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Properties

Label 2-175-1.1-c3-0-21
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  − 1.55·2-s + 4.96·3-s − 5.58·4-s − 7.71·6-s − 7·7-s + 21.1·8-s − 2.38·9-s + 29.8·11-s − 27.6·12-s − 90.7·13-s + 10.8·14-s + 11.7·16-s − 29.5·17-s + 3.70·18-s − 62.3·19-s − 34.7·21-s − 46.3·22-s − 90.6·23-s + 104.·24-s + 141.·26-s − 145.·27-s + 39.0·28-s + 193.·29-s − 152.·31-s − 187.·32-s + 147.·33-s + 45.9·34-s + ⋯
L(s)  = 1  − 0.549·2-s + 0.954·3-s − 0.697·4-s − 0.525·6-s − 0.377·7-s + 0.933·8-s − 0.0882·9-s + 0.817·11-s − 0.666·12-s − 1.93·13-s + 0.207·14-s + 0.184·16-s − 0.421·17-s + 0.0485·18-s − 0.752·19-s − 0.360·21-s − 0.449·22-s − 0.821·23-s + 0.891·24-s + 1.06·26-s − 1.03·27-s + 0.263·28-s + 1.23·29-s − 0.881·31-s − 1.03·32-s + 0.780·33-s + 0.232·34-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+1.55T+8T2 1 + 1.55T + 8T^{2}
3 1−4.96T+27T2 1 - 4.96T + 27T^{2}
11 1−29.8T+1.33e3T2 1 - 29.8T + 1.33e3T^{2}
13 1+90.7T+2.19e3T2 1 + 90.7T + 2.19e3T^{2}
17 1+29.5T+4.91e3T2 1 + 29.5T + 4.91e3T^{2}
19 1+62.3T+6.85e3T2 1 + 62.3T + 6.85e3T^{2}
23 1+90.6T+1.21e4T2 1 + 90.6T + 1.21e4T^{2}
29 1−193.T+2.43e4T2 1 - 193.T + 2.43e4T^{2}
31 1+152.T+2.97e4T2 1 + 152.T + 2.97e4T^{2}
37 1+102.T+5.06e4T2 1 + 102.T + 5.06e4T^{2}
41 1+266.T+6.89e4T2 1 + 266.T + 6.89e4T^{2}
43 1+387.T+7.95e4T2 1 + 387.T + 7.95e4T^{2}
47 1−152.T+1.03e5T2 1 - 152.T + 1.03e5T^{2}
53 1−81.5T+1.48e5T2 1 - 81.5T + 1.48e5T^{2}
59 1+235.T+2.05e5T2 1 + 235.T + 2.05e5T^{2}
61 1−510.T+2.26e5T2 1 - 510.T + 2.26e5T^{2}
67 1−347.T+3.00e5T2 1 - 347.T + 3.00e5T^{2}
71 1−317.T+3.57e5T2 1 - 317.T + 3.57e5T^{2}
73 1−709.T+3.89e5T2 1 - 709.T + 3.89e5T^{2}
79 1−1.06e3T+4.93e5T2 1 - 1.06e3T + 4.93e5T^{2}
83 1+503.T+5.71e5T2 1 + 503.T + 5.71e5T^{2}
89 1−482.T+7.04e5T2 1 - 482.T + 7.04e5T^{2}
97 1+481.T+9.12e5T2 1 + 481.T + 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

−11.87090338807152925565847919254, −10.29523773581558706612515267513, −9.556645178497382588969375651863, −8.783426614830285971857754400643, −7.936645098058105761761755501884, −6.76912014646939290192206236008, −4.99134747386727011341159491398, −3.73887235298642340680868516013, −2.18617878655924310818044910082, 0, 2.18617878655924310818044910082, 3.73887235298642340680868516013, 4.99134747386727011341159491398, 6.76912014646939290192206236008, 7.936645098058105761761755501884, 8.783426614830285971857754400643, 9.556645178497382588969375651863, 10.29523773581558706612515267513, 11.87090338807152925565847919254

Graph of the ZZ-function along the critical line