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Properties

Label 4-2548e2-1.1-c1e2-0-12
Degree 44
Conductor 64923046492304
Sign 11
Analytic cond. 413.954413.954
Root an. cond. 4.510644.51064
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 2·5-s + 21·9-s − 10·11-s − 2·13-s − 12·15-s − 3·17-s − 6·19-s + 23-s + 5·25-s + 54·27-s + 29-s + 8·31-s − 60·33-s − 3·37-s − 12·39-s − 3·41-s − 43-s − 42·45-s − 4·47-s − 18·51-s + 6·53-s + 20·55-s − 36·57-s − 5·59-s − 10·61-s + 4·65-s + ⋯
L(s)  = 1  + 3.46·3-s − 0.894·5-s + 7·9-s − 3.01·11-s − 0.554·13-s − 3.09·15-s − 0.727·17-s − 1.37·19-s + 0.208·23-s + 25-s + 10.3·27-s + 0.185·29-s + 1.43·31-s − 10.4·33-s − 0.493·37-s − 1.92·39-s − 0.468·41-s − 0.152·43-s − 6.26·45-s − 0.583·47-s − 2.52·51-s + 0.824·53-s + 2.69·55-s − 4.76·57-s − 0.650·59-s − 1.28·61-s + 0.496·65-s + ⋯

Functional equation

Λ(s)=(6492304s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6492304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6492304s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6492304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 64923046492304    =    24⋅74⋅1322^{4} \cdot 7^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 413.954413.954
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 6492304, ( :1/2,1/2), 1)(4,\ 6492304,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.9535045684.953504568
L(12)L(\frac12) ≈\approx 4.9535045684.953504568
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)
bad2 1 1
7 1 1
13C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good3C2C_2 (1−pT+pT2)2 ( 1 - p T + p T^{2} )^{2}
5C22C_2^2 1+2T−T2+2pT3+p2T4 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4}
11C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
17C22C_2^2 1+3T−8T2+3pT3+p2T4 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
19C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
23C22C_2^2 1−T−22T2−pT3+p2T4 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4}
29C22C_2^2 1−T−28T2−pT3+p2T4 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4}
31C22C_2^2 1−8T+33T2−8pT3+p2T4 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+3T−28T2+3pT3+p2T4 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+3T−32T2+3pT3+p2T4 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+T−42T2+pT3+p2T4 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4}
47C22C_2^2 1+4T−31T2+4pT3+p2T4 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−6T−17T2−6pT3+p2T4 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+5T−34T2+5pT3+p2T4 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4}
61C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
67C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
71C22C_2^2 1−11T+50T2−11pT3+p2T4 1 - 11 T + 50 T^{2} - 11 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+14T+123T2+14pT3+p2T4 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4}
79C2C_2 (1−17T+pT2)(1+13T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} )
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C22C_2^2 1−9T−8T2−9pT3+p2T4 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−T−96T2−pT3+p2T4 1 - T - 96 T^{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

−8.923781032136109116442443337297, −8.546453361308465543567623165293, −8.390234108052513344028370159535, −7.936638125327999264854198614958, −7.67852793221974912083613531525, −7.67701831751439767960377740049, −7.15112879071930192539196385682, −6.56897137624714705523716885909, −6.34209079238866235784715179709, −5.20981897835808206586528465806, −4.93748600024629697787379997784, −4.63590795681710940761514171071, −4.12969618109156314590874138866, −3.66027137218395960404644135829, −3.24271396632944491419866574990, −2.76188331593670224689884123995, −2.62352640094111975930607566512, −2.17413327539503297473762276189, −1.80764862383456807106165038138, −0.53352339913839038338613375499, 0.53352339913839038338613375499, 1.80764862383456807106165038138, 2.17413327539503297473762276189, 2.62352640094111975930607566512, 2.76188331593670224689884123995, 3.24271396632944491419866574990, 3.66027137218395960404644135829, 4.12969618109156314590874138866, 4.63590795681710940761514171071, 4.93748600024629697787379997784, 5.20981897835808206586528465806, 6.34209079238866235784715179709, 6.56897137624714705523716885909, 7.15112879071930192539196385682, 7.67701831751439767960377740049, 7.67852793221974912083613531525, 7.936638125327999264854198614958, 8.390234108052513344028370159535, 8.546453361308465543567623165293, 8.923781032136109116442443337297

Graph of the ZZ-function along the critical line