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Properties

Label 4-60e3-1.1-c1e2-0-18
Degree 44
Conductor 216000216000
Sign −1-1
Analytic cond. 13.772313.7723
Root an. cond. 1.926421.92642
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 5-s + 9-s + 2·12-s + 2·13-s − 15-s + 4·16-s − 2·20-s + 25-s − 27-s − 8·31-s − 2·36-s − 10·37-s − 2·39-s − 18·41-s − 4·43-s + 45-s − 4·48-s + 2·49-s − 4·52-s + 12·53-s + 2·60-s − 8·64-s + 2·65-s + 8·67-s − 75-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s + 0.577·12-s + 0.554·13-s − 0.258·15-s + 16-s − 0.447·20-s + 1/5·25-s − 0.192·27-s − 1.43·31-s − 1/3·36-s − 1.64·37-s − 0.320·39-s − 2.81·41-s − 0.609·43-s + 0.149·45-s − 0.577·48-s + 2/7·49-s − 0.554·52-s + 1.64·53-s + 0.258·60-s − 64-s + 0.248·65-s + 0.977·67-s − 0.115·75-s + ⋯

Functional equation

Λ(s)=(216000s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(216000s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 216000216000    =    26⋅33⋅532^{6} \cdot 3^{3} \cdot 5^{3}
Sign: −1-1
Analytic conductor: 13.772313.7723
Root analytic conductor: 1.926421.92642
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 216000, ( :1/2,1/2), −1)(4,\ 216000,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2C2C_2 1+pT2 1 + p T^{2}
3C1C_1 1+T 1 + T
5C1C_1 1−T 1 - T
good7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C2C_2×\timesC2C_2 (1+6T+pT2)(1+12T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1−98T2+p2T4 1 - 98 T^{2} + p^{2} T^{4}
61C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+16T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} )
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
97C22C_2^2 1−170T2+p2T4 1 - 170 T^{2} + 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

−8.724106438367421303345162639094, −8.542414096846067177198244976201, −7.982913945729478238750702527615, −7.23853334997921135605777421359, −6.88247170613678561810180280311, −6.37477083063926939926298893868, −5.67797429529676722720135763007, −5.24149311167588692091359783047, −5.09255145585540175993384578396, −4.21040799738456801763099304038, −3.68350046726164799970870121574, −3.22951288219260696620707097305, −2.04071424181164247240988898380, −1.31631268356932863582550653017, 0, 1.31631268356932863582550653017, 2.04071424181164247240988898380, 3.22951288219260696620707097305, 3.68350046726164799970870121574, 4.21040799738456801763099304038, 5.09255145585540175993384578396, 5.24149311167588692091359783047, 5.67797429529676722720135763007, 6.37477083063926939926298893868, 6.88247170613678561810180280311, 7.23853334997921135605777421359, 7.982913945729478238750702527615, 8.542414096846067177198244976201, 8.724106438367421303345162639094

Graph of the ZZ-function along the critical line