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Properties

Label 4-480e2-1.1-c3e2-0-4
Degree 44
Conductor 230400230400
Sign 11
Analytic cond. 802.074802.074
Root an. cond. 5.321745.32174
Motivic weight 33
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  + 6·3-s − 10·5-s + 4·7-s + 27·9-s + 40·11-s − 24·13-s − 60·15-s + 32·17-s + 116·19-s + 24·21-s − 12·23-s + 75·25-s + 108·27-s + 148·29-s + 76·31-s + 240·33-s − 40·35-s + 280·37-s − 144·39-s + 572·41-s + 520·43-s − 270·45-s − 340·47-s + 130·49-s + 192·51-s + 524·53-s − 400·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 0.215·7-s + 9-s + 1.09·11-s − 0.512·13-s − 1.03·15-s + 0.456·17-s + 1.40·19-s + 0.249·21-s − 0.108·23-s + 3/5·25-s + 0.769·27-s + 0.947·29-s + 0.440·31-s + 1.26·33-s − 0.193·35-s + 1.24·37-s − 0.591·39-s + 2.17·41-s + 1.84·43-s − 0.894·45-s − 1.05·47-s + 0.379·49-s + 0.527·51-s + 1.35·53-s − 0.980·55-s + ⋯

Functional equation

Λ(s)=(230400s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(230400s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 230400230400    =    210⋅32⋅522^{10} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 802.074802.074
Root analytic conductor: 5.321745.32174
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 230400, ( :3/2,3/2), 1)(4,\ 230400,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.3607066365.360706636
L(12)L(\frac12) ≈\approx 5.3607066365.360706636
L(52)L(\frac{5}{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
3C1C_1 (1−pT)2 ( 1 - p T )^{2}
5C1C_1 (1+pT)2 ( 1 + p T )^{2}
good7D4D_{4} 1−4T−114T2−4p3T3+p6T4 1 - 4 T - 114 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}
11C2C_2 (1−20T+p3T2)2 ( 1 - 20 T + p^{3} T^{2} )^{2}
13D4D_{4} 1+24T+3734T2+24p3T3+p6T4 1 + 24 T + 3734 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−32T+2846T2−32p3T3+p6T4 1 - 32 T + 2846 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−116T+16278T2−116p3T3+p6T4 1 - 116 T + 16278 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+12T+23566T2+12p3T3+p6T4 1 + 12 T + 23566 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−148T+41390T2−148p3T3+p6T4 1 - 148 T + 41390 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−76T−4098T2−76p3T3+p6T4 1 - 76 T - 4098 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−280T+100806T2−280p3T3+p6T4 1 - 280 T + 100806 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−572T+216422T2−572p3T3+p6T4 1 - 572 T + 216422 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−520T+213750T2−520p3T3+p6T4 1 - 520 T + 213750 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+340T+100670T2+340p3T3+p6T4 1 + 340 T + 100670 T^{2} + 340 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−524T+337454T2−524p3T3+p6T4 1 - 524 T + 337454 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+552T+435478T2+552p3T3+p6T4 1 + 552 T + 435478 T^{2} + 552 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−628T+163422T2−628p3T3+p6T4 1 - 628 T + 163422 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−168T+286982T2−168p3T3+p6T4 1 - 168 T + 286982 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+1248T+1053742T2+1248p3T3+p6T4 1 + 1248 T + 1053742 T^{2} + 1248 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−908T+723654T2−908p3T3+p6T4 1 - 908 T + 723654 T^{2} - 908 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+1052T+1223358T2+1052p3T3+p6T4 1 + 1052 T + 1223358 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−40T+938150T2−40p3T3+p6T4 1 - 40 T + 938150 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−828T+857734T2−828p3T3+p6T4 1 - 828 T + 857734 T^{2} - 828 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+316T−671034T2+316p3T3+p6T4 1 + 316 T - 671034 T^{2} + 316 p^{3} T^{3} + p^{6} 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

−10.89195999420503365865128832469, −10.13254441819733875365628237018, −9.848017801791531731264736708663, −9.436387704973736674137603408515, −8.843653158631719744454767642130, −8.750249254305893253716540901777, −7.84063531446713307099307528031, −7.82974908836104763408757624992, −7.26295291946146576499045064391, −6.97944477624608067420526082003, −6.07212395300377907053598849885, −5.81501618418828311382857964648, −4.66855025925024018782452245963, −4.61855209234005759330070485959, −3.85729975159760863567950790838, −3.47336047385439846872753620141, −2.75729727742228247318676678477, −2.32746941148447204563904868637, −1.13623247091394193554145993441, −0.841081131810377554802056908417, 0.841081131810377554802056908417, 1.13623247091394193554145993441, 2.32746941148447204563904868637, 2.75729727742228247318676678477, 3.47336047385439846872753620141, 3.85729975159760863567950790838, 4.61855209234005759330070485959, 4.66855025925024018782452245963, 5.81501618418828311382857964648, 6.07212395300377907053598849885, 6.97944477624608067420526082003, 7.26295291946146576499045064391, 7.82974908836104763408757624992, 7.84063531446713307099307528031, 8.750249254305893253716540901777, 8.843653158631719744454767642130, 9.436387704973736674137603408515, 9.848017801791531731264736708663, 10.13254441819733875365628237018, 10.89195999420503365865128832469

Graph of the ZZ-function along the critical line