↑

Properties

Label 4-8788-1.1-c1e2-0-3
Degree 44
Conductor 87888788
Sign −1-1
Analytic cond. 0.5603300.560330
Root an. cond. 0.8651890.865189
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 6·5-s − 4·7-s − 5·9-s + 6·11-s + 13-s + 16-s − 6·17-s − 4·19-s − 6·20-s + 6·23-s + 17·25-s − 4·28-s + 6·29-s − 4·31-s + 24·35-s − 5·36-s − 10·37-s + 6·44-s + 30·45-s − 49-s + 52-s − 6·53-s − 36·55-s + 20·63-s + 64-s − 6·65-s + ⋯
L(s)  = 1  + 1/2·4-s − 2.68·5-s − 1.51·7-s − 5/3·9-s + 1.80·11-s + 0.277·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 1.34·20-s + 1.25·23-s + 17/5·25-s − 0.755·28-s + 1.11·29-s − 0.718·31-s + 4.05·35-s − 5/6·36-s − 1.64·37-s + 0.904·44-s + 4.47·45-s − 1/7·49-s + 0.138·52-s − 0.824·53-s − 4.85·55-s + 2.51·63-s + 1/8·64-s − 0.744·65-s + ⋯

Functional equation

Λ(s)=(8788s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8788s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 87888788    =    22⋅1332^{2} \cdot 13^{3}
Sign: −1-1
Analytic conductor: 0.5603300.560330
Root analytic conductor: 0.8651890.865189
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 8788, ( :1/2,1/2), −1)(4,\ 8788,\ (\ :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)
bad2C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
13C1C_1 1−T 1 - T
good3C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
5C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
7C2C_2×\timesC2C_2 (1+T+pT2)(1+3T+pT2) ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
19C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
29C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
31C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2×\timesC2C_2 (1+3T+pT2)(1+7T+pT2) ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} )
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
47C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
53C2C_2×\timesC2C_2 (1+pT2)(1+6T+pT2) ( 1 + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−14T+pT2)(1+12T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−15T+pT2)(1+3T+pT2) ( 1 - 15 T + p T^{2} )( 1 + 3 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−10T+pT2)(1−8T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−12T+pT2)(1−6T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} )
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2×\timesC2C_2 (1+10T+pT2)(1+12T+pT2) ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{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

−16.7401598333, −16.3289342707, −16.1610422211, −15.3407982120, −15.2621223361, −14.7871828532, −14.1081230548, −13.5230676379, −12.6948580012, −12.1081086828, −12.0906172632, −11.2148699926, −11.1734099077, −10.6571590377, −9.25208611998, −9.14676247888, −8.28912505160, −8.14577120022, −6.89586188103, −6.76423132328, −6.28038385959, −4.97317601948, −3.91117219162, −3.64028761626, −2.87766967615, 0, 2.87766967615, 3.64028761626, 3.91117219162, 4.97317601948, 6.28038385959, 6.76423132328, 6.89586188103, 8.14577120022, 8.28912505160, 9.14676247888, 9.25208611998, 10.6571590377, 11.1734099077, 11.2148699926, 12.0906172632, 12.1081086828, 12.6948580012, 13.5230676379, 14.1081230548, 14.7871828532, 15.2621223361, 15.3407982120, 16.1610422211, 16.3289342707, 16.7401598333

Graph of the ZZ-function along the critical line