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Properties

Label 4-1110e2-1.1-c1e2-0-12
Degree 44
Conductor 12321001232100
Sign −1-1
Analytic cond. 78.559778.5597
Root an. cond. 2.977142.97714
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  − 2·3-s + 4-s − 2·7-s + 9-s − 2·12-s − 8·13-s + 16-s + 4·19-s + 4·21-s + 25-s + 4·27-s − 2·28-s + 10·31-s + 36-s + 2·37-s + 16·39-s − 2·43-s − 2·48-s − 11·49-s − 8·52-s − 8·57-s − 2·61-s − 2·63-s + 64-s − 8·67-s − 32·73-s − 2·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s − 0.577·12-s − 2.21·13-s + 1/4·16-s + 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.769·27-s − 0.377·28-s + 1.79·31-s + 1/6·36-s + 0.328·37-s + 2.56·39-s − 0.304·43-s − 0.288·48-s − 1.57·49-s − 1.10·52-s − 1.05·57-s − 0.256·61-s − 0.251·63-s + 1/8·64-s − 0.977·67-s − 3.74·73-s − 0.230·75-s + ⋯

Functional equation

Λ(s)=(1232100s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1232100s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12321001232100    =    22⋅32⋅52⋅3722^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}
Sign: −1-1
Analytic conductor: 78.559778.5597
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1232100, ( :1/2,1/2), −1)(4,\ 1232100,\ (\ :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)
bad2C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
3C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
37C1C_1 (1−T)2 ( 1 - T )^{2}
good7C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
11C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
13C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
17C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
19C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
31C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
41C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
43C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
47C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
73C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + 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)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1−17T+pT2)2 ( 1 - 17 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

−7.48472813821623192367156751152, −7.47402466508823037518119617944, −6.73483784989315938735561487128, −6.56330850429489211825831149583, −5.99940287722666102586658741510, −5.73299805716885755231121374807, −5.06620453918957560960032027985, −4.60963444019434721197685153570, −4.59767259681282669603710565857, −3.42767809661089504801834268834, −2.99576716331676506161499712038, −2.62659393830675313979656999849, −1.82989205635176673644713429623, −0.856493313433969578668782918754, 0, 0.856493313433969578668782918754, 1.82989205635176673644713429623, 2.62659393830675313979656999849, 2.99576716331676506161499712038, 3.42767809661089504801834268834, 4.59767259681282669603710565857, 4.60963444019434721197685153570, 5.06620453918957560960032027985, 5.73299805716885755231121374807, 5.99940287722666102586658741510, 6.56330850429489211825831149583, 6.73483784989315938735561487128, 7.47402466508823037518119617944, 7.48472813821623192367156751152

Graph of the ZZ-function along the critical line