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Properties

Label 4-60e2-1.1-c1e2-0-1
Degree 44
Conductor 36003600
Sign 11
Analytic cond. 0.2295390.229539
Root an. cond. 0.6921720.692172
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-s − 4-s + 2·5-s + 3·8-s + 9-s − 2·10-s − 4·13-s − 16-s + 4·17-s − 18-s − 2·20-s + 3·25-s + 4·26-s − 4·29-s − 5·32-s − 4·34-s − 36-s − 20·37-s + 6·40-s + 20·41-s + 2·45-s − 14·49-s − 3·50-s + 4·52-s − 20·53-s + 4·58-s − 4·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.10·13-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.447·20-s + 3/5·25-s + 0.784·26-s − 0.742·29-s − 0.883·32-s − 0.685·34-s − 1/6·36-s − 3.28·37-s + 0.948·40-s + 3.12·41-s + 0.298·45-s − 2·49-s − 0.424·50-s + 0.554·52-s − 2.74·53-s + 0.525·58-s − 0.512·61-s + ⋯

Functional equation

Λ(s)=(3600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 36003600    =    24⋅32⋅522^{4} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 0.2295390.229539
Root analytic conductor: 0.6921720.692172
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 3600, ( :1/2,1/2), 1)(4,\ 3600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.55892542810.5589254281
L(12)L(\frac12) ≈\approx 0.55892542810.5589254281
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+T+pT2 1 + T + p T^{2}
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5C1C_1 (1−T)2 ( 1 - T )^{2}
good7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p 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

−12.64617876135106218873747419153, −12.26787268519799195488690762647, −11.16070118695890574879383709815, −10.67892245123374144028858824682, −10.00510556501914029619044642425, −9.523451675812284265792380668213, −9.265565204604989643425731491439, −8.330811446905088069540074876671, −7.66488013441745380243230842523, −7.12360478295630625617848815878, −6.13587545605028508104380795249, −5.23920392624592057055772361749, −4.68831052899409581492283307982, −3.39274372810076015084631955767, −1.82346513010869405208007428430, 1.82346513010869405208007428430, 3.39274372810076015084631955767, 4.68831052899409581492283307982, 5.23920392624592057055772361749, 6.13587545605028508104380795249, 7.12360478295630625617848815878, 7.66488013441745380243230842523, 8.330811446905088069540074876671, 9.265565204604989643425731491439, 9.523451675812284265792380668213, 10.00510556501914029619044642425, 10.67892245123374144028858824682, 11.16070118695890574879383709815, 12.26787268519799195488690762647, 12.64617876135106218873747419153

Graph of the ZZ-function along the critical line