↑

Properties

Label 4-117e2-1.1-c1e2-0-0
Degree 44
Conductor 1368913689
Sign 11
Analytic cond. 0.8728220.872822
Root an. cond. 0.9665650.966565
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 9-s + 2·12-s + 5·13-s − 3·16-s + 6·17-s − 25-s + 4·27-s + 15·29-s − 36-s − 10·39-s − 2·43-s + 6·48-s − 13·49-s − 12·51-s − 5·52-s + 6·53-s + 19·61-s + 7·64-s − 6·68-s + 2·75-s − 20·79-s − 11·81-s − 30·87-s + 100-s − 6·101-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 1/3·9-s + 0.577·12-s + 1.38·13-s − 3/4·16-s + 1.45·17-s − 1/5·25-s + 0.769·27-s + 2.78·29-s − 1/6·36-s − 1.60·39-s − 0.304·43-s + 0.866·48-s − 1.85·49-s − 1.68·51-s − 0.693·52-s + 0.824·53-s + 2.43·61-s + 7/8·64-s − 0.727·68-s + 0.230·75-s − 2.25·79-s − 1.22·81-s − 3.21·87-s + 1/10·100-s − 0.597·101-s + ⋯

Functional equation

Λ(s)=(13689s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(13689s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1368913689    =    34⋅1323^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 0.8728220.872822
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 13689, ( :1/2,1/2), 1)(4,\ 13689,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.67515760740.6751576074
L(12)L(\frac12) ≈\approx 0.67515760740.6751576074
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)
bad3C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
13C2C_2 1−5T+pT2 1 - 5 T + p T^{2}
good2C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
5C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
7C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
11C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
17C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
19C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2×\timesC2C_2 (1−9T+pT2)(1−6T+pT2) ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} )
31C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
37C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
41C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−2T+pT2)(1+4T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (1−9T+pT2)(1+3T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} )
59C22C_2^2 1−35T2+p2T4 1 - 35 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−14T+pT2)(1−5T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} )
67C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
71C22C_2^2 1+13T2+p2T4 1 + 13 T^{2} + p^{2} T^{4}
73C22C_2^2 1−47T2+p2T4 1 - 47 T^{2} + p^{2} T^{4}
79C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
83C22C_2^2 1−119T2+p2T4 1 - 119 T^{2} + p^{2} T^{4}
89C22C_2^2 1+37T2+p2T4 1 + 37 T^{2} + p^{2} T^{4}
97C22C_2^2 1−14T2+p2T4 1 - 14 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

−11.35602805723046122206829210450, −10.68389180192618923629940403141, −10.15085227464949750149406598643, −9.830828044174449564877761972139, −8.901118960746985838598890682533, −8.417598580717825745872133683050, −8.033008851926041234750244332490, −6.93008699605088141542062183856, −6.52242754969305111991545394487, −5.89619929848292488548421093648, −5.26921996746328915551854401084, −4.67556083454843451240290976455, −3.85603569322153007789294912365, −2.89344923925415568822832195876, −1.09621016892456542075964254605, 1.09621016892456542075964254605, 2.89344923925415568822832195876, 3.85603569322153007789294912365, 4.67556083454843451240290976455, 5.26921996746328915551854401084, 5.89619929848292488548421093648, 6.52242754969305111991545394487, 6.93008699605088141542062183856, 8.033008851926041234750244332490, 8.417598580717825745872133683050, 8.901118960746985838598890682533, 9.830828044174449564877761972139, 10.15085227464949750149406598643, 10.68389180192618923629940403141, 11.35602805723046122206829210450

Graph of the ZZ-function along the critical line