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Properties

Label 4-332928-1.1-c1e2-0-21
Degree 44
Conductor 332928332928
Sign −1-1
Analytic cond. 21.227721.2277
Root an. cond. 2.146472.14647
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-s − 3·3-s + 4-s − 3·6-s + 8-s + 6·9-s − 3·11-s − 3·12-s + 16-s + 6·18-s − 5·19-s − 3·22-s − 3·24-s + 3·25-s − 9·27-s + 32-s + 9·33-s + 6·36-s − 5·38-s − 3·41-s + 10·43-s − 3·44-s − 3·48-s − 8·49-s + 3·50-s − 9·54-s + 15·57-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s + 0.353·8-s + 2·9-s − 0.904·11-s − 0.866·12-s + 1/4·16-s + 1.41·18-s − 1.14·19-s − 0.639·22-s − 0.612·24-s + 3/5·25-s − 1.73·27-s + 0.176·32-s + 1.56·33-s + 36-s − 0.811·38-s − 0.468·41-s + 1.52·43-s − 0.452·44-s − 0.433·48-s − 8/7·49-s + 0.424·50-s − 1.22·54-s + 1.98·57-s + ⋯

Functional equation

Λ(s)=(332928s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(332928s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 332928332928    =    27⋅32⋅1722^{7} \cdot 3^{2} \cdot 17^{2}
Sign: −1-1
Analytic conductor: 21.227721.2277
Root analytic conductor: 2.146472.14647
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 332928, ( :1/2,1/2), −1)(4,\ 332928,\ (\ :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)
bad2C1C_1 1−T 1 - T
3C2C_2 1+pT+pT2 1 + p T + p T^{2}
17C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good5C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
7C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
11C2C_2×\timesC2C_2 (1−2T+pT2)(1+5T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} )
13C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1+T+pT2)(1+4T+pT2) ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} )
23C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
29C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
31C22C_2^2 1+32T2+p2T4 1 + 32 T^{2} + p^{2} T^{4}
37C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1−2T+pT2)(1+5T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−11T+pT2)(1+T+pT2) ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} )
47C22C_2^2 1+51T2+p2T4 1 + 51 T^{2} + p^{2} T^{4}
53C22C_2^2 1−11T2+p2T4 1 - 11 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−3T+pT2)(1+pT2) ( 1 - 3 T + p T^{2} )( 1 + p T^{2} )
61C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C22C_2^2 1+75T2+p2T4 1 + 75 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−7T+pT2)(1+2T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C22C_2^2 1−104T2+p2T4 1 - 104 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1−9T+pT2)(1+12T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2×\timesC2C_2 (1−6T+pT2)(1+9T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−7T+pT2)(1−T+pT2) ( 1 - 7 T + p T^{2} )( 1 - 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

−8.604969221559244333634630913098, −7.69823812494879928556312846174, −7.59776998434374205435069602344, −6.89865749107197294369399543033, −6.45782929427348210743724639109, −6.13709332444057347666756704097, −5.65045951154400845042854387589, −5.10870910841514021482586673894, −4.81103346199117845966800164034, −4.28088665839059093669700037659, −3.71375658431355539997467716583, −2.84206545343609058705195086989, −2.15686340953137112061921451226, −1.19915912930715292553423329301, 0, 1.19915912930715292553423329301, 2.15686340953137112061921451226, 2.84206545343609058705195086989, 3.71375658431355539997467716583, 4.28088665839059093669700037659, 4.81103346199117845966800164034, 5.10870910841514021482586673894, 5.65045951154400845042854387589, 6.13709332444057347666756704097, 6.45782929427348210743724639109, 6.89865749107197294369399543033, 7.59776998434374205435069602344, 7.69823812494879928556312846174, 8.604969221559244333634630913098

Graph of the ZZ-function along the critical line