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Properties

Label 4-540800-1.1-c1e2-0-17
Degree 44
Conductor 540800540800
Sign 11
Analytic cond. 34.481834.4818
Root an. cond. 2.423242.42324
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 + 4-s + 2·5-s + 8-s − 5·9-s + 2·10-s − 5·11-s + 4·13-s + 16-s − 5·17-s − 5·18-s − 5·19-s + 2·20-s − 5·22-s + 10·23-s − 25-s + 4·26-s + 32-s − 5·34-s − 5·36-s + 9·37-s − 5·38-s + 2·40-s − 5·44-s − 10·45-s + 10·46-s + 10·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s − 5/3·9-s + 0.632·10-s − 1.50·11-s + 1.10·13-s + 1/4·16-s − 1.21·17-s − 1.17·18-s − 1.14·19-s + 0.447·20-s − 1.06·22-s + 2.08·23-s − 1/5·25-s + 0.784·26-s + 0.176·32-s − 0.857·34-s − 5/6·36-s + 1.47·37-s − 0.811·38-s + 0.316·40-s − 0.753·44-s − 1.49·45-s + 1.47·46-s + 10/7·49-s + ⋯

Functional equation

Λ(s)=(540800s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(540800s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 540800540800    =    27⋅52⋅1322^{7} \cdot 5^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 34.481834.4818
Root analytic conductor: 2.423242.42324
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 540800, ( :1/2,1/2), 1)(4,\ 540800,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6906917882.690691788
L(12)L(\frac12) ≈\approx 2.6906917882.690691788
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)
bad2C1C_1 1−T 1 - T
5C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
13C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good3C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2×\timesC2C_2 (1+pT2)(1+5T+pT2) ( 1 + p T^{2} )( 1 + 5 T + p T^{2} )
17C2C_2×\timesC2C_2 (1−2T+pT2)(1+7T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} )
19C2C_2×\timesC2C_2 (1−T+pT2)(1+6T+pT2) ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2×\timesC2C_2 (1−6T+pT2)(1−4T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} )
29C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
31C22C_2^2 1−37T2+p2T4 1 - 37 T^{2} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1−7T+pT2)(1−2T+pT2) ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} )
41C22C_2^2 1−7T2+p2T4 1 - 7 T^{2} + p^{2} T^{4}
43C22C_2^2 1+45T2+p2T4 1 + 45 T^{2} + p^{2} T^{4}
47C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
53C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
59C2C_2×\timesC2C_2 (1−9T+pT2)(1+4T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C22C_2^2 1−32T2+p2T4 1 - 32 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−8T+pT2)(1+2T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} )
71C22C_2^2 1+83T2+p2T4 1 + 83 T^{2} + p^{2} T^{4}
73C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−15T+pT2)(1−5T+pT2) ( 1 - 15 T + p T^{2} )( 1 - 5 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−16T+pT2)(1−6T+pT2) ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} )
89C22C_2^2 1−157T2+p2T4 1 - 157 T^{2} + p^{2} T^{4}
97C22C_2^2 1+170T2+p2T4 1 + 170 T^{2} + 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

−8.492336978829953026520466228567, −8.035586408751654110469652061173, −7.64791646569346460586478633013, −6.87627969783762347910928055543, −6.46796304996739428931435231301, −6.10610369765021072568123110118, −5.71551432890070618711992134228, −5.11604660689576680420305942249, −5.00495782937041374641078164791, −4.16879638878826673791512279611, −3.57173253019391291569431744188, −2.84690576720299293677237153014, −2.47526744211796281546010497613, −2.07464116870629392684308581672, −0.73033646217119766230883916812, 0.73033646217119766230883916812, 2.07464116870629392684308581672, 2.47526744211796281546010497613, 2.84690576720299293677237153014, 3.57173253019391291569431744188, 4.16879638878826673791512279611, 5.00495782937041374641078164791, 5.11604660689576680420305942249, 5.71551432890070618711992134228, 6.10610369765021072568123110118, 6.46796304996739428931435231301, 6.87627969783762347910928055543, 7.64791646569346460586478633013, 8.035586408751654110469652061173, 8.492336978829953026520466228567

Graph of the ZZ-function along the critical line