Properties

Label 4-2e12-1.1-c9e2-0-0
Degree 44
Conductor 40964096
Sign 11
Analytic cond. 1086.511086.51
Root an. cond. 5.741275.74127
Motivic weight 99
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  + 3.26e3·9-s − 1.19e6·17-s + 3.90e6·25-s + 6.86e7·41-s − 8.07e7·49-s + 8.44e8·73-s − 3.76e8·81-s − 2.17e9·89-s − 3.47e9·97-s + 2.29e9·113-s + 3.54e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.90e9·153-s + 157-s + 163-s + 167-s + 2.12e10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 0.165·9-s − 3.47·17-s + 2·25-s + 3.79·41-s − 2·49-s + 3.48·73-s − 0.972·81-s − 3.68·89-s − 3.98·97-s + 1.32·113-s + 0.150·121-s − 0.575·153-s + 2·169-s + ⋯

Functional equation

Λ(s)=(4096s/2ΓC(s)2L(s)=(Λ(10s)\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(4096s/2ΓC(s+9/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 40964096    =    2122^{12}
Sign: 11
Analytic conductor: 1086.511086.51
Root analytic conductor: 5.741275.74127
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 4096, ( :9/2,9/2), 1)(4,\ 4096,\ (\ :9/2, 9/2),\ 1)

Particular Values

L(5)L(5) \approx 1.5248571231.524857123
L(12)L(\frac12) \approx 1.5248571231.524857123
L(112)L(\frac{11}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
good3C22C_2^2 13266T2+p18T4 1 - 3266 T^{2} + p^{18} T^{4}
5C2C_2 (1p9T2)2 ( 1 - p^{9} T^{2} )^{2}
7C2C_2 (1+p9T2)2 ( 1 + p^{9} T^{2} )^{2}
11C22C_2^2 1354349618T2+p18T4 1 - 354349618 T^{2} + p^{18} T^{4}
13C2C_2 (1p9T2)2 ( 1 - p^{9} T^{2} )^{2}
17C2C_2 (1+597510T+p9T2)2 ( 1 + 597510 T + p^{9} T^{2} )^{2}
19C22C_2^2 1+335013705758T2+p18T4 1 + 335013705758 T^{2} + p^{18} T^{4}
23C2C_2 (1+p9T2)2 ( 1 + p^{9} T^{2} )^{2}
29C2C_2 (1p9T2)2 ( 1 - p^{9} T^{2} )^{2}
31C2C_2 (1+p9T2)2 ( 1 + p^{9} T^{2} )^{2}
37C2C_2 (1p9T2)2 ( 1 - p^{9} T^{2} )^{2}
41C2C_2 (134306362T+p9T2)2 ( 1 - 34306362 T + p^{9} T^{2} )^{2}
43C22C_2^2 1+1000250360894414T2+p18T4 1 + 1000250360894414 T^{2} + p^{18} T^{4}
47C2C_2 (1+p9T2)2 ( 1 + p^{9} T^{2} )^{2}
53C2C_2 (1p9T2)2 ( 1 - p^{9} T^{2} )^{2}
59C22C_2^2 110228968070290322T2+p18T4 1 - 10228968070290322 T^{2} + p^{18} T^{4}
61C2C_2 (1p9T2)2 ( 1 - p^{9} T^{2} )^{2}
67C22C_2^2 150467064407716994T2+p18T4 1 - 50467064407716994 T^{2} + p^{18} T^{4}
71C2C_2 (1+p9T2)2 ( 1 + p^{9} T^{2} )^{2}
73C2C_2 (1422324930T+p9T2)2 ( 1 - 422324930 T + p^{9} T^{2} )^{2}
79C2C_2 (1+p9T2)2 ( 1 + p^{9} T^{2} )^{2}
83C22C_2^2 1352960460737558306T2+p18T4 1 - 352960460737558306 T^{2} + p^{18} T^{4}
89C2C_2 (1+1089849006T+p9T2)2 ( 1 + 1089849006 T + p^{9} T^{2} )^{2}
97C2C_2 (1+1738254710T+p9T2)2 ( 1 + 1738254710 T + p^{9} T^{2} )^{2}
show more
show less
   L(s)=p j=14(1αj,pps)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

−13.03053767191208780186927824832, −12.84807370069876657290302759418, −12.44569336412597739285527916548, −11.17959200870430913064695055609, −11.10182810663012250082376202368, −10.83605369447645095706114592366, −9.646495822014850831218031628157, −9.309392875949209074459744187050, −8.647711668414114287852455210837, −8.185301553968613984526440695682, −7.16589091104376074649291661184, −6.71951468053183707218970546615, −6.22661171593006831491305194470, −5.19643130760178539449584535426, −4.43973078790012342746311078659, −4.11444714593756977299677698945, −2.76251509808669956280400975647, −2.37513433198687195154446743481, −1.33148871414915706942241167858, −0.38656767958788163960738348630, 0.38656767958788163960738348630, 1.33148871414915706942241167858, 2.37513433198687195154446743481, 2.76251509808669956280400975647, 4.11444714593756977299677698945, 4.43973078790012342746311078659, 5.19643130760178539449584535426, 6.22661171593006831491305194470, 6.71951468053183707218970546615, 7.16589091104376074649291661184, 8.185301553968613984526440695682, 8.647711668414114287852455210837, 9.309392875949209074459744187050, 9.646495822014850831218031628157, 10.83605369447645095706114592366, 11.10182810663012250082376202368, 11.17959200870430913064695055609, 12.44569336412597739285527916548, 12.84807370069876657290302759418, 13.03053767191208780186927824832

Graph of the ZZ-function along the critical line