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Properties

Label 4-2352e2-1.1-c1e2-0-44
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 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 6·5-s + 6·9-s + 18·15-s − 6·17-s + 17·25-s − 9·27-s + 14·37-s − 12·41-s − 8·43-s − 36·45-s + 6·47-s + 18·51-s − 6·59-s − 10·67-s − 51·75-s + 2·79-s + 9·81-s − 24·83-s + 36·85-s + 18·89-s + 18·101-s − 34·109-s − 42·111-s − 5·121-s + 36·123-s − 18·125-s + ⋯
L(s)  = 1  − 1.73·3-s − 2.68·5-s + 2·9-s + 4.64·15-s − 1.45·17-s + 17/5·25-s − 1.73·27-s + 2.30·37-s − 1.87·41-s − 1.21·43-s − 5.36·45-s + 0.875·47-s + 2.52·51-s − 0.781·59-s − 1.22·67-s − 5.88·75-s + 0.225·79-s + 81-s − 2.63·83-s + 3.90·85-s + 1.90·89-s + 1.79·101-s − 3.25·109-s − 3.98·111-s − 0.454·121-s + 3.24·123-s − 1.60·125-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: 22
Selberg data: (4, 5531904, ( :1/2,1/2), 1)(4,\ 5531904,\ (\ :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
3C2C_2 1+pT+pT2 1 + p T + p T^{2}
7 1 1
good5C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
11C22C_2^2 1+5T2+p2T4 1 + 5 T^{2} + p^{2} T^{4}
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
19C22C_2^2 1−35T2+p2T4 1 - 35 T^{2} + p^{2} T^{4}
23C22C_2^2 1−19T2+p2T4 1 - 19 T^{2} + p^{2} T^{4}
29C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
31C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
37C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
53C22C_2^2 1−79T2+p2T4 1 - 79 T^{2} + p^{2} T^{4}
59C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
61C22C_2^2 1+25T2+p2T4 1 + 25 T^{2} + p^{2} T^{4}
67C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
71C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
73C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
79C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
83C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
89C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
97C22C_2^2 1−146T2+p2T4 1 - 146 T^{2} + 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

−8.846627195615937834471654949274, −8.260881621114192979309572123975, −7.76067301561830944874172980745, −7.74965789236081186774395995966, −7.27062063980197453505704949253, −6.77760590509000362233881941072, −6.50001199665982583334120459203, −6.28367676954311908126695486635, −5.47522858693247873179393504889, −5.28179898313075499340992032338, −4.51024131722509948981894231522, −4.45026094027893276220404624101, −4.20950183504054508342572519817, −3.62627962375111411256766350452, −3.18093684708761575735433762568, −2.50368180196714403235931239629, −1.60956179076699655136018697458, −0.879943412796931761594590900225, 0, 0, 0.879943412796931761594590900225, 1.60956179076699655136018697458, 2.50368180196714403235931239629, 3.18093684708761575735433762568, 3.62627962375111411256766350452, 4.20950183504054508342572519817, 4.45026094027893276220404624101, 4.51024131722509948981894231522, 5.28179898313075499340992032338, 5.47522858693247873179393504889, 6.28367676954311908126695486635, 6.50001199665982583334120459203, 6.77760590509000362233881941072, 7.27062063980197453505704949253, 7.74965789236081186774395995966, 7.76067301561830944874172980745, 8.260881621114192979309572123975, 8.846627195615937834471654949274

Graph of the ZZ-function along the critical line