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Properties

Label 4-987696-1.1-c1e2-0-8
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 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s − 6·5-s + 4·6-s + 3·9-s + 12·10-s − 4·12-s + 12·15-s − 4·16-s − 2·17-s − 6·18-s − 19-s − 12·20-s + 17·25-s − 4·27-s − 24·30-s − 12·31-s + 8·32-s + 4·34-s + 6·36-s + 2·38-s − 18·45-s + 8·48-s + 11·49-s − 34·50-s + 4·51-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s − 2.68·5-s + 1.63·6-s + 9-s + 3.79·10-s − 1.15·12-s + 3.09·15-s − 16-s − 0.485·17-s − 1.41·18-s − 0.229·19-s − 2.68·20-s + 17/5·25-s − 0.769·27-s − 4.38·30-s − 2.15·31-s + 1.41·32-s + 0.685·34-s + 36-s + 0.324·38-s − 2.68·45-s + 1.15·48-s + 11/7·49-s − 4.80·50-s + 0.560·51-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: 22
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+pT+pT2 1 + p T + p T^{2}
3C1C_1 (1+T)2 ( 1 + T )^{2}
19C1C_1 1+T 1 + T
good5C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
7C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
11C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1+T+pT2)2 ( 1 + 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+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
37C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{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 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
61C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
67C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C2C_2 (1+11T+pT2)2 ( 1 + 11 T + p T^{2} )^{2}
79C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 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 (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

−7.48919464134144900735099782123, −7.47848778675296589128891353641, −7.27079151643991129277821474463, −6.41173993136661837405396202601, −6.25499216179811745555302525608, −5.36600607918508759890630649285, −4.89390739159963849894021052339, −4.42808617434113254239775941766, −3.81591641511052084615077615666, −3.79824334094358549621418265757, −2.80250883164613859029957691965, −1.86642316096709822967327164489, −1.02602590121900935054247645611, 0, 0, 1.02602590121900935054247645611, 1.86642316096709822967327164489, 2.80250883164613859029957691965, 3.79824334094358549621418265757, 3.81591641511052084615077615666, 4.42808617434113254239775941766, 4.89390739159963849894021052339, 5.36600607918508759890630649285, 6.25499216179811745555302525608, 6.41173993136661837405396202601, 7.27079151643991129277821474463, 7.47848778675296589128891353641, 7.48919464134144900735099782123

Graph of the ZZ-function along the critical line