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Properties

Label 2-1680-1.1-c3-0-54
Degree 22
Conductor 16801680
Sign −1-1
Analytic cond. 99.123299.1232
Root an. cond. 9.956069.95606
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 5·5-s + 7·7-s + 9·9-s + 2.93·11-s − 19.0·13-s − 15·15-s + 122.·17-s − 107.·19-s − 21·21-s − 210.·23-s + 25·25-s − 27·27-s + 95.4·29-s + 94.3·31-s − 8.81·33-s + 35·35-s + 97.1·37-s + 57.1·39-s − 491.·41-s + 43.0·43-s + 45·45-s − 473.·47-s + 49·49-s − 367.·51-s − 183.·53-s + 14.6·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.0805·11-s − 0.406·13-s − 0.258·15-s + 1.74·17-s − 1.29·19-s − 0.218·21-s − 1.90·23-s + 0.200·25-s − 0.192·27-s + 0.611·29-s + 0.546·31-s − 0.0464·33-s + 0.169·35-s + 0.431·37-s + 0.234·39-s − 1.87·41-s + 0.152·43-s + 0.149·45-s − 1.46·47-s + 0.142·49-s − 1.00·51-s − 0.476·53-s + 0.0360·55-s + ⋯

Functional equation

Λ(s)=(1680s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1680s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16801680    =    24⋅3⋅5⋅72^{4} \cdot 3 \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 99.123299.1232
Root analytic conductor: 9.956069.95606
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1680, ( :3/2), −1)(2,\ 1680,\ (\ :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+3T 1 + 3T
5 1−5T 1 - 5T
7 1−7T 1 - 7T
good11 1−2.93T+1.33e3T2 1 - 2.93T + 1.33e3T^{2}
13 1+19.0T+2.19e3T2 1 + 19.0T + 2.19e3T^{2}
17 1−122.T+4.91e3T2 1 - 122.T + 4.91e3T^{2}
19 1+107.T+6.85e3T2 1 + 107.T + 6.85e3T^{2}
23 1+210.T+1.21e4T2 1 + 210.T + 1.21e4T^{2}
29 1−95.4T+2.43e4T2 1 - 95.4T + 2.43e4T^{2}
31 1−94.3T+2.97e4T2 1 - 94.3T + 2.97e4T^{2}
37 1−97.1T+5.06e4T2 1 - 97.1T + 5.06e4T^{2}
41 1+491.T+6.89e4T2 1 + 491.T + 6.89e4T^{2}
43 1−43.0T+7.95e4T2 1 - 43.0T + 7.95e4T^{2}
47 1+473.T+1.03e5T2 1 + 473.T + 1.03e5T^{2}
53 1+183.T+1.48e5T2 1 + 183.T + 1.48e5T^{2}
59 1−760.T+2.05e5T2 1 - 760.T + 2.05e5T^{2}
61 1+198.T+2.26e5T2 1 + 198.T + 2.26e5T^{2}
67 1−309.T+3.00e5T2 1 - 309.T + 3.00e5T^{2}
71 1+665.T+3.57e5T2 1 + 665.T + 3.57e5T^{2}
73 1−621.T+3.89e5T2 1 - 621.T + 3.89e5T^{2}
79 1−24.7T+4.93e5T2 1 - 24.7T + 4.93e5T^{2}
83 1−406.T+5.71e5T2 1 - 406.T + 5.71e5T^{2}
89 1−261.T+7.04e5T2 1 - 261.T + 7.04e5T^{2}
97 1+1.00e3T+9.12e5T2 1 + 1.00e3T + 9.12e5T^{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

−8.365873331690865021355234294023, −7.965628669218771443711135700562, −6.82254018549056952184892875912, −6.13295442324356444696925055783, −5.37020213120630974111840141763, −4.56279232557202726139426967714, −3.56151909403718124243672165284, −2.26767437987736129067043877648, −1.31435524573579029096390567214, 0, 1.31435524573579029096390567214, 2.26767437987736129067043877648, 3.56151909403718124243672165284, 4.56279232557202726139426967714, 5.37020213120630974111840141763, 6.13295442324356444696925055783, 6.82254018549056952184892875912, 7.965628669218771443711135700562, 8.365873331690865021355234294023

Graph of the ZZ-function along the critical line