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Properties

Label 4-936e2-1.1-c1e2-0-32
Degree 44
Conductor 876096876096
Sign 11
Analytic cond. 55.860655.8606
Root an. cond. 2.733862.73386
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 6·7-s + 12·14-s − 4·16-s + 14·17-s − 8·23-s + 25-s + 12·28-s − 16·31-s − 8·32-s + 28·34-s − 4·41-s − 16·46-s + 14·47-s + 13·49-s + 2·50-s − 32·62-s − 8·64-s + 28·68-s + 6·71-s + 28·73-s − 20·79-s − 8·82-s − 16·92-s + 28·94-s + 16·97-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 2.26·7-s + 3.20·14-s − 16-s + 3.39·17-s − 1.66·23-s + 1/5·25-s + 2.26·28-s − 2.87·31-s − 1.41·32-s + 4.80·34-s − 0.624·41-s − 2.35·46-s + 2.04·47-s + 13/7·49-s + 0.282·50-s − 4.06·62-s − 64-s + 3.39·68-s + 0.712·71-s + 3.27·73-s − 2.25·79-s − 0.883·82-s − 1.66·92-s + 2.88·94-s + 1.62·97-s + ⋯

Functional equation

Λ(s)=(876096s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(876096s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 876096876096    =    26⋅34⋅1322^{6} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 55.860655.8606
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 876096, ( :1/2,1/2), 1)(4,\ 876096,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.2076786496.207678649
L(12)L(\frac12) ≈\approx 6.2076786496.207678649
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}
3 1 1
13C2C_2 1+T2 1 + T^{2}
good5C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
7C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
29C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43C22C_2^2 1−85T2+p2T4 1 - 85 T^{2} + p^{2} T^{4}
47C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
53C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C22C_2^2 1+78T2+p2T4 1 + 78 T^{2} + p^{2} T^{4}
61C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
71C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
73C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
79C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
83C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{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

−10.54879325857465538070230369061, −9.810929708715935350486376354643, −9.568344033548428929447615449757, −9.002976634679524268556987485372, −8.256892312600616901149994718986, −8.189352302905368400678882268362, −7.74795126446030632420585194532, −7.23009806150085706545676071888, −7.04750305861168250111128132207, −5.90549497669378764643639949139, −5.80182269904285993340750573937, −5.41375683031812336635310272242, −5.12351612177070882626560888552, −4.54168481132733101164581341578, −4.06336662775372832913137138087, −3.45500436521338596356623048772, −3.30337636812120140812732157124, −2.07875543361596919789728301552, −1.91542847886084698137045959320, −1.01251428109686507359472261915, 1.01251428109686507359472261915, 1.91542847886084698137045959320, 2.07875543361596919789728301552, 3.30337636812120140812732157124, 3.45500436521338596356623048772, 4.06336662775372832913137138087, 4.54168481132733101164581341578, 5.12351612177070882626560888552, 5.41375683031812336635310272242, 5.80182269904285993340750573937, 5.90549497669378764643639949139, 7.04750305861168250111128132207, 7.23009806150085706545676071888, 7.74795126446030632420585194532, 8.189352302905368400678882268362, 8.256892312600616901149994718986, 9.002976634679524268556987485372, 9.568344033548428929447615449757, 9.810929708715935350486376354643, 10.54879325857465538070230369061

Graph of the ZZ-function along the critical line