↑

Properties

Label 4-3192e2-1.1-c1e2-0-2
Degree 44
Conductor 1018886410188864
Sign 11
Analytic cond. 649.650649.650
Root an. cond. 5.048585.04858
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  + 2·3-s − 4·5-s − 2·7-s + 3·9-s − 8·15-s − 4·17-s + 2·19-s − 4·21-s + 4·25-s + 4·27-s + 12·29-s + 12·31-s + 8·35-s − 12·37-s − 12·41-s − 12·43-s − 12·45-s − 24·47-s + 3·49-s − 8·51-s + 4·53-s + 4·57-s − 16·59-s + 4·61-s − 6·63-s − 24·67-s − 12·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 0.755·7-s + 9-s − 2.06·15-s − 0.970·17-s + 0.458·19-s − 0.872·21-s + 4/5·25-s + 0.769·27-s + 2.22·29-s + 2.15·31-s + 1.35·35-s − 1.97·37-s − 1.87·41-s − 1.82·43-s − 1.78·45-s − 3.50·47-s + 3/7·49-s − 1.12·51-s + 0.549·53-s + 0.529·57-s − 2.08·59-s + 0.512·61-s − 0.755·63-s − 2.93·67-s − 1.40·73-s + ⋯

Functional equation

Λ(s)=(10188864s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10188864s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1018886410188864    =    26⋅32⋅72⋅1922^{6} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 649.650649.650
Root analytic conductor: 5.048585.04858
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 10188864, ( :1/2,1/2), 1)(4,\ 10188864,\ (\ :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
3C1C_1 (1−T)2 ( 1 - T )^{2}
7C1C_1 (1+T)2 ( 1 + T )^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good5D4D_{4} 1+4T+12T2+4pT3+p2T4 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
13C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
17D4D_{4} 1+4T+20T2+4pT3+p2T4 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
29D4D_{4} 1−12T+76T2−12pT3+p2T4 1 - 12 T + 76 T^{2} - 12 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−12T+90T2−12pT3+p2T4 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4}
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41D4D_{4} 1+12T+110T2+12pT3+p2T4 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+12T+114T2+12pT3+p2T4 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+24T+236T2+24pT3+p2T4 1 + 24 T + 236 T^{2} + 24 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−4T+60T2−4pT3+p2T4 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+16T+150T2+16pT3+p2T4 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67D4D_{4} 1+24T+270T2+24pT3+p2T4 1 + 24 T + 270 T^{2} + 24 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
73D4D_{4} 1+12T+174T2+12pT3+p2T4 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−16T+190T2−16pT3+p2T4 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+16T+212T2+16pT3+p2T4 1 + 16 T + 212 T^{2} + 16 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+28T+366T2+28pT3+p2T4 1 + 28 T + 366 T^{2} + 28 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+12T+198T2+12pT3+p2T4 1 + 12 T + 198 T^{2} + 12 p T^{3} + 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

−8.462399436032068907590945347015, −8.274935678345561594433718390332, −7.903796292839556892021560512037, −7.37413780590136710789175356611, −6.98997765969264461242392868366, −6.83477816890128458220995817469, −6.23352844012666151409034838664, −6.23143646501705730214391949789, −5.03506044716846449582163155312, −4.94701896220361361764655710890, −4.47071256086318044548912069066, −4.20547437352001516235749198810, −3.44354079713731384784731494977, −3.40258510178812708964369422359, −2.93877897115368570253503994234, −2.66814985854442093997786777427, −1.61911530681874039405030078714, −1.40567775873301748409140799060, 0, 0, 1.40567775873301748409140799060, 1.61911530681874039405030078714, 2.66814985854442093997786777427, 2.93877897115368570253503994234, 3.40258510178812708964369422359, 3.44354079713731384784731494977, 4.20547437352001516235749198810, 4.47071256086318044548912069066, 4.94701896220361361764655710890, 5.03506044716846449582163155312, 6.23143646501705730214391949789, 6.23352844012666151409034838664, 6.83477816890128458220995817469, 6.98997765969264461242392868366, 7.37413780590136710789175356611, 7.903796292839556892021560512037, 8.274935678345561594433718390332, 8.462399436032068907590945347015

Graph of the ZZ-function along the critical line