↑

Properties

Label 4-3528e2-1.1-c1e2-0-35
Degree 44
Conductor 1244678412446784
Sign 11
Analytic cond. 793.617793.617
Root an. cond. 5.307655.30765
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 5·13-s − 8·17-s + 5·19-s − 8·23-s + 5·25-s − 3·29-s + 2·31-s + 3·37-s + 6·41-s − 7·43-s − 12·47-s − 11·53-s − 55-s + 5·59-s − 20·61-s + 5·65-s + 7·67-s − 4·71-s + 73-s + 8·79-s + 7·83-s + 8·85-s − 6·89-s − 5·95-s − 25·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.38·13-s − 1.94·17-s + 1.14·19-s − 1.66·23-s + 25-s − 0.557·29-s + 0.359·31-s + 0.493·37-s + 0.937·41-s − 1.06·43-s − 1.75·47-s − 1.51·53-s − 0.134·55-s + 0.650·59-s − 2.56·61-s + 0.620·65-s + 0.855·67-s − 0.474·71-s + 0.117·73-s + 0.900·79-s + 0.768·83-s + 0.867·85-s − 0.635·89-s − 0.512·95-s − 2.53·97-s + ⋯

Functional equation

Λ(s)=(12446784s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(12446784s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1244678412446784    =    26⋅34⋅742^{6} \cdot 3^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 793.617793.617
Root analytic conductor: 5.307655.30765
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 12446784, ( :1/2,1/2), 1)(4,\ 12446784,\ (\ :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
3 1 1
7 1 1
good5C22C_2^2 1+T−4T2+pT3+p2T4 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4}
11D4D_{4} 1−T+8T2−pT3+p2T4 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4}
13D4D_{4} 1+5T+18T2+5pT3+p2T4 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4}
17C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
19D4D_{4} 1−5T+30T2−5pT3+p2T4 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4}
23C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
29D4D_{4} 1+3T+46T2+3pT3+p2T4 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4}
31C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
37D4D_{4} 1−3T+62T2−3pT3+p2T4 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−6T+34T2−6pT3+p2T4 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+7T+84T2+7pT3+p2T4 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4}
47C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
53D4D_{4} 1+11T+122T2+11pT3+p2T4 1 + 11 T + 122 T^{2} + 11 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−5T+110T2−5pT3+p2T4 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67D4D_{4} 1−7T+132T2−7pT3+p2T4 1 - 7 T + 132 T^{2} - 7 p T^{3} + p^{2} T^{4}
71C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
73D4D_{4} 1−T+132T2−pT3+p2T4 1 - T + 132 T^{2} - p T^{3} + p^{2} T^{4}
79D4D_{4} 1−8T+117T2−8pT3+p2T4 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−7T+164T2−7pT3+p2T4 1 - 7 T + 164 T^{2} - 7 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+6T+130T2+6pT3+p2T4 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+25T+336T2+25pT3+p2T4 1 + 25 T + 336 T^{2} + 25 p T^{3} + p^{2} T^{4}
show more
show less
   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

−8.199437375520808117057045065771, −8.106745005014989322563050040664, −7.56084715494105884112333032060, −7.45301541763017165200606598950, −6.71654288310414967634349763150, −6.70073632245401744492253588366, −6.23841162056677075120995915228, −5.83356918575981281550840563833, −5.18995316409897141238366334982, −4.85544098925798401277104215023, −4.67468365657476875553358733735, −4.12216009244573645365194056071, −3.78813135785413988787015943187, −3.21602223139018640327563799336, −2.70025665596136581460930474397, −2.38853213020453115209962611679, −1.74294431929387312457142758629, −1.23834149436161297774796709499, 0, 0, 1.23834149436161297774796709499, 1.74294431929387312457142758629, 2.38853213020453115209962611679, 2.70025665596136581460930474397, 3.21602223139018640327563799336, 3.78813135785413988787015943187, 4.12216009244573645365194056071, 4.67468365657476875553358733735, 4.85544098925798401277104215023, 5.18995316409897141238366334982, 5.83356918575981281550840563833, 6.23841162056677075120995915228, 6.70073632245401744492253588366, 6.71654288310414967634349763150, 7.45301541763017165200606598950, 7.56084715494105884112333032060, 8.106745005014989322563050040664, 8.199437375520808117057045065771

Graph of the ZZ-function along the critical line