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Properties

Label 8-1840e4-1.1-c1e4-0-1
Degree 88
Conductor 1.146×10131.146\times 10^{13}
Sign 11
Analytic cond. 46599.346599.3
Root an. cond. 3.833073.83307
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  − 12·9-s + 10·25-s − 4·29-s + 28·41-s − 18·49-s + 90·81-s − 68·101-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4·9-s + 2·25-s − 0.742·29-s + 4.37·41-s − 2.57·49-s + 10·81-s − 6.76·101-s − 4·121-s + 0.0887·127-s + 0.0873·131-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 + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

Λ(s)=((216⋅54⋅234)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅54⋅234)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{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: 216⋅54⋅2342^{16} \cdot 5^{4} \cdot 23^{4}
Sign: 11
Analytic conductor: 46599.346599.3
Root analytic conductor: 3.833073.83307
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅54⋅234, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.57991349930.5799134993
L(12)L(\frac12) ≈\approx 0.57991349930.5799134993
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
5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
good3C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
7C22C_2^2 (1+9T2+p2T4)2 ( 1 + 9 T^{2} + p^{2} T^{4} )^{2}
11C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
13C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
17C22C_2^2 (1−11T2+p2T4)2 ( 1 - 11 T^{2} + p^{2} T^{4} )^{2}
19C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
29C2C_2 (1+T+pT2)4 ( 1 + T + p T^{2} )^{4}
31C2C_2 (1−3T+pT2)2(1+3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2}
37C22C_2^2 (1−51T2+p2T4)2 ( 1 - 51 T^{2} + p^{2} T^{4} )^{2}
41C2C_2 (1−7T+pT2)4 ( 1 - 7 T + p T^{2} )^{4}
43C22C_2^2 (1+6T2+p2T4)2 ( 1 + 6 T^{2} + p^{2} T^{4} )^{2}
47C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
53C22C_2^2 (1+101T2+p2T4)2 ( 1 + 101 T^{2} + p^{2} T^{4} )^{2}
59C2C_2 (1−11T+pT2)2(1+11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2}
61C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
67C22C_2^2 (1−111T2+p2T4)2 ( 1 - 111 T^{2} + p^{2} T^{4} )^{2}
71C2C_2 (1−13T+pT2)2(1+13T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2}
73C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
79C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
83C22C_2^2 (1+41T2+p2T4)2 ( 1 + 41 T^{2} + p^{2} T^{4} )^{2}
89C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
97C22C_2^2 (1+174T2+p2T4)2 ( 1 + 174 T^{2} + p^{2} T^{4} )^{2}
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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

−6.52961426973584903350051045367, −6.39579841569836241510412957599, −6.01355556371422401659337887503, −5.93768865842394703826656149652, −5.75400720650370483794534513802, −5.60355237382305888174986188154, −5.26697566067275251932336283458, −5.17440272046197004690524038794, −4.92102137460758926290670495276, −4.90585800555690236179413115686, −4.28052466665405014407299457043, −4.15520192472543363413458049992, −3.90370225711260711889642521196, −3.78972420525058776239612961414, −3.20865053416013096390314620718, −3.12958522663944159721637292391, −2.79627492392428518324473270068, −2.72573984853211976124527586517, −2.66608284446246635060198402759, −2.35757200429384525315937316916, −1.84641117177676497775036772275, −1.52126339665491535471055152381, −1.02157955402819838808504326641, −0.66524876698138781753772870351, −0.18383230437144955010962535026, 0.18383230437144955010962535026, 0.66524876698138781753772870351, 1.02157955402819838808504326641, 1.52126339665491535471055152381, 1.84641117177676497775036772275, 2.35757200429384525315937316916, 2.66608284446246635060198402759, 2.72573984853211976124527586517, 2.79627492392428518324473270068, 3.12958522663944159721637292391, 3.20865053416013096390314620718, 3.78972420525058776239612961414, 3.90370225711260711889642521196, 4.15520192472543363413458049992, 4.28052466665405014407299457043, 4.90585800555690236179413115686, 4.92102137460758926290670495276, 5.17440272046197004690524038794, 5.26697566067275251932336283458, 5.60355237382305888174986188154, 5.75400720650370483794534513802, 5.93768865842394703826656149652, 6.01355556371422401659337887503, 6.39579841569836241510412957599, 6.52961426973584903350051045367

Graph of the ZZ-function along the critical line