Properties

Label 2-170-1.1-c1-0-3
Degree 22
Conductor 170170
Sign 11
Analytic cond. 1.357451.35745
Root an. cond. 1.165091.16509
Motivic weight 11
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 + 3·3-s + 4-s − 5-s − 3·6-s + 2·7-s − 8-s + 6·9-s + 10-s − 4·11-s + 3·12-s − 3·13-s − 2·14-s − 3·15-s + 16-s + 17-s − 6·18-s + 3·19-s − 20-s + 6·21-s + 4·22-s − 6·23-s − 3·24-s + 25-s + 3·26-s + 9·27-s + 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s + 0.755·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 1.20·11-s + 0.866·12-s − 0.832·13-s − 0.534·14-s − 0.774·15-s + 1/4·16-s + 0.242·17-s − 1.41·18-s + 0.688·19-s − 0.223·20-s + 1.30·21-s + 0.852·22-s − 1.25·23-s − 0.612·24-s + 1/5·25-s + 0.588·26-s + 1.73·27-s + 0.377·28-s + ⋯

Functional equation

Λ(s)=(170s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(170s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 170170    =    25172 \cdot 5 \cdot 17
Sign: 11
Analytic conductor: 1.357451.35745
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 170, ( :1/2), 1)(2,\ 170,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 1.3360256741.336025674
L(12)L(\frac12) \approx 1.3360256741.336025674
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+T 1 + T
5 1+T 1 + T
17 1T 1 - T
good3 1pT+pT2 1 - p T + p T^{2}
7 12T+pT2 1 - 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+3T+pT2 1 + 3 T + p T^{2}
19 13T+pT2 1 - 3 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 19T+pT2 1 - 9 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 16T+pT2 1 - 6 T + p T^{2}
47 1+13T+pT2 1 + 13 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 115T+pT2 1 - 15 T + p T^{2}
61 17T+pT2 1 - 7 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 19T+pT2 1 - 9 T + p T^{2}
73 1+3T+pT2 1 + 3 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 112T+pT2 1 - 12 T + p T^{2}
89 1+9T+pT2 1 + 9 T + p T^{2}
97 17T+pT2 1 - 7 T + p T^{2}
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   L(s)=p j=12(1αj,pps)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

−12.83373639226606959528283639348, −11.77392021354054025159681104411, −10.36366338935225999926837372006, −9.673175371223344610087496327723, −8.338175486309026415938497793980, −8.047148785250349636560357968225, −7.12227398289357983628483830063, −4.92015064777793909582073169694, −3.29460314858197068359451412690, −2.08881038737404835754425957871, 2.08881038737404835754425957871, 3.29460314858197068359451412690, 4.92015064777793909582073169694, 7.12227398289357983628483830063, 8.047148785250349636560357968225, 8.338175486309026415938497793980, 9.673175371223344610087496327723, 10.36366338935225999926837372006, 11.77392021354054025159681104411, 12.83373639226606959528283639348

Graph of the ZZ-function along the critical line