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Properties

Label 4-2352e2-1.1-c1e2-0-0
Degree 44
Conductor 55319045531904
Sign 11
Analytic cond. 352.718352.718
Root an. cond. 4.333684.33368
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  − 3·9-s − 14·13-s + 10·25-s − 2·37-s − 28·61-s − 14·73-s + 9·81-s + 28·97-s − 34·109-s + 42·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 121·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 9-s − 3.88·13-s + 2·25-s − 0.328·37-s − 3.58·61-s − 1.63·73-s + 81-s + 2.84·97-s − 3.25·109-s + 3.88·117-s − 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

Λ(s)=(5531904s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5531904s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 55319045531904    =    28⋅32⋅742^{8} \cdot 3^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 352.718352.718
Root analytic conductor: 4.333684.33368
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 5531904, ( :1/2,1/2), 1)(4,\ 5531904,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.15827803930.1582780393
L(12)L(\frac12) ≈\approx 0.15827803930.1582780393
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
3C2C_2 1+pT2 1 + p T^{2}
7 1 1
good5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
17C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
19C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
31C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
37C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
41C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
43C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
67C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
79C2C_2 (1−13T+pT2)(1+13T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} )
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
97C2C_2 (1−14T+pT2)2 ( 1 - 14 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

−9.057388009487985982741926637639, −9.053035966969722108950545627598, −8.366364291848424234648865922414, −8.000943590282717289561775407830, −7.41954774703763249464139903326, −7.38269028665345657070837334659, −7.07186959440517697353335820752, −6.33533873425176335963614904931, −6.25360107049326898120551386870, −5.45255514103565750986780914797, −5.15218771369243878379535029525, −4.80144729616672893954543188408, −4.65571740471496100887281170939, −4.01026648894047029139493730596, −3.03115392900874834067879813453, −3.03048460560840127825364430557, −2.51263898296052585980890995263, −2.10559207402222168616955538853, −1.24460249660826580575757902951, −0.13523685308600440766328821257, 0.13523685308600440766328821257, 1.24460249660826580575757902951, 2.10559207402222168616955538853, 2.51263898296052585980890995263, 3.03048460560840127825364430557, 3.03115392900874834067879813453, 4.01026648894047029139493730596, 4.65571740471496100887281170939, 4.80144729616672893954543188408, 5.15218771369243878379535029525, 5.45255514103565750986780914797, 6.25360107049326898120551386870, 6.33533873425176335963614904931, 7.07186959440517697353335820752, 7.38269028665345657070837334659, 7.41954774703763249464139903326, 8.000943590282717289561775407830, 8.366364291848424234648865922414, 9.053035966969722108950545627598, 9.057388009487985982741926637639

Graph of the ZZ-function along the critical line