↑

Properties

Label 4-3960e2-1.1-c1e2-0-8
Degree 44
Conductor 1568160015681600
Sign 11
Analytic cond. 999.872999.872
Root an. cond. 5.623235.62323
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 3·7-s − 2·11-s + 2·13-s − 7·17-s + 9·19-s − 6·23-s + 3·25-s + 5·29-s + 5·31-s + 6·35-s − 37-s + 20·41-s + 2·43-s + 6·47-s − 3·49-s + 13·53-s − 4·55-s + 2·59-s − 3·61-s + 4·65-s + 9·71-s + 18·73-s − 6·77-s + 10·79-s − 20·83-s − 14·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.13·7-s − 0.603·11-s + 0.554·13-s − 1.69·17-s + 2.06·19-s − 1.25·23-s + 3/5·25-s + 0.928·29-s + 0.898·31-s + 1.01·35-s − 0.164·37-s + 3.12·41-s + 0.304·43-s + 0.875·47-s − 3/7·49-s + 1.78·53-s − 0.539·55-s + 0.260·59-s − 0.384·61-s + 0.496·65-s + 1.06·71-s + 2.10·73-s − 0.683·77-s + 1.12·79-s − 2.19·83-s − 1.51·85-s + ⋯

Functional equation

Λ(s)=(15681600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(15681600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1568160015681600    =    26⋅34⋅52⋅1122^{6} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 999.872999.872
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 15681600, ( :1/2,1/2), 1)(4,\ 15681600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.0746749345.074674934
L(12)L(\frac12) ≈\approx 5.0746749345.074674934
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}
11C1C_1 (1+T)2 ( 1 + T )^{2}
good7D4D_{4} 1−3T+12T2−3pT3+p2T4 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4}
13C4C_4 1−2T+10T2−2pT3+p2T4 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+7T+42T2+7pT3+p2T4 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4}
19D4D_{4} 1−9T+54T2−9pT3+p2T4 1 - 9 T + 54 T^{2} - 9 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+6T+38T2+6pT3+p2T4 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−5T+60T2−5pT3+p2T4 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−5T+30T2−5pT3+p2T4 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+T−32T2+pT3+p2T4 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4}
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43D4D_{4} 1−2T+70T2−2pT3+p2T4 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−6T+86T2−6pT3+p2T4 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−13T+144T2−13pT3+p2T4 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−2T−34T2−2pT3+p2T4 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+3T+120T2+3pT3+p2T4 1 + 3 T + 120 T^{2} + 3 p T^{3} + p^{2} T^{4}
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71D4D_{4} 1−9T+158T2−9pT3+p2T4 1 - 9 T + 158 T^{2} - 9 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−18T+210T2−18pT3+p2T4 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−10T+166T2−10pT3+p2T4 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4}
83C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
89D4D_{4} 1−T+72T2−pT3+p2T4 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4}
97D4D_{4} 1−26T+346T2−26pT3+p2T4 1 - 26 T + 346 T^{2} - 26 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

−8.538635385706729659685077755658, −8.320278943682862826452854972727, −7.82429152551889613050976037380, −7.74282792418133725170019263166, −7.10616330976738281304077416408, −6.92801450965601876153003652734, −6.23338359420560625302644137487, −6.13569029005216695240034415597, −5.51667357525040713261715151201, −5.48716972949232146085652147968, −4.81601499484471995067949897508, −4.56197546287805922151796010638, −4.21112239957441019596349478859, −3.67880596935055625036903281520, −3.07887933862199104193716312538, −2.56445394667358042603938767457, −2.18178746378526174617723301059, −1.91391709410738428672465169812, −0.889077701804226795350455029179, −0.881744750176254648638780428010, 0.881744750176254648638780428010, 0.889077701804226795350455029179, 1.91391709410738428672465169812, 2.18178746378526174617723301059, 2.56445394667358042603938767457, 3.07887933862199104193716312538, 3.67880596935055625036903281520, 4.21112239957441019596349478859, 4.56197546287805922151796010638, 4.81601499484471995067949897508, 5.48716972949232146085652147968, 5.51667357525040713261715151201, 6.13569029005216695240034415597, 6.23338359420560625302644137487, 6.92801450965601876153003652734, 7.10616330976738281304077416408, 7.74282792418133725170019263166, 7.82429152551889613050976037380, 8.320278943682862826452854972727, 8.538635385706729659685077755658

Graph of the ZZ-function along the critical line