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Properties

Label 4-987696-1.1-c1e2-0-1
Degree 44
Conductor 987696987696
Sign −1-1
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 11

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 + 3·8-s + 3·9-s + 4·10-s + 2·12-s + 8·15-s − 16-s − 12·17-s − 3·18-s + 19-s + 4·20-s − 6·24-s + 2·25-s − 4·27-s − 8·30-s − 16·31-s − 5·32-s + 12·34-s − 3·36-s − 38-s − 12·40-s − 12·45-s + 2·48-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s − 1.78·5-s + 0.816·6-s + 1.06·8-s + 9-s + 1.26·10-s + 0.577·12-s + 2.06·15-s − 1/4·16-s − 2.91·17-s − 0.707·18-s + 0.229·19-s + 0.894·20-s − 1.22·24-s + 2/5·25-s − 0.769·27-s − 1.46·30-s − 2.87·31-s − 0.883·32-s + 2.05·34-s − 1/2·36-s − 0.162·38-s − 1.89·40-s − 1.78·45-s + 0.288·48-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: −1-1
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: 11
Selberg data: (4, 987696, ( :1/2,1/2), −1)(4,\ 987696,\ (\ :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)
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 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 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 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
89C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
97C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 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

−8.099644677533565214444467559710, −7.43259308735181576062150293394, −6.98236656892345640095319736227, −6.88281746236137995334549719629, −6.30774150521524867405980638808, −5.42974656732833448500123753190, −5.28029214515602867786662657712, −4.62864849822679992384179631378, −4.16838172205224943400021226417, −3.91841567788800315953565193679, −3.49609663069962101813276746331, −2.25179019930198933577153857080, −1.72203721066548968493435337820, −0.49711700208122723907813480784, 0, 0.49711700208122723907813480784, 1.72203721066548968493435337820, 2.25179019930198933577153857080, 3.49609663069962101813276746331, 3.91841567788800315953565193679, 4.16838172205224943400021226417, 4.62864849822679992384179631378, 5.28029214515602867786662657712, 5.42974656732833448500123753190, 6.30774150521524867405980638808, 6.88281746236137995334549719629, 6.98236656892345640095319736227, 7.43259308735181576062150293394, 8.099644677533565214444467559710

Graph of the ZZ-function along the critical line