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Properties

Label 4-936e2-1.1-c1e2-0-37
Degree 44
Conductor 876096876096
Sign −1-1
Analytic cond. 55.860655.8606
Root an. cond. 2.733862.73386
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  − 7-s − 13-s − 2·19-s − 6·25-s + 4·37-s − 43-s − 49-s + 7·61-s + 9·67-s + 6·79-s + 91-s − 17·97-s + 10·103-s + 24·109-s − 19·121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.277·13-s − 0.458·19-s − 6/5·25-s + 0.657·37-s − 0.152·43-s − 1/7·49-s + 0.896·61-s + 1.09·67-s + 0.675·79-s + 0.104·91-s − 1.72·97-s + 0.985·103-s + 2.29·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.173·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-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: −1-1
Analytic conductor: 55.860655.8606
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 876096, ( :1/2,1/2), −1)(4,\ 876096,\ (\ :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)
bad2 1 1
3 1 1
13C2C_2 1+T+pT2 1 + T + p T^{2}
good5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2×\timesC2C_2 (1−3T+pT2)(1+4T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
17C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1−3T+pT2)(1+5T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} )
23C22C_2^2 1+25T2+p2T4 1 + 25 T^{2} + p^{2} T^{4}
29C22C_2^2 1+43T2+p2T4 1 + 43 T^{2} + p^{2} T^{4}
31C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−3T+pT2)(1−T+pT2) ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} )
41C22C_2^2 1−7T2+p2T4 1 - 7 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+pT2)(1+T+pT2) ( 1 + p T^{2} )( 1 + T + p T^{2} )
47C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
53C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
59C22C_2^2 1−29T2+p2T4 1 - 29 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−5T+pT2)(1−2T+pT2) ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−8T+pT2)(1−T+pT2) ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} )
71C22C_2^2 1+103T2+p2T4 1 + 103 T^{2} + p^{2} T^{4}
73C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
79C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
83C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
89C22C_2^2 1+41T2+p2T4 1 + 41 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−2T+pT2)(1+19T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 19 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

−7.983737791080642132736814215363, −7.51854729411584168731948285343, −7.15093699372760938755876813366, −6.55607214421904824418352370269, −6.26234494744199123553130424064, −5.76162824470031636265799774780, −5.27296627730458637660873384431, −4.77234832397501308317155493789, −4.19694773271514163854185346296, −3.74019924531454921592039403196, −3.24290860878429919236845747336, −2.44900145190196715396620308329, −2.08417146527956992185093097602, −1.10282936938978031774324225005, 0, 1.10282936938978031774324225005, 2.08417146527956992185093097602, 2.44900145190196715396620308329, 3.24290860878429919236845747336, 3.74019924531454921592039403196, 4.19694773271514163854185346296, 4.77234832397501308317155493789, 5.27296627730458637660873384431, 5.76162824470031636265799774780, 6.26234494744199123553130424064, 6.55607214421904824418352370269, 7.15093699372760938755876813366, 7.51854729411584168731948285343, 7.983737791080642132736814215363

Graph of the ZZ-function along the critical line