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Properties

Label 4-987696-1.1-c1e2-0-3
Degree 44
Conductor 987696987696
Sign 11
Analytic cond. 62.976362.9763
Root an. cond. 2.817042.81704
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s − 4-s − 4·5-s + 2·6-s + 4·7-s − 3·8-s + 3·9-s − 4·10-s − 2·11-s − 2·12-s + 6·13-s + 4·14-s − 8·15-s − 16-s − 8·17-s + 3·18-s − 19-s + 4·20-s + 8·21-s − 2·22-s − 2·23-s − 6·24-s + 2·25-s + 6·26-s + 4·27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1/2·4-s − 1.78·5-s + 0.816·6-s + 1.51·7-s − 1.06·8-s + 9-s − 1.26·10-s − 0.603·11-s − 0.577·12-s + 1.66·13-s + 1.06·14-s − 2.06·15-s − 1/4·16-s − 1.94·17-s + 0.707·18-s − 0.229·19-s + 0.894·20-s + 1.74·21-s − 0.426·22-s − 0.417·23-s − 1.22·24-s + 2/5·25-s + 1.17·26-s + 0.769·27-s − 0.755·28-s + ⋯

Functional equation

Λ(s)=(987696s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(987696s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 987696987696    =    24⋅32⋅1932^{4} \cdot 3^{2} \cdot 19^{3}
Sign: 11
Analytic conductor: 62.976362.9763
Root analytic conductor: 2.817042.81704
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 987696, ( :1/2,1/2), 1)(4,\ 987696,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2842800232.284280023
L(12)L(\frac12) ≈\approx 2.2842800232.284280023
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)
bad2C2C_2 1−T+pT2 1 - T + p T^{2}
3C1C_1 (1−T)2 ( 1 - T )^{2}
19C1C_1 1+T 1 + T
good5C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
7C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
11C2C_2×\timesC2C_2 (1+pT2)(1+2T+pT2) ( 1 + p T^{2} )( 1 + 2 T + p T^{2} )
13C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
17C2C_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 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2×\timesC2C_2 (1+4T+pT2)(1+10T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−12T+pT2)(1+6T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−10T+pT2)(1+2T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2×\timesC2C_2 (1+4T+pT2)(1+16T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−16T+pT2)(1−10T+pT2) ( 1 - 16 T + p T^{2} )( 1 - 10 T + p T^{2} )
89C2C_2×\timesC2C_2 (1−18T+pT2)(1+2T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−12T+pT2)(1−10T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} )
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   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

−12.0798744008, −11.8739105144, −11.4085208449, −10.9289324032, −10.8415561337, −10.3317571499, −9.61000687688, −9.03211663271, −8.83746811854, −8.37436816344, −8.26351343286, −7.75989478086, −7.59950272568, −6.81360604684, −6.46787589567, −5.79402677262, −5.20430853708, −4.56991321081, −4.48660686111, −3.95264542154, −3.44365518363, −3.34310416861, −2.12198251427, −1.91788623711, −0.578917493177, 0.578917493177, 1.91788623711, 2.12198251427, 3.34310416861, 3.44365518363, 3.95264542154, 4.48660686111, 4.56991321081, 5.20430853708, 5.79402677262, 6.46787589567, 6.81360604684, 7.59950272568, 7.75989478086, 8.26351343286, 8.37436816344, 8.83746811854, 9.03211663271, 9.61000687688, 10.3317571499, 10.8415561337, 10.9289324032, 11.4085208449, 11.8739105144, 12.0798744008

Graph of the ZZ-function along the critical line