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Properties

Label 8-162e4-1.1-c2e4-0-0
Degree 88
Conductor 688747536688747536
Sign 11
Analytic cond. 379.664379.664
Root an. cond. 2.100992.10099
Motivic weight 22
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  + 2·4-s + 8·7-s − 16·13-s − 64·19-s − 32·25-s + 16·28-s − 88·31-s − 136·37-s + 80·43-s + 114·49-s − 32·52-s − 100·61-s − 8·64-s − 16·67-s − 64·73-s − 128·76-s + 152·79-s − 128·91-s − 352·97-s − 64·100-s + 56·103-s + 224·109-s + 46·121-s − 176·124-s + 127-s + 131-s − 512·133-s + ⋯
L(s)  = 1  + 1/2·4-s + 8/7·7-s − 1.23·13-s − 3.36·19-s − 1.27·25-s + 4/7·28-s − 2.83·31-s − 3.67·37-s + 1.86·43-s + 2.32·49-s − 0.615·52-s − 1.63·61-s − 1/8·64-s − 0.238·67-s − 0.876·73-s − 1.68·76-s + 1.92·79-s − 1.40·91-s − 3.62·97-s − 0.639·100-s + 0.543·103-s + 2.05·109-s + 0.380·121-s − 1.41·124-s + 0.00787·127-s + 0.00763·131-s − 3.84·133-s + ⋯

Functional equation

Λ(s)=((24⋅316)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((24⋅316)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅3162^{4} \cdot 3^{16}
Sign: 11
Analytic conductor: 379.664379.664
Root analytic conductor: 2.100992.10099
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅316, ( :1,1,1,1), 1)(8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.62517871850.6251787185
L(12)L(\frac12) ≈\approx 0.62517871850.6251787185
L(2)L(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−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
3 1 1
good5C23C_2^3 1+32T2+399T4+32p4T6+p8T8 1 + 32 T^{2} + 399 T^{4} + 32 p^{4} T^{6} + p^{8} T^{8}
7C22C_2^2 (1−4T−33T2−4p2T3+p4T4)2 ( 1 - 4 T - 33 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
11C22C_2^2×\timesC22C_2^2 (1−14T+75T2−14p2T3+p4T4)(1+14T+75T2+14p2T3+p4T4) ( 1 - 14 T + 75 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )( 1 + 14 T + 75 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )
13C22C_2^2 (1+8T−105T2+8p2T3+p4T4)2 ( 1 + 8 T - 105 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2}
17C22C_2^2 (1−416T2+p4T4)2 ( 1 - 416 T^{2} + p^{4} T^{4} )^{2}
19C2C_2 (1+16T+p2T2)4 ( 1 + 16 T + p^{2} T^{2} )^{4}
23C23C_2^3 1+770T2+313059T4+770p4T6+p8T8 1 + 770 T^{2} + 313059 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8}
29C23C_2^3 1+1664T2+2061615T4+1664p4T6+p8T8 1 + 1664 T^{2} + 2061615 T^{4} + 1664 p^{4} T^{6} + p^{8} T^{8}
31C22C_2^2 (1+44T+975T2+44p2T3+p4T4)2 ( 1 + 44 T + 975 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2}
37C2C_2 (1+34T+p2T2)4 ( 1 + 34 T + p^{2} T^{2} )^{4}
41C23C_2^3 1+1184T2−1423905T4+1184p4T6+p8T8 1 + 1184 T^{2} - 1423905 T^{4} + 1184 p^{4} T^{6} + p^{8} T^{8}
43C22C_2^2 (1−40T−249T2−40p2T3+p4T4)2 ( 1 - 40 T - 249 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2}
47C23C_2^3 1−2782T2+2859843T4−2782p4T6+p8T8 1 - 2782 T^{2} + 2859843 T^{4} - 2782 p^{4} T^{6} + p^{8} T^{8}
53C22C_2^2 (1−4160T2+p4T4)2 ( 1 - 4160 T^{2} + p^{4} T^{4} )^{2}
59C23C_2^3 1+5810T2+21638739T4+5810p4T6+p8T8 1 + 5810 T^{2} + 21638739 T^{4} + 5810 p^{4} T^{6} + p^{8} T^{8}
61C22C_2^2 (1+50T−1221T2+50p2T3+p4T4)2 ( 1 + 50 T - 1221 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2}
67C22C_2^2 (1+8T−4425T2+8p2T3+p4T4)2 ( 1 + 8 T - 4425 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2}
71C22C_2^2 (1−7490T2+p4T4)2 ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2}
73C2C_2 (1+16T+p2T2)4 ( 1 + 16 T + p^{2} T^{2} )^{4}
79C22C_2^2 (1−76T−465T2−76p2T3+p4T4)2 ( 1 - 76 T - 465 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2}
83C23C_2^3 1−334T2−47346765T4−334p4T6+p8T8 1 - 334 T^{2} - 47346765 T^{4} - 334 p^{4} T^{6} + p^{8} T^{8}
89C22C_2^2 (1−15680T2+p4T4)2 ( 1 - 15680 T^{2} + p^{4} T^{4} )^{2}
97C22C_2^2 (1+176T+21567T2+176p2T3+p4T4)2 ( 1 + 176 T + 21567 T^{2} + 176 p^{2} T^{3} + p^{4} 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

−9.110434110478828292079559110806, −8.727287624815488222493534994112, −8.717374288428360639008742618293, −8.526418836284378476621410743676, −8.056280778261669263405040850989, −7.81796861014634595473043464083, −7.44075688452594573551872043750, −7.20640416515618152299617012085, −7.00257818445030685737589840380, −6.90486468172323226368728828864, −6.20889367143516985852749912517, −6.07535911693213322830452563462, −5.66453604210382631405591824341, −5.52392317562612939667571792801, −4.97318677650644879623271920543, −4.82377070063963437675771575990, −4.33724813874447706699116597336, −3.98989440783965587342884810830, −3.85086579071850667810935227774, −3.28770246934692631876822675452, −2.62272613116582676311546725858, −2.16674850045852790685693800677, −1.79964609406910535143879502568, −1.79877935892192256795035608831, −0.24015488036411555722703447758, 0.24015488036411555722703447758, 1.79877935892192256795035608831, 1.79964609406910535143879502568, 2.16674850045852790685693800677, 2.62272613116582676311546725858, 3.28770246934692631876822675452, 3.85086579071850667810935227774, 3.98989440783965587342884810830, 4.33724813874447706699116597336, 4.82377070063963437675771575990, 4.97318677650644879623271920543, 5.52392317562612939667571792801, 5.66453604210382631405591824341, 6.07535911693213322830452563462, 6.20889367143516985852749912517, 6.90486468172323226368728828864, 7.00257818445030685737589840380, 7.20640416515618152299617012085, 7.44075688452594573551872043750, 7.81796861014634595473043464083, 8.056280778261669263405040850989, 8.526418836284378476621410743676, 8.717374288428360639008742618293, 8.727287624815488222493534994112, 9.110434110478828292079559110806

Graph of the ZZ-function along the critical line