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Properties

Label 2-3360-1.1-c1-0-28
Degree 22
Conductor 33603360
Sign −1-1
Analytic cond. 26.829726.8297
Root an. cond. 5.179745.17974
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s + 9-s − 2·11-s + 15-s − 6·17-s + 6·19-s + 21-s + 8·23-s + 25-s − 27-s + 6·29-s + 6·31-s + 2·33-s + 35-s − 10·37-s + 2·41-s + 4·43-s − 45-s − 8·47-s + 49-s + 6·51-s − 12·53-s + 2·55-s − 6·57-s − 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.258·15-s − 1.45·17-s + 1.37·19-s + 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.348·33-s + 0.169·35-s − 1.64·37-s + 0.312·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s − 1.64·53-s + 0.269·55-s − 0.794·57-s − 1.56·59-s + ⋯

Functional equation

Λ(s)=(3360s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3360s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33603360    =    25⋅3⋅5⋅72^{5} \cdot 3 \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 26.829726.8297
Root analytic conductor: 5.179745.17974
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3360, ( :1/2), −1)(2,\ 3360,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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+T 1 + T
5 1+T 1 + T
7 1+T 1 + T
good11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−16T+pT2 1 - 16 T + p 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

−8.220445638160044929496837312007, −7.37497321130478871197915841208, −6.76360727272757381454020390912, −6.09270049536733414349274513765, −4.92025339438886395369489047888, −4.72916901985484185209248139884, −3.39896233277134585610990366575, −2.71557781749145545205368216765, −1.24960858885676340498672394499, 0, 1.24960858885676340498672394499, 2.71557781749145545205368216765, 3.39896233277134585610990366575, 4.72916901985484185209248139884, 4.92025339438886395369489047888, 6.09270049536733414349274513765, 6.76360727272757381454020390912, 7.37497321130478871197915841208, 8.220445638160044929496837312007

Graph of the ZZ-function along the critical line