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Properties

Label 2-171-1.1-c1-0-4
Degree 22
Conductor 171171
Sign 11
Analytic cond. 1.365441.36544
Root an. cond. 1.168521.16852
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  + 2·2-s + 2·4-s − 5-s + 3·7-s − 2·10-s + 3·11-s − 6·13-s + 6·14-s − 4·16-s − 3·17-s − 19-s − 2·20-s + 6·22-s − 4·23-s − 4·25-s − 12·26-s + 6·28-s + 10·29-s + 2·31-s − 8·32-s − 6·34-s − 3·35-s + 8·37-s − 2·38-s + 8·41-s − 43-s + 6·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.447·5-s + 1.13·7-s − 0.632·10-s + 0.904·11-s − 1.66·13-s + 1.60·14-s − 16-s − 0.727·17-s − 0.229·19-s − 0.447·20-s + 1.27·22-s − 0.834·23-s − 4/5·25-s − 2.35·26-s + 1.13·28-s + 1.85·29-s + 0.359·31-s − 1.41·32-s − 1.02·34-s − 0.507·35-s + 1.31·37-s − 0.324·38-s + 1.24·41-s − 0.152·43-s + 0.904·44-s + ⋯

Functional equation

Λ(s)=(171s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(171s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171171    =    32⋅193^{2} \cdot 19
Sign: 11
Analytic conductor: 1.365441.36544
Root analytic conductor: 1.168521.16852
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 171, ( :1/2), 1)(2,\ 171,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1401740372.140174037
L(12)L(\frac12) ≈\approx 2.1401740372.140174037
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
bad3 1 1
19 1+T 1 + T
good2 1−pT+pT2 1 - p T + p T^{2} 1.2.ac
5 1+T+pT2 1 + T + p T^{2} 1.5.b
7 1−3T+pT2 1 - 3 T + p T^{2} 1.7.ad
11 1−3T+pT2 1 - 3 T + p T^{2} 1.11.ad
13 1+6T+pT2 1 + 6 T + p T^{2} 1.13.g
17 1+3T+pT2 1 + 3 T + p T^{2} 1.17.d
23 1+4T+pT2 1 + 4 T + p T^{2} 1.23.e
29 1−10T+pT2 1 - 10 T + p T^{2} 1.29.ak
31 1−2T+pT2 1 - 2 T + p T^{2} 1.31.ac
37 1−8T+pT2 1 - 8 T + p T^{2} 1.37.ai
41 1−8T+pT2 1 - 8 T + p T^{2} 1.41.ai
43 1+T+pT2 1 + T + p T^{2} 1.43.b
47 1+3T+pT2 1 + 3 T + p T^{2} 1.47.d
53 1−6T+pT2 1 - 6 T + p T^{2} 1.53.ag
59 1+pT2 1 + p T^{2} 1.59.a
61 1−7T+pT2 1 - 7 T + p T^{2} 1.61.ah
67 1−8T+pT2 1 - 8 T + p T^{2} 1.67.ai
71 1+12T+pT2 1 + 12 T + p T^{2} 1.71.m
73 1+11T+pT2 1 + 11 T + p T^{2} 1.73.l
79 1+pT2 1 + p T^{2} 1.79.a
83 1+4T+pT2 1 + 4 T + p T^{2} 1.83.e
89 1+10T+pT2 1 + 10 T + p T^{2} 1.89.k
97 1+2T+pT2 1 + 2 T + p T^{2} 1.97.c
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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

−12.70402288938109990188600775185, −11.84438281704762988883998671737, −11.42285722847365155593370717983, −9.886397115513209246726221997818, −8.513296718878561122779255009178, −7.33179869161333338136909211767, −6.12831923497636436478284774657, −4.75227634922127063518685270995, −4.20883425493377341015603476651, −2.44031011733442394942262196334, 2.44031011733442394942262196334, 4.20883425493377341015603476651, 4.75227634922127063518685270995, 6.12831923497636436478284774657, 7.33179869161333338136909211767, 8.513296718878561122779255009178, 9.886397115513209246726221997818, 11.42285722847365155593370717983, 11.84438281704762988883998671737, 12.70402288938109990188600775185

Graph of the ZZ-function along the critical line