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Properties

Label 8-13e8-1.1-c1e4-0-0
Degree 88
Conductor 815730721815730721
Sign 11
Analytic cond. 3.316313.31631
Root an. cond. 1.161661.16166
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  − 4·3-s + 4-s + 10·9-s − 4·12-s + 4·16-s − 6·17-s − 12·23-s − 14·25-s − 32·27-s − 6·29-s + 10·36-s + 16·43-s − 16·48-s + 14·49-s + 24·51-s − 12·53-s − 2·61-s + 11·64-s − 6·68-s + 48·69-s + 56·75-s + 16·79-s + 89·81-s + 24·87-s − 12·92-s − 14·100-s − 6·101-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s + 10/3·9-s − 1.15·12-s + 16-s − 1.45·17-s − 2.50·23-s − 2.79·25-s − 6.15·27-s − 1.11·29-s + 5/3·36-s + 2.43·43-s − 2.30·48-s + 2·49-s + 3.36·51-s − 1.64·53-s − 0.256·61-s + 11/8·64-s − 0.727·68-s + 5.77·69-s + 6.46·75-s + 1.80·79-s + 89/9·81-s + 2.57·87-s − 1.25·92-s − 7/5·100-s − 0.597·101-s + ⋯

Functional equation

Λ(s)=((138)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((138)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 13813^{8}
Sign: 11
Analytic conductor: 3.316313.31631
Root analytic conductor: 1.161661.16166
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 138, ( :1/2,1/2,1/2,1/2), 1)(8,\ 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.36972438700.3697243870
L(12)L(\frac12) ≈\approx 0.36972438700.3697243870
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)
bad13 1 1
good2C23C_2^3 1−T2−3T4−p2T6+p4T8 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8}
3C22C_2^2 (1+2T+T2+2pT3+p2T4)2 ( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
5C22C_2^2 (1+7T2+p2T4)2 ( 1 + 7 T^{2} + p^{2} T^{4} )^{2}
7C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
11C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
17C22C_2^2 (1+3T−8T2+3pT3+p2T4)2 ( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
19C22C_2^2×\timesC22C_2^2 (1−37T2+p2T4)(1+11T2+p2T4) ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} )
23C22C_2^2 (1+6T+13T2+6pT3+p2T4)2 ( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
29C22C_2^2 (1+3T−20T2+3pT3+p2T4)2 ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+50T2+p2T4)2 ( 1 + 50 T^{2} + p^{2} T^{4} )^{2}
37C23C_2^3 1+T2−1368T4+p2T6+p4T8 1 + T^{2} - 1368 T^{4} + p^{2} T^{6} + p^{4} T^{8}
41C23C_2^3 1−55T2+1344T4−55p2T6+p4T8 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8}
43C2C_2 (1−13T+pT2)2(1+5T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2}
47C22C_2^2 (1+82T2+p2T4)2 ( 1 + 82 T^{2} + p^{2} T^{4} )^{2}
53C2C_2 (1+3T+pT2)4 ( 1 + 3 T + p T^{2} )^{4}
59C23C_2^3 1−70T2+1419T4−70p2T6+p4T8 1 - 70 T^{2} + 1419 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8}
61C2C_2 (1−13T+pT2)2(1+14T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2}
67C22C_2^2×\timesC22C_2^2 (1−109T2+p2T4)(1−13T2+p2T4) ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} )
71C23C_2^3 1−130T2+11859T4−130p2T6+p4T8 1 - 130 T^{2} + 11859 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8}
73C22C_2^2 (1+143T2+p2T4)2 ( 1 + 143 T^{2} + p^{2} T^{4} )^{2}
79C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
83C22C_2^2 (1−26T2+p2T4)2 ( 1 - 26 T^{2} + p^{2} T^{4} )^{2}
89C23C_2^3 1−130T2+8979T4−130p2T6+p4T8 1 - 130 T^{2} + 8979 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8}
97C23C_2^3 1−146T2+11907T4−146p2T6+p4T8 1 - 146 T^{2} + 11907 T^{4} - 146 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

−9.349665201872342313588955562620, −9.293738423359049310958431943775, −9.199571952202992873146175998705, −8.276433748151106426137030070353, −8.138632461951013330312426057722, −8.108558979652241302615090395837, −7.45145844507463007729883656062, −7.34049854202209080245702459206, −7.25815811685853860469678321852, −6.96304453987406784930278395185, −6.15359377268972769751707187780, −6.03641800405607795597681235072, −5.91962893181555972810241465281, −5.86009954120082912247415771514, −5.78011015459741765677317590653, −4.92344190111112481042795812599, −4.82934182953175803873413901625, −4.34192001417676835383240037133, −3.91237810568063505552473766069, −3.75789500024280062620500480945, −3.55982381203645878803030763207, −2.16477055288425466478804862196, −2.16476351842961474470588166149, −1.80752134955969724882689588861, −0.48155172633364216074792975011, 0.48155172633364216074792975011, 1.80752134955969724882689588861, 2.16476351842961474470588166149, 2.16477055288425466478804862196, 3.55982381203645878803030763207, 3.75789500024280062620500480945, 3.91237810568063505552473766069, 4.34192001417676835383240037133, 4.82934182953175803873413901625, 4.92344190111112481042795812599, 5.78011015459741765677317590653, 5.86009954120082912247415771514, 5.91962893181555972810241465281, 6.03641800405607795597681235072, 6.15359377268972769751707187780, 6.96304453987406784930278395185, 7.25815811685853860469678321852, 7.34049854202209080245702459206, 7.45145844507463007729883656062, 8.108558979652241302615090395837, 8.138632461951013330312426057722, 8.276433748151106426137030070353, 9.199571952202992873146175998705, 9.293738423359049310958431943775, 9.349665201872342313588955562620

Graph of the ZZ-function along the critical line