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Properties

Label 4-88e4-1.1-c1e2-0-27
Degree 44
Conductor 5996953659969536
Sign 11
Analytic cond. 3823.703823.70
Root an. cond. 7.863597.86359
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·5-s − 2·7-s − 9-s − 2·13-s − 3·15-s − 4·17-s + 8·19-s − 2·21-s − 9·23-s + 25-s − 2·29-s + 7·31-s + 6·35-s + 11·37-s − 2·39-s − 6·41-s + 6·43-s + 3·45-s − 16·47-s + 6·49-s − 4·51-s − 8·53-s + 8·57-s − 5·59-s − 6·61-s + 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 0.755·7-s − 1/3·9-s − 0.554·13-s − 0.774·15-s − 0.970·17-s + 1.83·19-s − 0.436·21-s − 1.87·23-s + 1/5·25-s − 0.371·29-s + 1.25·31-s + 1.01·35-s + 1.80·37-s − 0.320·39-s − 0.937·41-s + 0.914·43-s + 0.447·45-s − 2.33·47-s + 6/7·49-s − 0.560·51-s − 1.09·53-s + 1.05·57-s − 0.650·59-s − 0.768·61-s + 0.251·63-s + ⋯

Functional equation

Λ(s)=(59969536s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(59969536s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5996953659969536    =    212⋅1142^{12} \cdot 11^{4}
Sign: 11
Analytic conductor: 3823.703823.70
Root analytic conductor: 7.863597.86359
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 59969536, ( :1/2,1/2), 1)(4,\ 59969536,\ (\ :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
11 1 1
good3D4D_{4} 1−T+2T2−pT3+p2T4 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4}
5C22C_2^2 1+3T+8T2+3pT3+p2T4 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
7D4D_{4} 1+2T−2T2+2pT3+p2T4 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
13C4C_4 1+2T+10T2+2pT3+p2T4 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4}
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23D4D_{4} 1+9T+62T2+9pT3+p2T4 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+2T+42T2+2pT3+p2T4 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−7T+70T2−7pT3+p2T4 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−11T+100T2−11pT3+p2T4 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+6T+74T2+6pT3+p2T4 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−6T+78T2−6pT3+p2T4 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4}
47C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
53D4D_{4} 1+8T+54T2+8pT3+p2T4 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+5T+18T2+5pT3+p2T4 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+6T+114T2+6pT3+p2T4 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−15T+186T2−15pT3+p2T4 1 - 15 T + 186 T^{2} - 15 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−5T+110T2−5pT3+p2T4 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+2T+130T2+2pT3+p2T4 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+14T+190T2+14pT3+p2T4 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+10T+174T2+10pT3+p2T4 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+7T+152T2+7pT3+p2T4 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−27T+372T2−27pT3+p2T4 1 - 27 T + 372 T^{2} - 27 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

−7.71012333098053052243334732074, −7.67200345678067202383208474597, −6.95391682314268620118293528732, −6.79036160709965208373862507876, −6.26968558320890098744015448234, −6.09392548294987421341637371512, −5.63728208451714129528673518946, −5.17936817038298565910034740892, −4.75013249513086591301762770704, −4.31589894052272916640617190924, −4.19624347706959284344875058661, −3.63640730214586215705516311289, −3.18840850231594885172768986287, −3.15116864201567348428547078141, −2.49210957468196495633600849714, −2.24803446888915899261214553519, −1.51611110382579419061686343950, −0.914593848108578311340633650519, 0, 0, 0.914593848108578311340633650519, 1.51611110382579419061686343950, 2.24803446888915899261214553519, 2.49210957468196495633600849714, 3.15116864201567348428547078141, 3.18840850231594885172768986287, 3.63640730214586215705516311289, 4.19624347706959284344875058661, 4.31589894052272916640617190924, 4.75013249513086591301762770704, 5.17936817038298565910034740892, 5.63728208451714129528673518946, 6.09392548294987421341637371512, 6.26968558320890098744015448234, 6.79036160709965208373862507876, 6.95391682314268620118293528732, 7.67200345678067202383208474597, 7.71012333098053052243334732074

Graph of the ZZ-function along the critical line