↑

Properties

Label 4-3960e2-1.1-c1e2-0-10
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 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·11-s − 8·19-s − 25-s − 4·29-s + 16·31-s − 12·41-s − 2·49-s + 4·55-s − 8·59-s − 20·61-s + 16·71-s − 8·79-s − 28·89-s + 16·95-s − 20·101-s + 28·109-s + 3·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s − 32·155-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.603·11-s − 1.83·19-s − 1/5·25-s − 0.742·29-s + 2.87·31-s − 1.87·41-s − 2/7·49-s + 0.539·55-s − 1.04·59-s − 2.56·61-s + 1.89·71-s − 0.900·79-s − 2.96·89-s + 1.64·95-s − 1.99·101-s + 2.68·109-s + 3/11·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-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: 22
Selberg data: (4, 15681600, ( :1/2,1/2), 1)(4,\ 15681600,\ (\ :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
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
11C1C_1 (1+T)2 ( 1 + T )^{2}
good7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
47C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
59C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73C22C_2^2 1−142T2+p2T4 1 - 142 T^{2} + p^{2} T^{4}
79C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
97C22C_2^2 1−178T2+p2T4 1 - 178 T^{2} + 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

−8.390077121471355053278739533051, −8.094005497106503424522686677155, −7.46098434412460710736999700070, −7.38423699253185673021647901859, −6.67664847513532635198327944940, −6.52940496700225220335212698709, −6.12650605190121258613463840327, −5.76071479032182738164383061399, −5.18471647514324138086839513350, −4.76669528937552966059545611232, −4.41388260512417558592231944777, −4.24087541192893065647347719401, −3.58617630609164311183359368232, −3.25374408795945736250518516221, −2.71120007214133344256004293538, −2.34095725083537300450962399157, −1.69763026056002695589171623761, −1.14268767624092829376826341943, 0, 0, 1.14268767624092829376826341943, 1.69763026056002695589171623761, 2.34095725083537300450962399157, 2.71120007214133344256004293538, 3.25374408795945736250518516221, 3.58617630609164311183359368232, 4.24087541192893065647347719401, 4.41388260512417558592231944777, 4.76669528937552966059545611232, 5.18471647514324138086839513350, 5.76071479032182738164383061399, 6.12650605190121258613463840327, 6.52940496700225220335212698709, 6.67664847513532635198327944940, 7.38423699253185673021647901859, 7.46098434412460710736999700070, 8.094005497106503424522686677155, 8.390077121471355053278739533051

Graph of the ZZ-function along the critical line