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Properties

Label 4-987696-1.1-c1e2-0-12
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  + 2·2-s − 2·3-s + 2·4-s + 2·5-s − 4·6-s + 3·9-s + 4·10-s − 4·12-s − 4·15-s − 4·16-s + 6·17-s + 6·18-s + 19-s + 4·20-s − 7·25-s − 4·27-s − 8·30-s − 4·31-s − 8·32-s + 12·34-s + 6·36-s + 2·38-s + 6·45-s + 8·48-s − 5·49-s − 14·50-s − 12·51-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s + 0.894·5-s − 1.63·6-s + 9-s + 1.26·10-s − 1.15·12-s − 1.03·15-s − 16-s + 1.45·17-s + 1.41·18-s + 0.229·19-s + 0.894·20-s − 7/5·25-s − 0.769·27-s − 1.46·30-s − 0.718·31-s − 1.41·32-s + 2.05·34-s + 36-s + 0.324·38-s + 0.894·45-s + 1.15·48-s − 5/7·49-s − 1.97·50-s − 1.68·51-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: −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−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−T+pT2)2 ( 1 - T + p T^{2} )^{2}
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−3T+pT2)2 ( 1 - 3 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 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{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 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−7T+pT2)2 ( 1 - 7 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+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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

−7.75525037135579232003734806978, −7.06801910593054313669065331224, −7.04173846168624158743830695656, −6.11989947342759976206118717321, −6.02567308265222601671827257647, −5.52130085797761183807247002982, −5.49960710074578838011452836914, −4.80392893944635697455333037352, −4.29287803140015808576580756413, −3.90431057417865746907593992810, −3.22481329079945918066760847835, −2.76056898471702515117641585129, −1.85542976329431191785537331479, −1.37927155401612470737240964655, 0, 1.37927155401612470737240964655, 1.85542976329431191785537331479, 2.76056898471702515117641585129, 3.22481329079945918066760847835, 3.90431057417865746907593992810, 4.29287803140015808576580756413, 4.80392893944635697455333037352, 5.49960710074578838011452836914, 5.52130085797761183807247002982, 6.02567308265222601671827257647, 6.11989947342759976206118717321, 7.04173846168624158743830695656, 7.06801910593054313669065331224, 7.75525037135579232003734806978

Graph of the ZZ-function along the critical line