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Properties

Label 8-392e4-1.1-c1e4-0-2
Degree 88
Conductor 2361262489623612624896
Sign 11
Analytic cond. 95.995995.9959
Root an. cond. 1.769211.76921
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  + 4·4-s − 8·9-s − 8·11-s + 12·16-s − 20·25-s − 32·36-s + 24·43-s − 32·44-s + 32·64-s + 32·81-s + 64·99-s − 80·100-s − 24·107-s − 32·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s − 96·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 96·172-s + ⋯
L(s)  = 1  + 2·4-s − 8/3·9-s − 2.41·11-s + 3·16-s − 4·25-s − 5.33·36-s + 3.65·43-s − 4.82·44-s + 4·64-s + 32/9·81-s + 6.43·99-s − 8·100-s − 2.32·107-s − 3.01·113-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 8·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 7.31·172-s + ⋯

Functional equation

Λ(s)=((212⋅78)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((212⋅78)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 212⋅782^{12} \cdot 7^{8}
Sign: 11
Analytic conductor: 95.995995.9959
Root analytic conductor: 1.769211.76921
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 212⋅78, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{12} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.87833917250.8783391725
L(12)L(\frac12) ≈\approx 0.87833917250.8783391725
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)
bad2C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
7 1 1
good3C4×C2C_4\times C_2 1+8T2+32T4+8p2T6+p4T8 1 + 8 T^{2} + 32 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8}
5C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
11C22C_2^2 (1+4T+8T2+4pT3+p2T4)2 ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
13C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
17C4×C2C_4\times C_2 1−48T2+1152T4−48p2T6+p4T8 1 - 48 T^{2} + 1152 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8}
19C4×C2C_4\times C_2 1−24T2+288T4−24p2T6+p4T8 1 - 24 T^{2} + 288 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8}
23C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
29C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
31C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
37C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
41C4×C2C_4\times C_2 1−96T2+4608T4−96p2T6+p4T8 1 - 96 T^{2} + 4608 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8}
43C22C_2^2 (1−12T+72T2−12pT3+p2T4)2 ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
47C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
53C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
59C4×C2C_4\times C_2 1+120T2+7200T4+120p2T6+p4T8 1 + 120 T^{2} + 7200 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8}
61C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
67C22C_2^2 (1+62T2+p2T4)2 ( 1 + 62 T^{2} + p^{2} T^{4} )^{2}
71C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
73C4×C2C_4\times C_2 1−48T2+1152T4−48p2T6+p4T8 1 - 48 T^{2} + 1152 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8}
79C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
83C4×C2C_4\times C_2 1+72T2+2592T4+72p2T6+p4T8 1 + 72 T^{2} + 2592 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8}
89C4×C2C_4\times C_2 1+144T2+10368T4+144p2T6+p4T8 1 + 144 T^{2} + 10368 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8}
97C4×C2C_4\times C_2 1+240T2+28800T4+240p2T6+p4T8 1 + 240 T^{2} + 28800 T^{4} + 240 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

−7.945536706990464478146177546286, −7.87610034453133981741459986903, −7.82093062013373173431685175971, −7.58420339354765717340524314315, −7.26772986397110680071738196151, −7.17152183556223483653360433100, −6.39200259162056675645166996625, −6.37675623192398542502351789208, −6.20494577969483761611031721415, −5.89480714158431894335098077380, −5.66146869234467549316520572774, −5.42634059451941202387515861568, −5.37123198709216509579861239908, −5.12989219918026085971034929763, −4.54746904003177883314268759710, −3.89467172415305145333909008026, −3.78752791798539252294808926140, −3.69171182682501508444853310717, −2.85810297230598775477696347360, −2.73370984822445473800108118628, −2.54091915502585696429376100927, −2.49852174616488983608155380727, −1.96742849332664588346930030476, −1.42251241089855163724047909662, −0.31759997894220413338792641919, 0.31759997894220413338792641919, 1.42251241089855163724047909662, 1.96742849332664588346930030476, 2.49852174616488983608155380727, 2.54091915502585696429376100927, 2.73370984822445473800108118628, 2.85810297230598775477696347360, 3.69171182682501508444853310717, 3.78752791798539252294808926140, 3.89467172415305145333909008026, 4.54746904003177883314268759710, 5.12989219918026085971034929763, 5.37123198709216509579861239908, 5.42634059451941202387515861568, 5.66146869234467549316520572774, 5.89480714158431894335098077380, 6.20494577969483761611031721415, 6.37675623192398542502351789208, 6.39200259162056675645166996625, 7.17152183556223483653360433100, 7.26772986397110680071738196151, 7.58420339354765717340524314315, 7.82093062013373173431685175971, 7.87610034453133981741459986903, 7.945536706990464478146177546286

Graph of the ZZ-function along the critical line