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Properties

Label 8-4080e4-1.1-c1e4-0-3
Degree 88
Conductor 2.771×10142.771\times 10^{14}
Sign 11
Analytic cond. 1.12654×1061.12654\times 10^{6}
Root an. cond. 5.707795.70779
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·9-s − 8·11-s − 8·19-s − 10·25-s + 16·29-s + 8·31-s − 4·41-s + 16·49-s + 20·59-s − 36·71-s − 32·79-s + 3·81-s − 32·89-s + 16·99-s + 20·101-s + 48·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + ⋯
L(s)  = 1  − 2/3·9-s − 2.41·11-s − 1.83·19-s − 2·25-s + 2.97·29-s + 1.43·31-s − 0.624·41-s + 16/7·49-s + 2.60·59-s − 4.27·71-s − 3.60·79-s + 1/3·81-s − 3.39·89-s + 1.60·99-s + 1.99·101-s + 4.59·109-s − 0.363·121-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 + 1.84·169-s + ⋯

Functional equation

Λ(s)=((216⋅34⋅54⋅174)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅34⋅54⋅174)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅34⋅54⋅1742^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}
Sign: 11
Analytic conductor: 1.12654×1061.12654\times 10^{6}
Root analytic conductor: 5.707795.70779
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅34⋅54⋅174, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.17051446750.1705144675
L(12)L(\frac12) ≈\approx 0.17051446750.1705144675
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
3C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good7D4×C2D_4\times C_2 1−16T2+142T4−16p2T6+p4T8 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}
11C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
13D4×C2D_4\times C_2 1−24T2+302T4−24p2T6+p4T8 1 - 24 T^{2} + 302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8}
19D4D_{4} (1+4T+22T2+4pT3+p2T4)2 ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
23C22C_2^2 (1−30T2+p2T4)2 ( 1 - 30 T^{2} + p^{2} T^{4} )^{2}
29C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
31C4C_4 (1−4T+46T2−4pT3+p2T4)2 ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
37C2C_2 (1−12T+pT2)2(1+12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}
41D4D_{4} (1+2T+38T2+2pT3+p2T4)2 ( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1+80T2+4798T4+80p2T6+p4T8 1 + 80 T^{2} + 4798 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−76T2+2982T4−76p2T6+p4T8 1 - 76 T^{2} + 2982 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−44T2+4822T4−44p2T6+p4T8 1 - 44 T^{2} + 4822 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1−10T+98T2−10pT3+p2T4)2 ( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}
61C22C_2^2 (1+102T2+p2T4)2 ( 1 + 102 T^{2} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−128T2+8574T4−128p2T6+p4T8 1 - 128 T^{2} + 8574 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1+18T+218T2+18pT3+p2T4)2 ( 1 + 18 T + 218 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1−40T2+8638T4−40p2T6+p4T8 1 - 40 T^{2} + 8638 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1+16T+142T2+16pT3+p2T4)2 ( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−220T2+22998T4−220p2T6+p4T8 1 - 220 T^{2} + 22998 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8}
89D4D_{4} (1+16T+222T2+16pT3+p2T4)2 ( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−248T2+29694T4−248p2T6+p4T8 1 - 248 T^{2} + 29694 T^{4} - 248 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

−6.02244475709297635281973488973, −5.70134575522069334691078882688, −5.59072527763094925640632060388, −5.38313141692913691597224538846, −5.17843912155961614641938952091, −5.09300375656129953604384451321, −4.70766003212774932856010075898, −4.42181694466282766178358907042, −4.37241611052468460928019853961, −4.27089181925853376177795912031, −4.10610048722845622415751108265, −3.79966083414469113715176292498, −3.45176441543282188055113069834, −3.03120219789291241890020678953, −3.01425772458401786756405067313, −2.80217576298623238814174035331, −2.67782010055908602708073282445, −2.41172681570718965548775681708, −2.15056542184496250130093119113, −1.98247643508177530252271915590, −1.68185555287550977864723945373, −1.10301304985688485936536160564, −1.04622488441678741255469473672, −0.47714848154106191539011068639, −0.082846234740168725565072584165, 0.082846234740168725565072584165, 0.47714848154106191539011068639, 1.04622488441678741255469473672, 1.10301304985688485936536160564, 1.68185555287550977864723945373, 1.98247643508177530252271915590, 2.15056542184496250130093119113, 2.41172681570718965548775681708, 2.67782010055908602708073282445, 2.80217576298623238814174035331, 3.01425772458401786756405067313, 3.03120219789291241890020678953, 3.45176441543282188055113069834, 3.79966083414469113715176292498, 4.10610048722845622415751108265, 4.27089181925853376177795912031, 4.37241611052468460928019853961, 4.42181694466282766178358907042, 4.70766003212774932856010075898, 5.09300375656129953604384451321, 5.17843912155961614641938952091, 5.38313141692913691597224538846, 5.59072527763094925640632060388, 5.70134575522069334691078882688, 6.02244475709297635281973488973

Graph of the ZZ-function along the critical line