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Properties

Label 4-855e2-1.1-c1e2-0-13
Degree 44
Conductor 731025731025
Sign −1-1
Analytic cond. 46.610746.6107
Root an. cond. 2.612892.61289
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  − 2·4-s + 10·13-s − 19-s + 25-s + 3·31-s − 12·37-s − 10·43-s − 10·49-s − 20·52-s + 2·61-s + 8·64-s + 4·67-s − 8·73-s + 2·76-s + 2·79-s − 22·97-s − 2·100-s − 36·103-s − 7·109-s − 9·121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 24·148-s + 149-s + ⋯
L(s)  = 1  − 4-s + 2.77·13-s − 0.229·19-s + 1/5·25-s + 0.538·31-s − 1.97·37-s − 1.52·43-s − 1.42·49-s − 2.77·52-s + 0.256·61-s + 64-s + 0.488·67-s − 0.936·73-s + 0.229·76-s + 0.225·79-s − 2.23·97-s − 1/5·100-s − 3.54·103-s − 0.670·109-s − 0.818·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.97·148-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 731025731025    =    34⋅52⋅1923^{4} \cdot 5^{2} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 46.610746.6107
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 731025, ( :1/2,1/2), −1)(4,\ 731025,\ (\ :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)
bad3 1 1
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
19C2C_2 1+T+pT2 1 + T + p T^{2}
good2C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C22C_2^2 1+9T2+p2T4 1 + 9 T^{2} + p^{2} T^{4}
13C2C_2 (1−6T+pT2)(1−4T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} )
17C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
23C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
29C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
31C2C_2×\timesC2C_2 (1−3T+pT2)(1+pT2) ( 1 - 3 T + p T^{2} )( 1 + p T^{2} )
37C2C_2×\timesC2C_2 (1+4T+pT2)(1+8T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+4T+pT2)(1+6T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
47C22C_2^2 1−71T2+p2T4 1 - 71 T^{2} + p^{2} T^{4}
53C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
59C22C_2^2 1+61T2+p2T4 1 + 61 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−13T+pT2)(1+11T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−6T+pT2)(1+2T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )
71C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−8T+pT2)(1+16T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
83C22C_2^2 1+93T2+p2T4 1 + 93 T^{2} + p^{2} T^{4}
89C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
97C2C_2×\timesC2C_2 (1+4T+pT2)(1+18T+pT2) ( 1 + 4 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

−8.270731226826548888974275331436, −7.979861869097349205620026627173, −6.87244029092208968082688570656, −6.80574947248602853190266223077, −6.35450331544968665653471087535, −5.65924042232185999500062358436, −5.41049972917035471336152487853, −4.77929711168029645016524439593, −4.28187877298872610960741606633, −3.71984530305062655813366610869, −3.48894881547595653665387029712, −2.76768644633230574350606151134, −1.67643204003326032024474779265, −1.23589929378979944378282330870, 0, 1.23589929378979944378282330870, 1.67643204003326032024474779265, 2.76768644633230574350606151134, 3.48894881547595653665387029712, 3.71984530305062655813366610869, 4.28187877298872610960741606633, 4.77929711168029645016524439593, 5.41049972917035471336152487853, 5.65924042232185999500062358436, 6.35450331544968665653471087535, 6.80574947248602853190266223077, 6.87244029092208968082688570656, 7.979861869097349205620026627173, 8.270731226826548888974275331436

Graph of the ZZ-function along the critical line