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Properties

Label 4-855e2-1.1-c1e2-0-1
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 4·5-s − 4·11-s + 5·16-s − 2·19-s − 12·20-s + 11·25-s − 12·29-s − 8·31-s − 4·41-s − 12·44-s + 10·49-s + 16·55-s − 20·61-s + 3·64-s + 16·71-s − 6·76-s + 16·79-s − 20·80-s + 20·89-s + 8·95-s + 33·100-s + 12·109-s − 36·116-s − 10·121-s − 24·124-s − 24·125-s + ⋯
L(s)  = 1  + 3/2·4-s − 1.78·5-s − 1.20·11-s + 5/4·16-s − 0.458·19-s − 2.68·20-s + 11/5·25-s − 2.22·29-s − 1.43·31-s − 0.624·41-s − 1.80·44-s + 10/7·49-s + 2.15·55-s − 2.56·61-s + 3/8·64-s + 1.89·71-s − 0.688·76-s + 1.80·79-s − 2.23·80-s + 2.11·89-s + 0.820·95-s + 3.29·100-s + 1.14·109-s − 3.34·116-s − 0.909·121-s − 2.15·124-s − 2.14·125-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 1.1693700541.169370054
L(12)L(\frac12) ≈\approx 1.1693700541.169370054
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+4T+pT2 1 + 4 T + p T^{2}
19C1C_1 (1+T)2 ( 1 + T )^{2}
good2C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
89C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
97C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
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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

−10.79825486131797391212962993649, −10.28861374002305199843058850931, −9.422755247394498177531821990707, −9.128616988427608653322791127689, −8.619296314052961747359287112903, −7.905911542458527280977634980468, −7.85720717402839161488270301940, −7.36596128272417927145266405831, −7.24480507230017162053532898912, −6.58309485197631130609406322773, −6.20532620623201522542730246763, −5.41736537745405259602797701921, −5.28509586863728655778752439984, −4.45431662823273891089033482771, −3.96246734792274788932724653669, −3.32898224247665400544275299909, −3.13914195681282117764075202651, −2.19232002144848551942770727194, −1.87447109145379178149974946641, −0.49308327442812810118920035591, 0.49308327442812810118920035591, 1.87447109145379178149974946641, 2.19232002144848551942770727194, 3.13914195681282117764075202651, 3.32898224247665400544275299909, 3.96246734792274788932724653669, 4.45431662823273891089033482771, 5.28509586863728655778752439984, 5.41736537745405259602797701921, 6.20532620623201522542730246763, 6.58309485197631130609406322773, 7.24480507230017162053532898912, 7.36596128272417927145266405831, 7.85720717402839161488270301940, 7.905911542458527280977634980468, 8.619296314052961747359287112903, 9.128616988427608653322791127689, 9.422755247394498177531821990707, 10.28861374002305199843058850931, 10.79825486131797391212962993649

Graph of the ZZ-function along the critical line