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Properties

Label 2-175-1.1-c3-0-17
Degree 22
Conductor 175175
Sign −1-1
Analytic cond. 10.325310.3253
Root an. cond. 3.213303.21330
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  − 5.41·2-s + 4.65·3-s + 21.3·4-s − 25.2·6-s + 7·7-s − 72.0·8-s − 5.31·9-s − 52.2·11-s + 99.2·12-s − 30.6·13-s − 37.8·14-s + 219.·16-s − 37.2·17-s + 28.7·18-s + 80.2·19-s + 32.5·21-s + 282.·22-s − 25.8·23-s − 335.·24-s + 165.·26-s − 150.·27-s + 149.·28-s + 20.9·29-s − 314.·31-s − 613.·32-s − 243.·33-s + 201.·34-s + ⋯
L(s)  = 1  − 1.91·2-s + 0.896·3-s + 2.66·4-s − 1.71·6-s + 0.377·7-s − 3.18·8-s − 0.196·9-s − 1.43·11-s + 2.38·12-s − 0.654·13-s − 0.723·14-s + 3.43·16-s − 0.531·17-s + 0.376·18-s + 0.968·19-s + 0.338·21-s + 2.74·22-s − 0.234·23-s − 2.85·24-s + 1.25·26-s − 1.07·27-s + 1.00·28-s + 0.134·29-s − 1.82·31-s − 3.38·32-s − 1.28·33-s + 1.01·34-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −1-1
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 175, ( :3/2), −1)(2,\ 175,\ (\ :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)
bad5 1 1
7 1−7T 1 - 7T
good2 1+5.41T+8T2 1 + 5.41T + 8T^{2}
3 1−4.65T+27T2 1 - 4.65T + 27T^{2}
11 1+52.2T+1.33e3T2 1 + 52.2T + 1.33e3T^{2}
13 1+30.6T+2.19e3T2 1 + 30.6T + 2.19e3T^{2}
17 1+37.2T+4.91e3T2 1 + 37.2T + 4.91e3T^{2}
19 1−80.2T+6.85e3T2 1 - 80.2T + 6.85e3T^{2}
23 1+25.8T+1.21e4T2 1 + 25.8T + 1.21e4T^{2}
29 1−20.9T+2.43e4T2 1 - 20.9T + 2.43e4T^{2}
31 1+314.T+2.97e4T2 1 + 314.T + 2.97e4T^{2}
37 1+197.T+5.06e4T2 1 + 197.T + 5.06e4T^{2}
41 1−11.3T+6.89e4T2 1 - 11.3T + 6.89e4T^{2}
43 1−33.8T+7.95e4T2 1 - 33.8T + 7.95e4T^{2}
47 1−361.T+1.03e5T2 1 - 361.T + 1.03e5T^{2}
53 1+153.T+1.48e5T2 1 + 153.T + 1.48e5T^{2}
59 1+616T+2.05e5T2 1 + 616T + 2.05e5T^{2}
61 1−15.2T+2.26e5T2 1 - 15.2T + 2.26e5T^{2}
67 1−166.T+3.00e5T2 1 - 166.T + 3.00e5T^{2}
71 1+952T+3.57e5T2 1 + 952T + 3.57e5T^{2}
73 1−148.T+3.89e5T2 1 - 148.T + 3.89e5T^{2}
79 1−857.T+4.93e5T2 1 - 857.T + 4.93e5T^{2}
83 1+660.T+5.71e5T2 1 + 660.T + 5.71e5T^{2}
89 1+45.7T+7.04e5T2 1 + 45.7T + 7.04e5T^{2}
97 1+1.68e3T+9.12e5T2 1 + 1.68e3T + 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

−11.28505152687293393368280636519, −10.49534608207034099304100668135, −9.495296333070899946858132608086, −8.733178217326145290307301973842, −7.82242308001519031268868003098, −7.27516159588567079769821899657, −5.55668888252589109026295256726, −2.99085211354196282624221812754, −1.99036491990729153464656318823, 0, 1.99036491990729153464656318823, 2.99085211354196282624221812754, 5.55668888252589109026295256726, 7.27516159588567079769821899657, 7.82242308001519031268868003098, 8.733178217326145290307301973842, 9.495296333070899946858132608086, 10.49534608207034099304100668135, 11.28505152687293393368280636519

Graph of the ZZ-function along the critical line