↑

Properties

Label 4-1040e2-1.1-c1e2-0-77
Degree 44
Conductor 10816001081600
Sign 11
Analytic cond. 68.963768.9637
Root an. cond. 2.881742.88174
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 − 2·5-s − 4·7-s + 6·11-s + 2·13-s + 4·15-s + 2·19-s + 8·21-s − 6·23-s + 3·25-s + 2·27-s − 12·29-s − 10·31-s − 12·33-s + 8·35-s − 8·37-s − 4·39-s − 10·43-s − 12·47-s − 2·49-s − 12·55-s − 4·57-s + 6·59-s + 4·61-s − 4·65-s + 8·67-s + 12·69-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 1.51·7-s + 1.80·11-s + 0.554·13-s + 1.03·15-s + 0.458·19-s + 1.74·21-s − 1.25·23-s + 3/5·25-s + 0.384·27-s − 2.22·29-s − 1.79·31-s − 2.08·33-s + 1.35·35-s − 1.31·37-s − 0.640·39-s − 1.52·43-s − 1.75·47-s − 2/7·49-s − 1.61·55-s − 0.529·57-s + 0.781·59-s + 0.512·61-s − 0.496·65-s + 0.977·67-s + 1.44·69-s + ⋯

Functional equation

Λ(s)=(1081600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1081600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 10816001081600    =    28⋅52⋅1322^{8} \cdot 5^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 68.963768.9637
Root analytic conductor: 2.881742.88174
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 1081600, ( :1/2,1/2), 1)(4,\ 1081600,\ (\ :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
5C1C_1 (1+T)2 ( 1 + T )^{2}
13C1C_1 (1−T)2 ( 1 - T )^{2}
good3D4D_{4} 1+2T+4T2+2pT3+p2T4 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4}
7C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
11D4D_{4} 1−6T+28T2−6pT3+p2T4 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
19D4D_{4} 1−2T+12T2−2pT3+p2T4 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+6T+52T2+6pT3+p2T4 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+12T+82T2+12pT3+p2T4 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+10T+60T2+10pT3+p2T4 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4}
37C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
41C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
43D4D_{4} 1+10T+84T2+10pT3+p2T4 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4}
47C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
53C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
59D4D_{4} 1−6T−20T2−6pT3+p2T4 1 - 6 T - 20 T^{2} - 6 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−4T+18T2−4pT3+p2T4 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−8T+42T2−8pT3+p2T4 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+6T+148T2+6pT3+p2T4 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4}
73C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
79D4D_{4} 1+4T+54T2+4pT3+p2T4 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}
83C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
89D4D_{4} 1+12T+166T2+12pT3+p2T4 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
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

−9.532663191880723444619536247441, −9.520396530595932899143950961913, −8.967206648728870014215894131024, −8.540717878316464630724059987299, −8.085339355611054246272212776661, −7.49276571100856719837184004871, −6.96712216720779269762232073081, −6.79838215939637904678030754983, −6.15940545510306890101837070572, −6.12996587625445438988118560282, −5.32936653089787902577483318484, −5.24556846747375416793353123411, −4.25980771066364625189641702523, −3.81607625168915196341785738898, −3.41420358823914627451734117820, −3.33321581223917908753820993425, −1.98037084213370900213557926370, −1.41141287518027260408649480549, 0, 0, 1.41141287518027260408649480549, 1.98037084213370900213557926370, 3.33321581223917908753820993425, 3.41420358823914627451734117820, 3.81607625168915196341785738898, 4.25980771066364625189641702523, 5.24556846747375416793353123411, 5.32936653089787902577483318484, 6.12996587625445438988118560282, 6.15940545510306890101837070572, 6.79838215939637904678030754983, 6.96712216720779269762232073081, 7.49276571100856719837184004871, 8.085339355611054246272212776661, 8.540717878316464630724059987299, 8.967206648728870014215894131024, 9.520396530595932899143950961913, 9.532663191880723444619536247441

Graph of the ZZ-function along the critical line