 1+2pT+p3T2 1 + 2 p 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

−8.735229703157067682633599621954, −8.045205815811440097860446477827, −7.56239864962174165790575892292, −6.28499740097241640823993570310, −5.49479052150955976739072968344, −4.76123073111374433511542300960, −3.92748470922967557762775377781, −2.32444286707874308619545067967, −1.43649291851529844981664610270, 0, 1.43649291851529844981664610270, 2.32444286707874308619545067967, 3.92748470922967557762775377781, 4.76123073111374433511542300960, 5.49479052150955976739072968344, 6.28499740097241640823993570310, 7.56239864962174165790575892292, 8.045205815811440097860446477827, 8.735229703157067682633599621954

Graph of the ZZ-function along the critical line