Properties

Label 4-1280e2-1.1-c1e2-0-67
Degree 44
Conductor 16384001638400
Sign 11
Analytic cond. 104.465104.465
Root an. cond. 3.197003.19700
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·5-s − 2·7-s + 2·9-s − 6·13-s − 8·15-s − 6·17-s − 12·19-s + 4·21-s − 6·23-s + 11·25-s − 6·27-s − 8·35-s − 6·37-s + 12·39-s − 12·41-s − 6·43-s + 8·45-s + 18·47-s + 2·49-s + 12·51-s − 10·53-s + 24·57-s − 20·59-s + 24·61-s − 4·63-s − 24·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s − 0.755·7-s + 2/3·9-s − 1.66·13-s − 2.06·15-s − 1.45·17-s − 2.75·19-s + 0.872·21-s − 1.25·23-s + 11/5·25-s − 1.15·27-s − 1.35·35-s − 0.986·37-s + 1.92·39-s − 1.87·41-s − 0.914·43-s + 1.19·45-s + 2.62·47-s + 2/7·49-s + 1.68·51-s − 1.37·53-s + 3.17·57-s − 2.60·59-s + 3.07·61-s − 0.503·63-s − 2.97·65-s + ⋯

Functional equation

Λ(s)=(1638400s/2ΓC(s)2L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1638400s/2ΓC(s+1/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 16384001638400    =    216522^{16} \cdot 5^{2}
Sign: 11
Analytic conductor: 104.465104.465
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 1638400, ( :1/2,1/2), 1)(4,\ 1638400,\ (\ :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(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)Isogeny Class over Fp\mathbf{F}_p
bad2 1 1
5C2C_2 14T+pT2 1 - 4 T + p T^{2}
good3C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} 2.3.c_c
7C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} 2.7.c_c
11C22C_2^2 16T2+p2T4 1 - 6 T^{2} + p^{2} T^{4} 2.11.a_ag
13C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} 2.13.g_s
17C2C_2 (12T+pT2)(1+8T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) 2.17.g_s
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2} 2.19.m_cw
23C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} 2.23.g_s
29C22C_2^2 154T2+p2T4 1 - 54 T^{2} + p^{2} T^{4} 2.29.a_acc
31C22C_2^2 126T2+p2T4 1 - 26 T^{2} + p^{2} T^{4} 2.31.a_aba
37C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} 2.37.g_s
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2} 2.41.m_eo
43C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} 2.43.g_s
47C22C_2^2 118T+162T218pT3+p2T4 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} 2.47.as_gg
53C2C_2 (14T+pT2)(1+14T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) 2.53.k_by
59C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2} 2.59.u_ik
61C2C_2 (112T+pT2)2 ( 1 - 12 T + p T^{2} )^{2} 2.61.ay_kg
67C22C_2^2 1+18T+162T2+18pT3+p2T4 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} 2.67.s_gg
71C22C_2^2 1106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4} 2.71.a_aec
73C2C_2 (16T+pT2)(1+16T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) 2.73.k_by
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2} 2.79.a_gc
83C22C_2^2 16T+18T26pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} 2.83.ag_s
89C2C_2 (1pT2)2 ( 1 - p T^{2} )^{2} 2.89.a_agw
97C22C_2^2 1+14T+98T2+14pT3+p2T4 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} 2.97.o_du
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   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

−9.606299836933068250284171331592, −9.199410844352709356249726533898, −8.632318355985944144310071390822, −8.573553673398465119919932228310, −7.74162940427514495117161920140, −7.14168588225264393809283129143, −6.74614447926115849110520562244, −6.55790017168689678450898063968, −5.98330834278771211460823296822, −5.97872457856260773200563781721, −5.24034670808618061461712700234, −4.99151588020140837812434783222, −4.23933687504990012576776971278, −4.20491201335763035945327198834, −3.10793775980364593750376803825, −2.46298842795261715058988739993, −1.98380104940191069438811831423, −1.76459039528730533636464057857, 0, 0, 1.76459039528730533636464057857, 1.98380104940191069438811831423, 2.46298842795261715058988739993, 3.10793775980364593750376803825, 4.20491201335763035945327198834, 4.23933687504990012576776971278, 4.99151588020140837812434783222, 5.24034670808618061461712700234, 5.97872457856260773200563781721, 5.98330834278771211460823296822, 6.55790017168689678450898063968, 6.74614447926115849110520562244, 7.14168588225264393809283129143, 7.74162940427514495117161920140, 8.573553673398465119919932228310, 8.632318355985944144310071390822, 9.199410844352709356249726533898, 9.606299836933068250284171331592

Graph of the ZZ-function along the critical line