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Properties

Label 4-1200e2-1.1-c2e2-0-1
Degree 44
Conductor 14400001440000
Sign 11
Analytic cond. 1069.131069.13
Root an. cond. 5.718185.71818
Motivic weight 22
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  − 3·9-s + 12·13-s − 36·17-s − 56·29-s − 60·37-s + 4·41-s + 86·49-s − 204·53-s − 148·61-s − 264·73-s + 9·81-s + 28·89-s − 48·97-s − 40·101-s + 172·109-s − 252·113-s − 36·117-s + 230·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 108·153-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1/3·9-s + 0.923·13-s − 2.11·17-s − 1.93·29-s − 1.62·37-s + 4/41·41-s + 1.75·49-s − 3.84·53-s − 2.42·61-s − 3.61·73-s + 1/9·81-s + 0.314·89-s − 0.494·97-s − 0.396·101-s + 1.57·109-s − 2.23·113-s − 0.307·117-s + 1.90·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.705·153-s + 0.00636·157-s + 0.00613·163-s + ⋯

Functional equation

Λ(s)=(1440000s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(1440000s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14400001440000    =    28⋅32⋅542^{8} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 1069.131069.13
Root analytic conductor: 5.718185.71818
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1440000, ( :1,1), 1)(4,\ 1440000,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.010859026130.01085902613
L(12)L(\frac12) ≈\approx 0.010859026130.01085902613
L(2)L(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
3C2C_2 1+pT2 1 + p T^{2}
5 1 1
good7C22C_2^2 1−86T2+p4T4 1 - 86 T^{2} + p^{4} T^{4}
11C22C_2^2 1−230T2+p4T4 1 - 230 T^{2} + p^{4} T^{4}
13C2C_2 (1−6T+p2T2)2 ( 1 - 6 T + p^{2} T^{2} )^{2}
17C2C_2 (1+18T+p2T2)2 ( 1 + 18 T + p^{2} T^{2} )^{2}
19C22C_2^2 1−530T2+p4T4 1 - 530 T^{2} + p^{4} T^{4}
23C22C_2^2 1−1010T2+p4T4 1 - 1010 T^{2} + p^{4} T^{4}
29C2C_2 (1+28T+p2T2)2 ( 1 + 28 T + p^{2} T^{2} )^{2}
31C22C_2^2 1+430T2+p4T4 1 + 430 T^{2} + p^{4} T^{4}
37C2C_2 (1+30T+p2T2)2 ( 1 + 30 T + p^{2} T^{2} )^{2}
41C2C_2 (1−2T+p2T2)2 ( 1 - 2 T + p^{2} T^{2} )^{2}
43C22C_2^2 1+190T2+p4T4 1 + 190 T^{2} + p^{4} T^{4}
47C22C_2^2 1−1346T2+p4T4 1 - 1346 T^{2} + p^{4} T^{4}
53C2C_2 (1+102T+p2T2)2 ( 1 + 102 T + p^{2} T^{2} )^{2}
59C22C_2^2 1+3130T2+p4T4 1 + 3130 T^{2} + p^{4} T^{4}
61C2C_2 (1+74T+p2T2)2 ( 1 + 74 T + p^{2} T^{2} )^{2}
67C22C_2^2 1+430T2+p4T4 1 + 430 T^{2} + p^{4} T^{4}
71C22C_2^2 1−10034T2+p4T4 1 - 10034 T^{2} + p^{4} T^{4}
73C2C_2 (1+132T+p2T2)2 ( 1 + 132 T + p^{2} T^{2} )^{2}
79C22C_2^2 1−1682T2+p4T4 1 - 1682 T^{2} + p^{4} T^{4}
83C22C_2^2 1+94T2+p4T4 1 + 94 T^{2} + p^{4} T^{4}
89C2C_2 (1−14T+p2T2)2 ( 1 - 14 T + p^{2} T^{2} )^{2}
97C2C_2 (1+24T+p2T2)2 ( 1 + 24 T + p^{2} T^{2} )^{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

−9.716018936452984202390891814254, −9.045492322141150931408124317744, −8.989618646708631818833942500363, −8.782253626936753878170281264989, −8.190557119119297016984229695489, −7.55994306026124732453743996103, −7.43623616154388132538336921207, −6.83767705572325181423463983435, −6.36640512535190446898946115511, −5.98583209974019983062895041069, −5.74519278107940721781054856408, −4.81696726017546119962620155191, −4.80584202155084393737432738159, −4.02103559248388976424761877455, −3.70758625064693816471739048699, −3.04519113977483184084028260975, −2.56292298281281345009573820355, −1.68828380870196673499368885279, −1.53169448199448096316653308641, −0.02942861356119341813152622712, 0.02942861356119341813152622712, 1.53169448199448096316653308641, 1.68828380870196673499368885279, 2.56292298281281345009573820355, 3.04519113977483184084028260975, 3.70758625064693816471739048699, 4.02103559248388976424761877455, 4.80584202155084393737432738159, 4.81696726017546119962620155191, 5.74519278107940721781054856408, 5.98583209974019983062895041069, 6.36640512535190446898946115511, 6.83767705572325181423463983435, 7.43623616154388132538336921207, 7.55994306026124732453743996103, 8.190557119119297016984229695489, 8.782253626936753878170281264989, 8.989618646708631818833942500363, 9.045492322141150931408124317744, 9.716018936452984202390891814254

Graph of the ZZ-function along the critical line