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Properties

Label 4-4050e2-1.1-c1e2-0-21
Degree 44
Conductor 1640250016402500
Sign 11
Analytic cond. 1045.831045.83
Root an. cond. 5.686775.68677
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 4·7-s − 4·8-s + 4·13-s − 8·14-s + 5·16-s + 10·19-s − 8·26-s + 12·28-s + 4·31-s − 6·32-s − 8·37-s − 20·38-s + 18·41-s + 10·43-s − 12·47-s + 4·49-s + 12·52-s − 12·53-s − 16·56-s − 6·59-s + 16·61-s − 8·62-s + 7·64-s − 14·67-s + 12·71-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.51·7-s − 1.41·8-s + 1.10·13-s − 2.13·14-s + 5/4·16-s + 2.29·19-s − 1.56·26-s + 2.26·28-s + 0.718·31-s − 1.06·32-s − 1.31·37-s − 3.24·38-s + 2.81·41-s + 1.52·43-s − 1.75·47-s + 4/7·49-s + 1.66·52-s − 1.64·53-s − 2.13·56-s − 0.781·59-s + 2.04·61-s − 1.01·62-s + 7/8·64-s − 1.71·67-s + 1.42·71-s + ⋯

Functional equation

Λ(s)=(16402500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(16402500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1640250016402500    =    22⋅38⋅542^{2} \cdot 3^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 1045.831045.83
Root analytic conductor: 5.686775.68677
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 16402500, ( :1/2,1/2), 1)(4,\ 16402500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3469044192.346904419
L(12)L(\frac12) ≈\approx 2.3469044192.346904419
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 (1+T)2 ( 1 + T )^{2}
3 1 1
5 1 1
good7D4D_{4} 1−4T+12T2−4pT3+p2T4 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
13D4D_{4} 1−4T+24T2−4pT3+p2T4 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
19D4D_{4} 1−10T+3pT2−10pT3+p2T4 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+40T2+p2T4 1 + 40 T^{2} + p^{2} T^{4}
29C22C_2^2 1+52T2+p2T4 1 + 52 T^{2} + p^{2} T^{4}
31D4D_{4} 1−4T+60T2−4pT3+p2T4 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+8T+36T2+8pT3+p2T4 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4}
41C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
43D4D_{4} 1−10T+105T2−10pT3+p2T4 1 - 10 T + 105 T^{2} - 10 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+12T+106T2+12pT3+p2T4 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+12T+136T2+12pT3+p2T4 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+6T+121T2+6pT3+p2T4 1 + 6 T + 121 T^{2} + 6 p T^{3} + p^{2} T^{4}
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C22C_2^2 1+14T+129T2+14pT3+p2T4 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−12T+124T2−12pT3+p2T4 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4}
73C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
79D4D_{4} 1−4T−54T2−4pT3+p2T4 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+6T+169T2+6pT3+p2T4 1 + 6 T + 169 T^{2} + 6 p T^{3} + p^{2} T^{4}
89C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
97D4D_{4} 1+2T+99T2+2pT3+p2T4 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4}
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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

−8.565122914265422413180519727831, −8.236331521113562060104445980852, −7.84280256884599765358098708901, −7.79560081188642338553291523229, −7.27353266277905106986250896092, −7.09917935813281311790445618491, −6.34152847393911674712762172820, −6.28586800201560105273862277356, −5.61616455572404207731135870286, −5.43891589921856121528684256221, −4.95828145449799586796628422329, −4.50019367787690069354192185985, −4.01923011670138581226104531694, −3.50140165235192517846193406946, −2.94195459979969887835445949365, −2.72717946068575890670220038511, −1.86452554272646001298394071269, −1.58123396705195023043790620993, −1.08780858008953016579553312217, −0.65850855089594296065068152211, 0.65850855089594296065068152211, 1.08780858008953016579553312217, 1.58123396705195023043790620993, 1.86452554272646001298394071269, 2.72717946068575890670220038511, 2.94195459979969887835445949365, 3.50140165235192517846193406946, 4.01923011670138581226104531694, 4.50019367787690069354192185985, 4.95828145449799586796628422329, 5.43891589921856121528684256221, 5.61616455572404207731135870286, 6.28586800201560105273862277356, 6.34152847393911674712762172820, 7.09917935813281311790445618491, 7.27353266277905106986250896092, 7.79560081188642338553291523229, 7.84280256884599765358098708901, 8.236331521113562060104445980852, 8.565122914265422413180519727831

Graph of the ZZ-function along the critical line