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Properties

Label 4-3240e2-1.1-c1e2-0-35
Degree 44
Conductor 1049760010497600
Sign 11
Analytic cond. 669.336669.336
Root an. cond. 5.086405.08640
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·7-s − 4·17-s − 4·19-s − 10·23-s + 3·25-s + 6·29-s + 12·31-s − 4·35-s − 12·37-s − 10·41-s + 4·43-s − 14·47-s − 5·49-s − 4·53-s − 12·59-s − 6·61-s − 2·67-s + 8·73-s − 4·79-s − 6·83-s + 8·85-s − 14·89-s + 8·95-s + 4·97-s − 4·101-s + 20·103-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s − 0.970·17-s − 0.917·19-s − 2.08·23-s + 3/5·25-s + 1.11·29-s + 2.15·31-s − 0.676·35-s − 1.97·37-s − 1.56·41-s + 0.609·43-s − 2.04·47-s − 5/7·49-s − 0.549·53-s − 1.56·59-s − 0.768·61-s − 0.244·67-s + 0.936·73-s − 0.450·79-s − 0.658·83-s + 0.867·85-s − 1.48·89-s + 0.820·95-s + 0.406·97-s − 0.398·101-s + 1.97·103-s + ⋯

Functional equation

Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10497600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1049760010497600    =    26⋅38⋅522^{6} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 669.336669.336
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 10497600, ( :1/2,1/2), 1)(4,\ 10497600,\ (\ :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
5C1C_1 (1+T)2 ( 1 + T )^{2}
good7D4D_{4} 1−2T+9T2−2pT3+p2T4 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19D4D_{4} 1+4T+18T2+4pT3+p2T4 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+10T+65T2+10pT3+p2T4 1 + 10 T + 65 T^{2} + 10 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−6T+43T2−6pT3+p2T4 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−12T+74T2−12pT3+p2T4 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4}
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41D4D_{4} 1+10T+83T2+10pT3+p2T4 1 + 10 T + 83 T^{2} + 10 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−4T−6T2−4pT3+p2T4 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+14T+137T2+14pT3+p2T4 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+4T+14T2+4pT3+p2T4 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+12T+130T2+12pT3+p2T4 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4}
61C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
67D4D_{4} 1+2T−15T2+2pT3+p2T4 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
73D4D_{4} 1−8T+66T2−8pT3+p2T4 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+4T+138T2+4pT3+p2T4 1 + 4 T + 138 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}
89D4D_{4} 1+14T+131T2+14pT3+p2T4 1 + 14 T + 131 T^{2} + 14 p T^{3} + p^{2} T^{4}
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

−8.321954330565190670546770237618, −8.269388673049869867898993227212, −7.69799642451249789294204202953, −7.62620403829308572052042681192, −6.82501363862108435672879883240, −6.58729483272453982115367604885, −6.31437347731658171487128154550, −6.03260523213820954563537902392, −5.08397461220960252514209057093, −5.04924821575820287588360703708, −4.47352745114991563769336165146, −4.37672227492823335198910506060, −3.74112630462186499594093168576, −3.39645753062151709406053779632, −2.76685277250510753791582560652, −2.35525540850745789774399018372, −1.62008066329913966364782445589, −1.39616136812889307768968616451, 0, 0, 1.39616136812889307768968616451, 1.62008066329913966364782445589, 2.35525540850745789774399018372, 2.76685277250510753791582560652, 3.39645753062151709406053779632, 3.74112630462186499594093168576, 4.37672227492823335198910506060, 4.47352745114991563769336165146, 5.04924821575820287588360703708, 5.08397461220960252514209057093, 6.03260523213820954563537902392, 6.31437347731658171487128154550, 6.58729483272453982115367604885, 6.82501363862108435672879883240, 7.62620403829308572052042681192, 7.69799642451249789294204202953, 8.269388673049869867898993227212, 8.321954330565190670546770237618

Graph of the ZZ-function along the critical line