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Properties

Label 4-91e4-1.1-c1e2-0-8
Degree 44
Conductor 6857496168574961
Sign 11
Analytic cond. 4372.394372.39
Root an. cond. 8.131678.13167
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4·4-s + 3·8-s − 9-s + 6·11-s + 3·16-s + 6·17-s − 3·18-s + 6·19-s + 18·22-s − 12·23-s − 5·25-s + 10·31-s + 6·32-s + 18·34-s − 4·36-s + 4·37-s + 18·38-s − 16·43-s + 24·44-s − 36·46-s − 6·47-s − 15·50-s − 6·53-s + 6·59-s − 6·61-s + 30·62-s + ⋯
L(s)  = 1  + 2.12·2-s + 2·4-s + 1.06·8-s − 1/3·9-s + 1.80·11-s + 3/4·16-s + 1.45·17-s − 0.707·18-s + 1.37·19-s + 3.83·22-s − 2.50·23-s − 25-s + 1.79·31-s + 1.06·32-s + 3.08·34-s − 2/3·36-s + 0.657·37-s + 2.91·38-s − 2.43·43-s + 3.61·44-s − 5.30·46-s − 0.875·47-s − 2.12·50-s − 0.824·53-s + 0.781·59-s − 0.768·61-s + 3.81·62-s + ⋯

Functional equation

Λ(s)=(68574961s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(68574961s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6857496168574961    =    74⋅1347^{4} \cdot 13^{4}
Sign: 11
Analytic conductor: 4372.394372.39
Root analytic conductor: 8.131678.13167
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 68574961, ( :1/2,1/2), 1)(4,\ 68574961,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 11.3781621811.37816218
L(12)L(\frac12) ≈\approx 11.3781621811.37816218
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)
bad7 1 1
13 1 1
good2C22C_2^2 1−3T+5T2−3pT3+p2T4 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4}
3C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
5C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
17D4D_{4} 1−6T+23T2−6pT3+p2T4 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
19C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
23D4D_{4} 1+12T+77T2+12pT3+p2T4 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
31C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
37D4D_{4} 1−4T+33T2−4pT3+p2T4 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
43C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
47D4D_{4} 1+6T+83T2+6pT3+p2T4 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+6T+95T2+6pT3+p2T4 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−6T+107T2−6pT3+p2T4 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4}
61C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
67C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
71C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
73D4D_{4} 1−8T+117T2−8pT3+p2T4 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−8T+129T2−8pT3+p2T4 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1+173T2+p2T4 1 + 173 T^{2} + p^{2} T^{4}
97D4D_{4} 1+8T+30T2+8pT3+p2T4 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4}
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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

−7.982820559881118595989684120727, −7.73496349705862751558404028915, −7.03618598381248319923111028360, −6.69005494630249353699739735354, −6.41351457488829962799862626931, −6.20328545158585072117424451383, −5.65978229613637070655340511471, −5.62405833519709966136408854330, −5.12242155758843402766723540319, −4.89862679230168212830244837030, −4.21939438976793778270733002014, −4.17052913989804499796135433640, −3.83523993381332403445586364991, −3.44986878532506369747052712632, −3.00538890566313823792091989657, −2.94719526213056545435382393792, −2.01256227550936978261495537370, −1.66539717911064327220235980572, −1.24658112869423572569226566663, −0.54379000206957633115181158789, 0.54379000206957633115181158789, 1.24658112869423572569226566663, 1.66539717911064327220235980572, 2.01256227550936978261495537370, 2.94719526213056545435382393792, 3.00538890566313823792091989657, 3.44986878532506369747052712632, 3.83523993381332403445586364991, 4.17052913989804499796135433640, 4.21939438976793778270733002014, 4.89862679230168212830244837030, 5.12242155758843402766723540319, 5.62405833519709966136408854330, 5.65978229613637070655340511471, 6.20328545158585072117424451383, 6.41351457488829962799862626931, 6.69005494630249353699739735354, 7.03618598381248319923111028360, 7.73496349705862751558404028915, 7.982820559881118595989684120727

Graph of the ZZ-function along the critical line