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Properties

Label 4-259200-1.1-c1e2-0-41
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 + 5·19-s − 3·20-s + 3·23-s + 4·25-s − 9·29-s − 32-s − 5·38-s + 3·40-s − 9·43-s − 3·46-s − 15·47-s + 4·49-s − 4·50-s + 18·53-s + 9·58-s + 64-s + 18·67-s − 9·71-s − 27·73-s + 5·76-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 + 1.14·19-s − 0.670·20-s + 0.625·23-s + 4/5·25-s − 1.67·29-s − 0.176·32-s − 0.811·38-s + 0.474·40-s − 1.37·43-s − 0.442·46-s − 2.18·47-s + 4/7·49-s − 0.565·50-s + 2.47·53-s + 1.18·58-s + 1/8·64-s + 2.19·67-s − 1.06·71-s − 3.16·73-s + 0.573·76-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}
good7C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
11C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
13C22C_2^2 1−19T2+p2T4 1 - 19 T^{2} + p^{2} T^{4}
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2×\timesC2C_2 (1−7T+pT2)(1+2T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2×\timesC2C_2 (1−6T+pT2)(1+3T+pT2) ( 1 - 6 T + 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 1−17T2+p2T4 1 - 17 T^{2} + p^{2} T^{4}
37C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
41C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+T+pT2)(1+8T+pT2) ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2×\timesC2C_2 (1+3T+pT2)(1+12T+pT2) ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
59C22C_2^2 1+80T2+p2T4 1 + 80 T^{2} + p^{2} T^{4}
61C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−14T+pT2)(1−4T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} )
71C2C_2×\timesC2C_2 (1+pT2)(1+9T+pT2) ( 1 + p T^{2} )( 1 + 9 T + p T^{2} )
73C2C_2×\timesC2C_2 (1+11T+pT2)(1+16T+pT2) ( 1 + 11 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C22C_2^2 1+103T2+p2T4 1 + 103 T^{2} + p^{2} T^{4}
83C22C_2^2 1−113T2+p2T4 1 - 113 T^{2} + p^{2} T^{4}
89C22C_2^2 1−43T2+p2T4 1 - 43 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1+T+pT2)(1+8T+pT2) ( 1 + T + p T^{2} )( 1 + 8 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

−8.697195746157920782875702812837, −8.255504638981699164869591310184, −7.70968573373049697344916592480, −7.34107847737531780917400384171, −7.07166907084479305676430039793, −6.51718624391523389418725774430, −5.71974941571566655766109684626, −5.34729709925935676627870313075, −4.69327716580066576617467360574, −4.02815771306921680916527359686, −3.44958169279328725318537128187, −3.07375791172403503107853326193, −2.09245008436728335076150952409, −1.15529697078920214006424913776, 0, 1.15529697078920214006424913776, 2.09245008436728335076150952409, 3.07375791172403503107853326193, 3.44958169279328725318537128187, 4.02815771306921680916527359686, 4.69327716580066576617467360574, 5.34729709925935676627870313075, 5.71974941571566655766109684626, 6.51718624391523389418725774430, 7.07166907084479305676430039793, 7.34107847737531780917400384171, 7.70968573373049697344916592480, 8.255504638981699164869591310184, 8.697195746157920782875702812837

Graph of the ZZ-function along the critical line