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Properties

Label 2-1710-1.1-c3-0-66
Degree 22
Conductor 17101710
Sign −1-1
Analytic cond. 100.893100.893
Root an. cond. 10.044510.0445
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  − 2·2-s + 4·4-s + 5·5-s + 22.2·7-s − 8·8-s − 10·10-s − 41.4·11-s − 40.8·13-s − 44.5·14-s + 16·16-s + 61.7·17-s + 19·19-s + 20·20-s + 82.9·22-s − 85.4·23-s + 25·25-s + 81.7·26-s + 89.1·28-s + 46.4·29-s + 196.·31-s − 32·32-s − 123.·34-s + 111.·35-s − 278.·37-s − 38·38-s − 40·40-s − 344.·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.20·7-s − 0.353·8-s − 0.316·10-s − 1.13·11-s − 0.872·13-s − 0.851·14-s + 0.250·16-s + 0.881·17-s + 0.229·19-s + 0.223·20-s + 0.803·22-s − 0.774·23-s + 0.200·25-s + 0.616·26-s + 0.601·28-s + 0.297·29-s + 1.13·31-s − 0.176·32-s − 0.623·34-s + 0.538·35-s − 1.23·37-s − 0.162·38-s − 0.158·40-s − 1.31·41-s + ⋯

Functional equation

Λ(s)=(1710s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1710s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17101710    =    2⋅32⋅5⋅192 \cdot 3^{2} \cdot 5 \cdot 19
Sign: −1-1
Analytic conductor: 100.893100.893
Root analytic conductor: 10.044510.0445
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1710, ( :3/2), −1)(2,\ 1710,\ (\ :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+2T 1 + 2T
3 1 1
5 1−5T 1 - 5T
19 1−19T 1 - 19T
good7 1−22.2T+343T2 1 - 22.2T + 343T^{2}
11 1+41.4T+1.33e3T2 1 + 41.4T + 1.33e3T^{2}
13 1+40.8T+2.19e3T2 1 + 40.8T + 2.19e3T^{2}
17 1−61.7T+4.91e3T2 1 - 61.7T + 4.91e3T^{2}
23 1+85.4T+1.21e4T2 1 + 85.4T + 1.21e4T^{2}
29 1−46.4T+2.43e4T2 1 - 46.4T + 2.43e4T^{2}
31 1−196.T+2.97e4T2 1 - 196.T + 2.97e4T^{2}
37 1+278.T+5.06e4T2 1 + 278.T + 5.06e4T^{2}
41 1+344.T+6.89e4T2 1 + 344.T + 6.89e4T^{2}
43 1−359.T+7.95e4T2 1 - 359.T + 7.95e4T^{2}
47 1+407.T+1.03e5T2 1 + 407.T + 1.03e5T^{2}
53 1+172.T+1.48e5T2 1 + 172.T + 1.48e5T^{2}
59 1+526.T+2.05e5T2 1 + 526.T + 2.05e5T^{2}
61 1−130.T+2.26e5T2 1 - 130.T + 2.26e5T^{2}
67 1+395.T+3.00e5T2 1 + 395.T + 3.00e5T^{2}
71 1+1.16e3T+3.57e5T2 1 + 1.16e3T + 3.57e5T^{2}
73 1−814.T+3.89e5T2 1 - 814.T + 3.89e5T^{2}
79 1+229.T+4.93e5T2 1 + 229.T + 4.93e5T^{2}
83 1+320.T+5.71e5T2 1 + 320.T + 5.71e5T^{2}
89 1−1.27e3T+7.04e5T2 1 - 1.27e3T + 7.04e5T^{2}
97 1+1.46e3T+9.12e5T2 1 + 1.46e3T + 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

−8.410098131734911788285138532899, −7.897323498164370058845847925549, −7.29606029689753142337700952453, −6.17770223496622092397089061326, −5.25141601203826296740342128421, −4.70076143429152189215957821873, −3.15631627204239547582789495389, −2.21093885036388537804670013407, −1.34413016530373120416242598234, 0, 1.34413016530373120416242598234, 2.21093885036388537804670013407, 3.15631627204239547582789495389, 4.70076143429152189215957821873, 5.25141601203826296740342128421, 6.17770223496622092397089061326, 7.29606029689753142337700952453, 7.897323498164370058845847925549, 8.410098131734911788285138532899

Graph of the ZZ-function along the critical line