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Properties

Label 4-855e2-1.1-c1e2-0-12
Degree 44
Conductor 731025731025
Sign 11
Analytic cond. 46.610746.6107
Root an. cond. 2.612892.61289
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  + 2·4-s + 5-s + 4·7-s + 6·11-s + 4·13-s − 7·19-s + 2·20-s − 6·23-s + 8·28-s + 3·29-s + 10·31-s + 4·35-s + 16·37-s − 6·41-s + 4·43-s + 12·44-s + 6·47-s − 2·49-s + 8·52-s − 6·53-s + 6·55-s − 9·59-s + 7·61-s − 8·64-s + 4·65-s − 2·67-s − 9·71-s + ⋯
L(s)  = 1  + 4-s + 0.447·5-s + 1.51·7-s + 1.80·11-s + 1.10·13-s − 1.60·19-s + 0.447·20-s − 1.25·23-s + 1.51·28-s + 0.557·29-s + 1.79·31-s + 0.676·35-s + 2.63·37-s − 0.937·41-s + 0.609·43-s + 1.80·44-s + 0.875·47-s − 2/7·49-s + 1.10·52-s − 0.824·53-s + 0.809·55-s − 1.17·59-s + 0.896·61-s − 64-s + 0.496·65-s − 0.244·67-s − 1.06·71-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 731025731025    =    34⋅52⋅1923^{4} \cdot 5^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 46.610746.6107
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 731025, ( :1/2,1/2), 1)(4,\ 731025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.2699688904.269968890
L(12)L(\frac12) ≈\approx 4.2699688904.269968890
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)
bad3 1 1
5C2C_2 1−T+T2 1 - T + T^{2}
19C2C_2 1+7T+pT2 1 + 7 T + p T^{2}
good2C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
7C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
13C22C_2^2 1−4T+3T2−4pT3+p2T4 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
23C22C_2^2 1+6T+13T2+6pT3+p2T4 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−3T−20T2−3pT3+p2T4 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4}
31C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C22C_2^2 1+6T−5T2+6pT3+p2T4 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−4T−27T2−4pT3+p2T4 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−6T−11T2−6pT3+p2T4 1 - 6 T - 11 T^{2} - 6 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+9T+22T2+9pT3+p2T4 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−7T−12T2−7pT3+p2T4 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+2T−63T2+2pT3+p2T4 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+9T+10T2+9pT3+p2T4 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4}
73C22C_2^2 1−4T−57T2−4pT3+p2T4 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−7T−30T2−7pT3+p2T4 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1−3T−80T2−3pT3+p2T4 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−10T+3T2−10pT3+p2T4 1 - 10 T + 3 T^{2} - 10 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

−10.37802296429431719248202563402, −10.16143255715236096694498936611, −9.463538113949743957476755824073, −9.130161875272418961155897558854, −8.592302065686632975588167916371, −8.323181365297273262509568131784, −7.84503221359509373921561065675, −7.55507865849627031912550530546, −6.70377090797413947112980043411, −6.36209800037672362467075038845, −6.20669072455171087582507040555, −5.94818136233015567114212422462, −4.86305315376714423181502804439, −4.59648242966108528162028824072, −4.09549348528459852481621896174, −3.66166160214414539189139822770, −2.63261284754577781174387588484, −2.26097898184724496658355889148, −1.51536073064056676997525217643, −1.19796066188443671074681822521, 1.19796066188443671074681822521, 1.51536073064056676997525217643, 2.26097898184724496658355889148, 2.63261284754577781174387588484, 3.66166160214414539189139822770, 4.09549348528459852481621896174, 4.59648242966108528162028824072, 4.86305315376714423181502804439, 5.94818136233015567114212422462, 6.20669072455171087582507040555, 6.36209800037672362467075038845, 6.70377090797413947112980043411, 7.55507865849627031912550530546, 7.84503221359509373921561065675, 8.323181365297273262509568131784, 8.592302065686632975588167916371, 9.130161875272418961155897558854, 9.463538113949743957476755824073, 10.16143255715236096694498936611, 10.37802296429431719248202563402

Graph of the ZZ-function along the critical line