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Properties

Label 4-1091-1.1-c1e2-0-0
Degree 44
Conductor 10911091
Sign −1-1
Analytic cond. 0.06956310.0695631
Root an. cond. 0.5135640.513564
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  − 2·2-s − 3·3-s + 4-s − 3·5-s + 6·6-s + 3·9-s + 6·10-s − 3·11-s − 3·12-s + 5·13-s + 9·15-s + 16-s − 6·18-s − 8·19-s − 3·20-s + 6·22-s + 23-s + 4·25-s − 10·26-s − 4·29-s − 18·30-s − 31-s + 2·32-s + 9·33-s + 3·36-s − 7·37-s + 16·38-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s + 9-s + 1.89·10-s − 0.904·11-s − 0.866·12-s + 1.38·13-s + 2.32·15-s + 1/4·16-s − 1.41·18-s − 1.83·19-s − 0.670·20-s + 1.27·22-s + 0.208·23-s + 4/5·25-s − 1.96·26-s − 0.742·29-s − 3.28·30-s − 0.179·31-s + 0.353·32-s + 1.56·33-s + 1/2·36-s − 1.15·37-s + 2.59·38-s + ⋯

Functional equation

Λ(s)=(1091s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1091 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1091s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1091 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 10911091
Sign: −1-1
Analytic conductor: 0.06956310.0695631
Root analytic conductor: 0.5135640.513564
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1091, ( :1/2,1/2), −1)(4,\ 1091,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad1091C1C_1×\timesC2C_2 (1−T)(1+21T+pT2) ( 1 - T )( 1 + 21 T + p T^{2} )
good2D4D_{4} 1+pT+3T2+p2T3+p2T4 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}
3C2C_2 (1+pT2)(1+pT+pT2) ( 1 + p T^{2} )( 1 + p T + p T^{2} )
5D4D_{4} 1+3T+pT2+3pT3+p2T4 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4}
7C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
11D4D_{4} 1+3T+10T2+3pT3+p2T4 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4}
13D4D_{4} 1−5T+19T2−5pT3+p2T4 1 - 5 T + 19 T^{2} - 5 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
19D4D_{4} 1+8T+52T2+8pT3+p2T4 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1−5T+pT2)(1+4T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} )
29D4D_{4} 1+4T+18T2+4pT3+p2T4 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+T−42T2+pT3+p2T4 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4}
37D4D_{4} 1+7T+58T2+7pT3+p2T4 1 + 7 T + 58 T^{2} + 7 p T^{3} + p^{2} T^{4}
41C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+pT2)(1+5T+pT2) ( 1 + p T^{2} )( 1 + 5 T + p T^{2} )
47D4D_{4} 1+2T+60T2+2pT3+p2T4 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−12T+134T2−12pT3+p2T4 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+3T+83T2+3pT3+p2T4 1 + 3 T + 83 T^{2} + 3 p T^{3} + p^{2} T^{4}
61C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
67D4D_{4} 1−7T+109T2−7pT3+p2T4 1 - 7 T + 109 T^{2} - 7 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
73D4D_{4} 1−T+85T2−pT3+p2T4 1 - T + 85 T^{2} - p T^{3} + p^{2} T^{4}
79D4D_{4} 1−3T−45T2−3pT3+p2T4 1 - 3 T - 45 T^{2} - 3 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−T−86T2−pT3+p2T4 1 - T - 86 T^{2} - p T^{3} + p^{2} T^{4}
89D4D_{4} 1+2T−24T2+2pT3+p2T4 1 + 2 T - 24 T^{2} + 2 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−2T−2T2−2pT3+p2T4 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−19.8087339679, −19.2927698249, −18.7588261401, −18.4252657333, −17.9325643653, −17.3940434538, −16.8904747008, −16.4983732150, −15.8702950967, −15.3577164109, −14.8289717289, −13.5671344301, −12.9058210635, −12.2374732596, −11.5873611136, −11.0546519721, −10.7201337516, −10.0391382416, −8.86324040146, −8.45331855624, −7.82014337708, −6.75157269231, −5.99604896663, −5.03981995913, −3.82540183589, 0, 3.82540183589, 5.03981995913, 5.99604896663, 6.75157269231, 7.82014337708, 8.45331855624, 8.86324040146, 10.0391382416, 10.7201337516, 11.0546519721, 11.5873611136, 12.2374732596, 12.9058210635, 13.5671344301, 14.8289717289, 15.3577164109, 15.8702950967, 16.4983732150, 16.8904747008, 17.3940434538, 17.9325643653, 18.4252657333, 18.7588261401, 19.2927698249, 19.8087339679

Graph of the ZZ-function along the critical line