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Properties

Label 8-200e4-1.1-c1e4-0-1
Degree 88
Conductor 16000000001600000000
Sign 11
Analytic cond. 6.504716.50471
Root an. cond. 1.263721.26372
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  − 2·2-s + 4·3-s + 2·4-s − 8·6-s − 4·8-s + 4·9-s + 8·12-s + 8·16-s − 8·18-s − 16·24-s − 4·27-s − 8·31-s − 8·32-s + 8·36-s + 8·37-s − 8·41-s − 28·43-s + 32·48-s + 20·49-s + 32·53-s + 8·54-s + 16·62-s + 8·64-s − 36·67-s + 8·71-s − 16·72-s − 16·74-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.30·3-s + 4-s − 3.26·6-s − 1.41·8-s + 4/3·9-s + 2.30·12-s + 2·16-s − 1.88·18-s − 3.26·24-s − 0.769·27-s − 1.43·31-s − 1.41·32-s + 4/3·36-s + 1.31·37-s − 1.24·41-s − 4.26·43-s + 4.61·48-s + 20/7·49-s + 4.39·53-s + 1.08·54-s + 2.03·62-s + 64-s − 4.39·67-s + 0.949·71-s − 1.88·72-s − 1.85·74-s + ⋯

Functional equation

Λ(s)=((212⋅58)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((212⋅58)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 212⋅582^{12} \cdot 5^{8}
Sign: 11
Analytic conductor: 6.504716.50471
Root analytic conductor: 1.263721.26372
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 212⋅58, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{12} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.1640259381.164025938
L(12)L(\frac12) ≈\approx 1.1640259381.164025938
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)
bad2C22C_2^2 1+pT+pT2+p2T3+p2T4 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4}
5 1 1
good3D4D_{4} (1−2T+4T2−2pT3+p2T4)2 ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
7D4×C2D_4\times C_2 1−20T2+186T4−20p2T6+p4T8 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
11C22C_2^2 (1−18T2+p2T4)2 ( 1 - 18 T^{2} + p^{2} T^{4} )^{2}
13C22C_2^2 (1+14T2+p2T4)2 ( 1 + 14 T^{2} + p^{2} T^{4} )^{2}
17C22C_2^2 (1−22T2+p2T4)2 ( 1 - 22 T^{2} + p^{2} T^{4} )^{2}
19D4×C2D_4\times C_2 1−20T2+54T4−20p2T6+p4T8 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
23D4×C2D_4\times C_2 1−36T2+1274T4−36p2T6+p4T8 1 - 36 T^{2} + 1274 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1−10T2+p2T4)2 ( 1 - 10 T^{2} + p^{2} T^{4} )^{2}
31D4D_{4} (1+4T+54T2+4pT3+p2T4)2 ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
37C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
41D4D_{4} (1+4T+74T2+4pT3+p2T4)2 ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
43D4D_{4} (1+14T+132T2+14pT3+p2T4)2 ( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}
47D4×C2D_4\times C_2 1−132T2+8474T4−132p2T6+p4T8 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8}
53D4D_{4} (1−16T+158T2−16pT3+p2T4)2 ( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1−180T2+14294T4−180p2T6+p4T8 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8}
61D4×C2D_4\times C_2 1−140T2+11574T4−140p2T6+p4T8 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
67D4D_{4} (1+18T+212T2+18pT3+p2T4)2 ( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}
71D4D_{4} (1−4T+134T2−4pT3+p2T4)2 ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1−236T2+23814T4−236p2T6+p4T8 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1−16T+174T2−16pT3+p2T4)2 ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}
83D4D_{4} (1−6T+172T2−6pT3+p2T4)2 ( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
89D4D_{4} (1+4T+134T2+4pT3+p2T4)2 ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−140T2+16806T4−140p2T6+p4T8 1 - 140 T^{2} + 16806 T^{4} - 140 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

−9.121893544177709379473274233582, −8.611198104545520429956874499974, −8.574460175233462015115753246520, −8.545406457889532750429173903124, −8.396693322336625231350986220045, −7.989946772885703948650253364531, −7.56978981097575224447493628148, −7.50281848193954513407267736321, −7.10968471180960990171383998362, −6.93615623970837667152620247320, −6.37533703346812715166520916945, −6.32053956450899452943993228132, −5.69285397887016362610639577947, −5.65793535687404674032941093397, −5.08516966199271487860207398171, −5.00899091981260562276640246658, −4.15046156057961027481250194208, −3.95361207165823130152554861631, −3.54362985360426655564845850290, −3.11540259944749621306411411312, −3.02714902970405647398462517339, −2.62640780843944343045930744331, −2.00285850582407157288296123012, −1.91673788953325424389475291069, −0.72645289390318715026944689259, 0.72645289390318715026944689259, 1.91673788953325424389475291069, 2.00285850582407157288296123012, 2.62640780843944343045930744331, 3.02714902970405647398462517339, 3.11540259944749621306411411312, 3.54362985360426655564845850290, 3.95361207165823130152554861631, 4.15046156057961027481250194208, 5.00899091981260562276640246658, 5.08516966199271487860207398171, 5.65793535687404674032941093397, 5.69285397887016362610639577947, 6.32053956450899452943993228132, 6.37533703346812715166520916945, 6.93615623970837667152620247320, 7.10968471180960990171383998362, 7.50281848193954513407267736321, 7.56978981097575224447493628148, 7.989946772885703948650253364531, 8.396693322336625231350986220045, 8.545406457889532750429173903124, 8.574460175233462015115753246520, 8.611198104545520429956874499974, 9.121893544177709379473274233582

Graph of the ZZ-function along the critical line