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Properties

Label 8-483e4-1.1-c1e4-0-2
Degree 88
Conductor 5442375752154423757521
Sign 11
Analytic cond. 221.256221.256
Root an. cond. 1.963861.96386
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  − 2·3-s + 5·4-s − 6·5-s + 6·7-s + 2·9-s − 10·12-s + 12·15-s + 12·16-s + 12·17-s − 30·20-s − 12·21-s + 5·25-s − 6·27-s + 30·28-s − 36·35-s + 10·36-s + 32·37-s + 8·41-s + 26·43-s − 12·45-s + 32·47-s − 24·48-s + 18·49-s − 24·51-s − 10·59-s + 60·60-s + 12·63-s + ⋯
L(s)  = 1  − 1.15·3-s + 5/2·4-s − 2.68·5-s + 2.26·7-s + 2/3·9-s − 2.88·12-s + 3.09·15-s + 3·16-s + 2.91·17-s − 6.70·20-s − 2.61·21-s + 25-s − 1.15·27-s + 5.66·28-s − 6.08·35-s + 5/3·36-s + 5.26·37-s + 1.24·41-s + 3.96·43-s − 1.78·45-s + 4.66·47-s − 3.46·48-s + 18/7·49-s − 3.36·51-s − 1.30·59-s + 7.74·60-s + 1.51·63-s + ⋯

Functional equation

Λ(s)=((34⋅74⋅234)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((34⋅74⋅234)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅74⋅2343^{4} \cdot 7^{4} \cdot 23^{4}
Sign: 11
Analytic conductor: 221.256221.256
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅74⋅234, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.2099962323.209996232
L(12)L(\frac12) ≈\approx 3.2099962323.209996232
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)
bad3C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
7C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
23C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good2D4×C2D_4\times C_2 1−5T2+13T4−5p2T6+p4T8 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8}
5D4D_{4} (1+3T+11T2+3pT3+p2T4)2 ( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
11C22C_2^2 (1−17T2+p2T4)2 ( 1 - 17 T^{2} + p^{2} T^{4} )^{2}
13D4×C2D_4\times C_2 1−5T2−207T4−5p2T6+p4T8 1 - 5 T^{2} - 207 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8}
17D4D_{4} (1−6T+23T2−6pT3+p2T4)2 ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
19D4×C2D_4\times C_2 1−34T2+49pT4−34p2T6+p4T8 1 - 34 T^{2} + 49 p T^{4} - 34 p^{2} T^{6} + p^{4} T^{8}
29D4×C2D_4\times C_2 1−74T2+2731T4−74p2T6+p4T8 1 - 74 T^{2} + 2731 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8}
31D4×C2D_4\times C_2 1−34T2+211T4−34p2T6+p4T8 1 - 34 T^{2} + 211 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8}
37D4D_{4} (1−16T+133T2−16pT3+p2T4)2 ( 1 - 16 T + 133 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}
41D4D_{4} (1−4T+81T2−4pT3+p2T4)2 ( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
43D4D_{4} (1−13T+127T2−13pT3+p2T4)2 ( 1 - 13 T + 127 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2}
47C2C_2 (1−8T+pT2)4 ( 1 - 8 T + p T^{2} )^{4}
53D4×C2D_4\times C_2 1+15T2+713T4+15p2T6+p4T8 1 + 15 T^{2} + 713 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1+5T+23T2+5pT3+p2T4)2 ( 1 + 5 T + 23 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}
61D4×C2D_4\times C_2 1−209T2+18081T4−209p2T6+p4T8 1 - 209 T^{2} + 18081 T^{4} - 209 p^{2} T^{6} + p^{4} T^{8}
67D4D_{4} (1+19T+223T2+19pT3+p2T4)2 ( 1 + 19 T + 223 T^{2} + 19 p T^{3} + p^{2} T^{4} )^{2}
71D4×C2D_4\times C_2 1−149T2+14101T4−149p2T6+p4T8 1 - 149 T^{2} + 14101 T^{4} - 149 p^{2} T^{6} + p^{4} T^{8}
73D4×C2D_4\times C_2 1−210T2+20963T4−210p2T6+p4T8 1 - 210 T^{2} + 20963 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1+20T+253T2+20pT3+p2T4)2 ( 1 + 20 T + 253 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2}
83D4D_{4} (1−8T+137T2−8pT3+p2T4)2 ( 1 - 8 T + 137 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
89D4D_{4} (1+5T+153T2+5pT3+p2T4)2 ( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−130T2+7363T4−130p2T6+p4T8 1 - 130 T^{2} + 7363 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.81184580704691757893158130539, −7.78810662816369486143713918226, −7.39312133864889674666895858711, −7.38960518737465808985714979799, −7.26074160231560375929796028441, −7.09300225927295986012089836964, −6.24950225500305866874104666780, −6.03698500824422165967690122463, −5.86613417956197558177144932963, −5.76675511588821245096844928521, −5.64323863464469169906565641734, −5.61698431371832350317012482678, −4.64404557229709651744000741985, −4.40040607583075241343838917013, −4.24542615017828239053790774945, −4.19718093654593616556289346795, −4.13164799423517797233267763376, −3.29935330698309202818785611910, −3.18407411443623148358256842240, −2.60044587185931908194168802920, −2.49494862096002136280773891981, −2.12642090823846021802669485036, −1.41308452259816050191723358517, −1.07632849649512588977219094797, −0.798829303941988403988768756800, 0.798829303941988403988768756800, 1.07632849649512588977219094797, 1.41308452259816050191723358517, 2.12642090823846021802669485036, 2.49494862096002136280773891981, 2.60044587185931908194168802920, 3.18407411443623148358256842240, 3.29935330698309202818785611910, 4.13164799423517797233267763376, 4.19718093654593616556289346795, 4.24542615017828239053790774945, 4.40040607583075241343838917013, 4.64404557229709651744000741985, 5.61698431371832350317012482678, 5.64323863464469169906565641734, 5.76675511588821245096844928521, 5.86613417956197558177144932963, 6.03698500824422165967690122463, 6.24950225500305866874104666780, 7.09300225927295986012089836964, 7.26074160231560375929796028441, 7.38960518737465808985714979799, 7.39312133864889674666895858711, 7.78810662816369486143713918226, 7.81184580704691757893158130539

Graph of the ZZ-function along the critical line