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Properties

Label 2-175-1.1-c3-0-10
Degree 22
Conductor 175175
Sign 11
Analytic cond. 10.325310.3253
Root an. cond. 3.213303.21330
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 8·3-s − 7·4-s − 8·6-s − 7·7-s + 15·8-s + 37·9-s + 12·11-s − 56·12-s + 78·13-s + 7·14-s + 41·16-s + 94·17-s − 37·18-s + 40·19-s − 56·21-s − 12·22-s − 32·23-s + 120·24-s − 78·26-s + 80·27-s + 49·28-s − 50·29-s − 248·31-s − 161·32-s + 96·33-s − 94·34-s + ⋯
L(s)  = 1  − 0.353·2-s + 1.53·3-s − 7/8·4-s − 0.544·6-s − 0.377·7-s + 0.662·8-s + 1.37·9-s + 0.328·11-s − 1.34·12-s + 1.66·13-s + 0.133·14-s + 0.640·16-s + 1.34·17-s − 0.484·18-s + 0.482·19-s − 0.581·21-s − 0.116·22-s − 0.290·23-s + 1.02·24-s − 0.588·26-s + 0.570·27-s + 0.330·28-s − 0.320·29-s − 1.43·31-s − 0.889·32-s + 0.506·33-s − 0.474·34-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: 11
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), 1)(2,\ 175,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.0954449402.095444940
L(12)L(\frac12) ≈\approx 2.0954449402.095444940
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+pT 1 + p T
good2 1+T+p3T2 1 + T + p^{3} T^{2}
3 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
11 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
13 1−6pT+p3T2 1 - 6 p T + p^{3} T^{2}
17 1−94T+p3T2 1 - 94 T + p^{3} T^{2}
19 1−40T+p3T2 1 - 40 T + p^{3} T^{2}
23 1+32T+p3T2 1 + 32 T + p^{3} T^{2}
29 1+50T+p3T2 1 + 50 T + p^{3} T^{2}
31 1+8pT+p3T2 1 + 8 p T + p^{3} T^{2}
37 1−434T+p3T2 1 - 434 T + p^{3} T^{2}
41 1−402T+p3T2 1 - 402 T + p^{3} T^{2}
43 1−68T+p3T2 1 - 68 T + p^{3} T^{2}
47 1+536T+p3T2 1 + 536 T + p^{3} T^{2}
53 1+22T+p3T2 1 + 22 T + p^{3} T^{2}
59 1+560T+p3T2 1 + 560 T + p^{3} T^{2}
61 1+278T+p3T2 1 + 278 T + p^{3} T^{2}
67 1−164T+p3T2 1 - 164 T + p^{3} T^{2}
71 1−672T+p3T2 1 - 672 T + p^{3} T^{2}
73 1+82T+p3T2 1 + 82 T + p^{3} T^{2}
79 1+1000T+p3T2 1 + 1000 T + p^{3} T^{2}
83 1−448T+p3T2 1 - 448 T + p^{3} T^{2}
89 1+870T+p3T2 1 + 870 T + p^{3} T^{2}
97 1+1026T+p3T2 1 + 1026 T + p^{3} 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

−12.69551318958635373038341377079, −11.03883557541485018602384750904, −9.694672046690131244583399120674, −9.258248143118897162077758202479, −8.268671407679627604996565551323, −7.60368248294959681558600152727, −5.85222442009799353217035975865, −4.04989709957727718247483837114, −3.25436067910591054760461208736, −1.30528193135988779285210397147, 1.30528193135988779285210397147, 3.25436067910591054760461208736, 4.04989709957727718247483837114, 5.85222442009799353217035975865, 7.60368248294959681558600152727, 8.268671407679627604996565551323, 9.258248143118897162077758202479, 9.694672046690131244583399120674, 11.03883557541485018602384750904, 12.69551318958635373038341377079

Graph of the ZZ-function along the critical line