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Properties

Label 4-987696-1.1-c1e2-0-5
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 yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s − 3·9-s + 19-s + 3·25-s + 2·43-s + 13·49-s + 14·61-s + 18·63-s − 6·73-s + 9·81-s + 15·121-s + 127-s + 131-s − 6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s − 3·171-s + 173-s − 18·175-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.26·7-s − 9-s + 0.229·19-s + 3/5·25-s + 0.304·43-s + 13/7·49-s + 1.79·61-s + 2.26·63-s − 0.702·73-s + 81-s + 1.36·121-s + 0.0887·127-s + 0.0873·131-s − 0.520·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s − 0.229·171-s + 0.0760·173-s − 1.36·175-s + 0.0747·179-s + 0.0743·181-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: yes
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)
bad2 1 1
3C2C_2 1+pT2 1 + p T^{2}
19C1C_1 1−T 1 - T
good5C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
7C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
11C22C_2^2 1−15T2+p2T4 1 - 15 T^{2} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C22C_2^2 1+29T2+p2T4 1 + 29 T^{2} + p^{2} T^{4}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
31C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
37C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
41C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
43C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
47C22C_2^2 1−87T2+p2T4 1 - 87 T^{2} + p^{2} T^{4}
53C22C_2^2 1+78T2+p2T4 1 + 78 T^{2} + p^{2} T^{4}
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 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
73C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
79C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1+86T2+p2T4 1 + 86 T^{2} + p^{2} T^{4}
89C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
97C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 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

−7.961129469380962181989895212878, −7.32305130052660723092311766345, −6.94425131620653819275823424387, −6.59760480537648271802059262796, −6.11070418133103761585213486013, −5.83615360060713533534812825033, −5.32506833441713424107768067043, −4.76444957510243367255024010339, −4.05768926443047187818056846955, −3.52186728479255860529351470583, −3.12901467819300418738126057534, −2.75614081137912399940820930647, −2.08483614218035674769636783112, −0.854302633003477795640210094843, 0, 0.854302633003477795640210094843, 2.08483614218035674769636783112, 2.75614081137912399940820930647, 3.12901467819300418738126057534, 3.52186728479255860529351470583, 4.05768926443047187818056846955, 4.76444957510243367255024010339, 5.32506833441713424107768067043, 5.83615360060713533534812825033, 6.11070418133103761585213486013, 6.59760480537648271802059262796, 6.94425131620653819275823424387, 7.32305130052660723092311766345, 7.961129469380962181989895212878

Graph of the ZZ-function along the critical line