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Properties

Label 4-259200-1.1-c1e2-0-40
Degree 44
Conductor 259200259200
Sign −1-1
Analytic cond. 16.526816.5268
Root an. cond. 2.016262.01626
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 + 4-s − 3·5-s − 8-s + 3·10-s + 16-s + 19-s − 3·20-s − 3·23-s + 4·25-s − 3·29-s − 32-s − 38-s + 3·40-s + 19·43-s + 3·46-s − 12·47-s − 10·49-s − 4·50-s + 15·53-s + 3·58-s + 64-s − 17·67-s + 15·71-s + 4·73-s + 76-s − 3·80-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 1/4·16-s + 0.229·19-s − 0.670·20-s − 0.625·23-s + 4/5·25-s − 0.557·29-s − 0.176·32-s − 0.162·38-s + 0.474·40-s + 2.89·43-s + 0.442·46-s − 1.75·47-s − 1.42·49-s − 0.565·50-s + 2.06·53-s + 0.393·58-s + 1/8·64-s − 2.07·67-s + 1.78·71-s + 0.468·73-s + 0.114·76-s − 0.335·80-s + ⋯

Functional equation

Λ(s)=(259200s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(259200s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 259200259200    =    27⋅34⋅522^{7} \cdot 3^{4} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 16.526816.5268
Root analytic conductor: 2.016262.01626
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 259200, ( :1/2,1/2), −1)(4,\ 259200,\ (\ :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
3 1 1
5C2C_2 1+3T+pT2 1 + 3 T + p T^{2}
good7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1−2T+pT2)(1+T+pT2) ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} )
23C2C_2×\timesC2C_2 (1+pT2)(1+3T+pT2) ( 1 + p T^{2} )( 1 + 3 T + p T^{2} )
29C2C_2×\timesC2C_2 (1−3T+pT2)(1+6T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
37C22C_2^2 1+7T2+p2T4 1 + 7 T^{2} + p^{2} T^{4}
41C22C_2^2 1−80T2+p2T4 1 - 80 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−11T+pT2)(1−8T+pT2) ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} )
47C2C_2×\timesC2C_2 (1+3T+pT2)(1+9T+pT2) ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−9T+pT2)(1−6T+pT2) ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} )
59C22C_2^2 1−89T2+p2T4 1 - 89 T^{2} + p^{2} T^{4}
61C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1+4T+pT2)(1+13T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−12T+pT2)(1−3T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 3 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−5T+pT2)(1+T+pT2) ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
89C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−14T+pT2)(1+10T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 10 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

−8.592024887147004238522478109400, −8.191195454658669617417835931001, −7.81293625892389359448959552865, −7.32063813036665890781783507491, −7.12734422063696952927030048463, −6.29167561017585931333796614613, −5.99135520826587030680866696404, −5.23616651406848519937671396223, −4.67460018755356754408633544587, −3.95085073190493848317073200703, −3.67797931095645385521420582580, −2.87648326950427116269956643477, −2.18190094213517380836854554891, −1.12540824385951466905642356085, 0, 1.12540824385951466905642356085, 2.18190094213517380836854554891, 2.87648326950427116269956643477, 3.67797931095645385521420582580, 3.95085073190493848317073200703, 4.67460018755356754408633544587, 5.23616651406848519937671396223, 5.99135520826587030680866696404, 6.29167561017585931333796614613, 7.12734422063696952927030048463, 7.32063813036665890781783507491, 7.81293625892389359448959552865, 8.191195454658669617417835931001, 8.592024887147004238522478109400

Graph of the ZZ-function along the critical line