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Properties

Label 2-158-1.1-c1-0-6
Degree 22
Conductor 158158
Sign −1-1
Analytic cond. 1.261631.26163
Root an. cond. 1.123221.12322
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 4-s − 3·5-s − 3·6-s − 3·7-s + 8-s + 6·9-s − 3·10-s − 2·11-s − 3·12-s − 5·13-s − 3·14-s + 9·15-s + 16-s + 6·17-s + 6·18-s − 3·20-s + 9·21-s − 2·22-s − 2·23-s − 3·24-s + 4·25-s − 5·26-s − 9·27-s − 3·28-s + 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s − 1.22·6-s − 1.13·7-s + 0.353·8-s + 2·9-s − 0.948·10-s − 0.603·11-s − 0.866·12-s − 1.38·13-s − 0.801·14-s + 2.32·15-s + 1/4·16-s + 1.45·17-s + 1.41·18-s − 0.670·20-s + 1.96·21-s − 0.426·22-s − 0.417·23-s − 0.612·24-s + 4/5·25-s − 0.980·26-s − 1.73·27-s − 0.566·28-s + 1.11·29-s + ⋯

Functional equation

Λ(s)=(158s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(158s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 158158    =    2⋅792 \cdot 79
Sign: −1-1
Analytic conductor: 1.261631.26163
Root analytic conductor: 1.123221.12322
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 158, ( :1/2), −1)(2,\ 158,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
79 1−T 1 - T
good3 1+pT+pT2 1 + p T + p T^{2} 1.3.d
5 1+3T+pT2 1 + 3 T + p T^{2} 1.5.d
7 1+3T+pT2 1 + 3 T + p T^{2} 1.7.d
11 1+2T+pT2 1 + 2 T + p T^{2} 1.11.c
13 1+5T+pT2 1 + 5 T + p T^{2} 1.13.f
17 1−6T+pT2 1 - 6 T + p T^{2} 1.17.ag
19 1+pT2 1 + p T^{2} 1.19.a
23 1+2T+pT2 1 + 2 T + p T^{2} 1.23.c
29 1−6T+pT2 1 - 6 T + p T^{2} 1.29.ag
31 1+10T+pT2 1 + 10 T + p T^{2} 1.31.k
37 1+10T+pT2 1 + 10 T + p T^{2} 1.37.k
41 1−2T+pT2 1 - 2 T + p T^{2} 1.41.ac
43 1−4T+pT2 1 - 4 T + p T^{2} 1.43.ae
47 1+3T+pT2 1 + 3 T + p T^{2} 1.47.d
53 1+12T+pT2 1 + 12 T + p T^{2} 1.53.m
59 1+T+pT2 1 + T + p T^{2} 1.59.b
61 1−12T+pT2 1 - 12 T + p T^{2} 1.61.am
67 1+8T+pT2 1 + 8 T + p T^{2} 1.67.i
71 1+3T+pT2 1 + 3 T + p T^{2} 1.71.d
73 1+6T+pT2 1 + 6 T + p T^{2} 1.73.g
83 1−14T+pT2 1 - 14 T + p T^{2} 1.83.ao
89 1+7T+pT2 1 + 7 T + p T^{2} 1.89.h
97 1+11T+pT2 1 + 11 T + p T^{2} 1.97.l
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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

−12.34878022289235219551388536645, −11.76229819783503690986272534976, −10.67275650162718779873054836109, −9.871463690263605580552418845308, −7.66138306465432851890178277032, −6.92116149525582118764831835825, −5.69924671275910195432130153357, −4.76932431908835403735443234199, −3.43543530646035424633948117891, 0, 3.43543530646035424633948117891, 4.76932431908835403735443234199, 5.69924671275910195432130153357, 6.92116149525582118764831835825, 7.66138306465432851890178277032, 9.871463690263605580552418845308, 10.67275650162718779873054836109, 11.76229819783503690986272534976, 12.34878022289235219551388536645

Graph of the ZZ-function along the critical line