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Properties

Label 4-588e2-1.1-c1e2-0-21
Degree 44
Conductor 345744345744
Sign −1-1
Analytic cond. 22.044922.0449
Root an. cond. 2.166842.16684
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  − 2·2-s + 2·4-s − 3·9-s − 4·16-s + 6·18-s + 8·25-s − 14·29-s + 8·32-s − 6·36-s − 16·50-s + 10·53-s + 28·58-s − 8·64-s + 9·81-s + 16·100-s − 20·106-s − 2·113-s − 28·116-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + 149-s + 151-s + 157-s − 18·162-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 9-s − 16-s + 1.41·18-s + 8/5·25-s − 2.59·29-s + 1.41·32-s − 36-s − 2.26·50-s + 1.37·53-s + 3.67·58-s − 64-s + 81-s + 8/5·100-s − 1.94·106-s − 0.188·113-s − 2.59·116-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 1.41·162-s + ⋯

Functional equation

Λ(s)=(345744s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(345744s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 345744345744    =    24⋅32⋅742^{4} \cdot 3^{2} \cdot 7^{4}
Sign: −1-1
Analytic conductor: 22.044922.0449
Root analytic conductor: 2.166842.16684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 345744, ( :1/2,1/2), −1)(4,\ 345744,\ (\ :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}
3C2C_2 1+pT2 1 + p T^{2}
7 1 1
good5C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
11C22C_2^2 1+p2T4 1 + p^{2} T^{4}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 1+p2T4 1 + p^{2} T^{4}
29C2C_2 (1+4T+pT2)(1+10T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C22C_2^2 1−80T2+p2T4 1 - 80 T^{2} + p^{2} T^{4}
43C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−14T+pT2)(1+4T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
71C22C_2^2 1+p2T4 1 + p^{2} T^{4}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1+160T2+p2T4 1 + 160 T^{2} + p^{2} T^{4}
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 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

−8.675283624900062511911811378656, −8.165842182931156078591588403838, −7.56253085249857986653649458051, −7.43401045105870059031051173050, −6.73681136581834265637925026910, −6.39296396190320632860668721558, −5.59206924634252637463300387111, −5.35483700669784098380769784887, −4.64926701107503784437807501566, −3.94848150715579718577789265617, −3.32917196730094299212080403177, −2.57865934666678914372372469635, −2.00275142514673925542798845843, −1.07413156948196004775276684427, 0, 1.07413156948196004775276684427, 2.00275142514673925542798845843, 2.57865934666678914372372469635, 3.32917196730094299212080403177, 3.94848150715579718577789265617, 4.64926701107503784437807501566, 5.35483700669784098380769784887, 5.59206924634252637463300387111, 6.39296396190320632860668721558, 6.73681136581834265637925026910, 7.43401045105870059031051173050, 7.56253085249857986653649458051, 8.165842182931156078591588403838, 8.675283624900062511911811378656

Graph of the ZZ-function along the critical line