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Properties

Label 4-600e2-1.1-c1e2-0-30
Degree 44
Conductor 360000360000
Sign −1-1
Analytic cond. 22.953922.9539
Root an. cond. 2.188842.18884
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 8·17-s − 12·29-s + 16·37-s − 20·41-s + 2·49-s + 24·53-s + 4·61-s + 16·73-s + 81-s − 20·89-s − 16·97-s − 4·101-s − 4·109-s − 24·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + 167-s − 26·169-s + ⋯
L(s)  = 1  + 1/3·9-s − 1.94·17-s − 2.22·29-s + 2.63·37-s − 3.12·41-s + 2/7·49-s + 3.29·53-s + 0.512·61-s + 1.87·73-s + 1/9·81-s − 2.11·89-s − 1.62·97-s − 0.398·101-s − 0.383·109-s − 2.25·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + ⋯

Functional equation

Λ(s)=(360000s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(360000s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 360000360000    =    26⋅32⋅542^{6} \cdot 3^{2} \cdot 5^{4}
Sign: −1-1
Analytic conductor: 22.953922.9539
Root analytic conductor: 2.188842.18884
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 360000, ( :1/2,1/2), −1)(4,\ 360000,\ (\ :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)
bad2 1 1
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5 1 1
good7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−8T+pT2)2 ( 1 - 8 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 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C2C_2 (1−12T+pT2)2 ( 1 - 12 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 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
79C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
83C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
89C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
97C2C_2 (1+8T+pT2)2 ( 1 + 8 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

−8.425273316407858927756969056783, −8.204559252191821560558835851641, −7.37238433524374824283811858439, −7.18372888449204428119332901427, −6.58708874819826533674259473804, −6.30782253827108715004959607025, −5.36757388021193461608897977383, −5.36371477889983409574050795450, −4.48113990818320943365008579851, −3.96453757467671714692485940103, −3.70436878140437791475645367729, −2.56579130253629562111590352122, −2.27167150313420976169553892108, −1.34965769440900465302455436948, 0, 1.34965769440900465302455436948, 2.27167150313420976169553892108, 2.56579130253629562111590352122, 3.70436878140437791475645367729, 3.96453757467671714692485940103, 4.48113990818320943365008579851, 5.36371477889983409574050795450, 5.36757388021193461608897977383, 6.30782253827108715004959607025, 6.58708874819826533674259473804, 7.18372888449204428119332901427, 7.37238433524374824283811858439, 8.204559252191821560558835851641, 8.425273316407858927756969056783

Graph of the ZZ-function along the critical line