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Properties

Label 2-174-1.1-c1-0-2
Degree 22
Conductor 174174
Sign 11
Analytic cond. 1.389391.38939
Root an. cond. 1.178721.17872
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-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s − 4·11-s + 12-s + 6·13-s + 2·15-s + 16-s − 2·17-s − 18-s + 4·19-s + 2·20-s + 4·22-s − 24-s − 25-s − 6·26-s + 27-s − 29-s − 2·30-s − 4·31-s − 32-s − 4·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s + 1.66·13-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.185·29-s − 0.365·30-s − 0.718·31-s − 0.176·32-s − 0.696·33-s + ⋯

Functional equation

Λ(s)=(174s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(174s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 174174    =    2⋅3⋅292 \cdot 3 \cdot 29
Sign: 11
Analytic conductor: 1.389391.38939
Root analytic conductor: 1.178721.17872
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 174, ( :1/2), 1)(2,\ 174,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1382223791.138222379
L(12)L(\frac12) ≈\approx 1.1382223791.138222379
L(32)L(\frac{3}{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)Isogeny Class over Fp\mathbf{F}_p
bad2 1+T 1 + T
3 1−T 1 - T
29 1+T 1 + T
good5 1−2T+pT2 1 - 2 T + p T^{2} 1.5.ac
7 1+pT2 1 + p T^{2} 1.7.a
11 1+4T+pT2 1 + 4 T + p T^{2} 1.11.e
13 1−6T+pT2 1 - 6 T + p T^{2} 1.13.ag
17 1+2T+pT2 1 + 2 T + p T^{2} 1.17.c
19 1−4T+pT2 1 - 4 T + p T^{2} 1.19.ae
23 1+pT2 1 + p T^{2} 1.23.a
31 1+4T+pT2 1 + 4 T + p T^{2} 1.31.e
37 1+6T+pT2 1 + 6 T + p T^{2} 1.37.g
41 1−6T+pT2 1 - 6 T + p T^{2} 1.41.ag
43 1+12T+pT2 1 + 12 T + p T^{2} 1.43.m
47 1+8T+pT2 1 + 8 T + p T^{2} 1.47.i
53 1+6T+pT2 1 + 6 T + p T^{2} 1.53.g
59 1−8T+pT2 1 - 8 T + p T^{2} 1.59.ai
61 1−10T+pT2 1 - 10 T + p T^{2} 1.61.ak
67 1+4T+pT2 1 + 4 T + p T^{2} 1.67.e
71 1+8T+pT2 1 + 8 T + p T^{2} 1.71.i
73 1−2T+pT2 1 - 2 T + p T^{2} 1.73.ac
79 1−4T+pT2 1 - 4 T + p T^{2} 1.79.ae
83 1+pT2 1 + p T^{2} 1.83.a
89 1−14T+pT2 1 - 14 T + p T^{2} 1.89.ao
97 1−18T+pT2 1 - 18 T + p T^{2} 1.97.as
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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

−13.08350822284660829527532209241, −11.46160956762346043433851090643, −10.51228751975807998983720994202, −9.664379188026696343396565275536, −8.703567572426077992686024733312, −7.83419862053560271569394084997, −6.51962651722825426517927501982, −5.35523044786598344694970550055, −3.32512902126658354016325680833, −1.81977103661582150142638563575, 1.81977103661582150142638563575, 3.32512902126658354016325680833, 5.35523044786598344694970550055, 6.51962651722825426517927501982, 7.83419862053560271569394084997, 8.703567572426077992686024733312, 9.664379188026696343396565275536, 10.51228751975807998983720994202, 11.46160956762346043433851090643, 13.08350822284660829527532209241

Graph of the ZZ-function along the critical line