↑

Properties

Label 4-3528e2-1.1-c1e2-0-2
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 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 12·13-s + 2·17-s + 4·19-s − 4·23-s + 5·25-s + 20·29-s − 8·31-s − 6·37-s − 4·41-s − 8·43-s − 8·47-s − 10·53-s − 12·59-s − 2·61-s + 24·65-s − 12·67-s + 24·71-s − 14·73-s + 8·79-s + 24·83-s − 4·85-s + 2·89-s − 8·95-s − 20·97-s + 6·101-s + 2·109-s + ⋯
L(s)  = 1  − 0.894·5-s − 3.32·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 25-s + 3.71·29-s − 1.43·31-s − 0.986·37-s − 0.624·41-s − 1.21·43-s − 1.16·47-s − 1.37·53-s − 1.56·59-s − 0.256·61-s + 2.97·65-s − 1.46·67-s + 2.84·71-s − 1.63·73-s + 0.900·79-s + 2.63·83-s − 0.433·85-s + 0.211·89-s − 0.820·95-s − 2.03·97-s + 0.597·101-s + 0.191·109-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: 00
Selberg data: (4, 12446784, ( :1/2,1/2), 1)(4,\ 12446784,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.25663835470.2566383547
L(12)L(\frac12) ≈\approx 0.25663835470.2566383547
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+2T−T2+2pT3+p2T4 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
13C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
17C22C_2^2 1−2T−13T2−2pT3+p2T4 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−4T−3T2−4pT3+p2T4 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+4T−7T2+4pT3+p2T4 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4}
29C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
31C22C_2^2 1+8T+33T2+8pT3+p2T4 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+6T−T2+6pT3+p2T4 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4}
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C22C_2^2 1+8T+17T2+8pT3+p2T4 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+10T+47T2+10pT3+p2T4 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+12T+85T2+12pT3+p2T4 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+2T−57T2+2pT3+p2T4 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+12T+77T2+12pT3+p2T4 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C22C_2^2 1+14T+123T2+14pT3+p2T4 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−8T−15T2−8pT3+p2T4 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4}
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C22C_2^2 1−2T−85T2−2pT3+p2T4 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4}
97C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
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

−8.640999272341286951049326553436, −8.154117198051418666831477395359, −8.039157493234952738651435269704, −7.71084010956113871197564615937, −7.31666385090932706879795966349, −6.86697632945751690128319984656, −6.73219651372610855731367190477, −6.33412977954345192539242047627, −5.59848273141288266590724538708, −5.13972949171431704581934301483, −4.90874190317743168797615358570, −4.65586883738171795913397678717, −4.37328711784558422094157111598, −3.44944934689807826946092778836, −3.26544837931114062552187486371, −2.88148956732844158701413511855, −2.35005770868575698380149320852, −1.81349383031692935641239444720, −1.08633607958420632713005377265, −0.16400404974601194804692622465, 0.16400404974601194804692622465, 1.08633607958420632713005377265, 1.81349383031692935641239444720, 2.35005770868575698380149320852, 2.88148956732844158701413511855, 3.26544837931114062552187486371, 3.44944934689807826946092778836, 4.37328711784558422094157111598, 4.65586883738171795913397678717, 4.90874190317743168797615358570, 5.13972949171431704581934301483, 5.59848273141288266590724538708, 6.33412977954345192539242047627, 6.73219651372610855731367190477, 6.86697632945751690128319984656, 7.31666385090932706879795966349, 7.71084010956113871197564615937, 8.039157493234952738651435269704, 8.154117198051418666831477395359, 8.640999272341286951049326553436

Graph of the ZZ-function along the critical line