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Properties

Label 4-1760e2-1.1-c1e2-0-14
Degree 44
Conductor 30976003097600
Sign −1-1
Analytic cond. 197.505197.505
Root an. cond. 3.748823.74882
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s − 2·11-s − 8·17-s + 25-s + 12·41-s − 4·43-s − 10·49-s − 24·59-s + 24·67-s + 8·73-s + 27·81-s − 4·83-s + 12·89-s − 4·97-s + 12·99-s + 12·107-s + 28·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2·9-s − 0.603·11-s − 1.94·17-s + 1/5·25-s + 1.87·41-s − 0.609·43-s − 1.42·49-s − 3.12·59-s + 2.93·67-s + 0.936·73-s + 3·81-s − 0.439·83-s + 1.27·89-s − 0.406·97-s + 1.20·99-s + 1.16·107-s + 2.63·113-s + 3/11·121-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 + 3.88·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(3097600s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3097600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3097600s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3097600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 30976003097600    =    210⋅52⋅1122^{10} \cdot 5^{2} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 197.505197.505
Root analytic conductor: 3.748823.74882
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 3097600, ( :1/2,1/2), −1)(4,\ 3097600,\ (\ :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
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
11C1C_1 (1+T)2 ( 1 + T )^{2}
good3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{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+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
67C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
71C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
73C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
79C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{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

−7.36247134612527590927307037370, −6.85667269899173093324578324851, −6.30961469790096626723393594297, −6.21087114261569312686272580596, −5.72971066068134741309412572095, −5.17000472187588479255187728599, −4.83284777784030043470160695571, −4.45821261643777188910209572887, −3.79836956390090775269445181604, −3.12595338350487188300141328594, −2.96374376382014471097320958505, −2.17666388867997363005053217178, −2.03399759604601396643465282622, −0.71531896275518796286339281459, 0, 0.71531896275518796286339281459, 2.03399759604601396643465282622, 2.17666388867997363005053217178, 2.96374376382014471097320958505, 3.12595338350487188300141328594, 3.79836956390090775269445181604, 4.45821261643777188910209572887, 4.83284777784030043470160695571, 5.17000472187588479255187728599, 5.72971066068134741309412572095, 6.21087114261569312686272580596, 6.30961469790096626723393594297, 6.85667269899173093324578324851, 7.36247134612527590927307037370

Graph of the ZZ-function along the critical line