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Properties

Label 16-975e8-1.1-c1e8-0-7
Degree 1616
Conductor 8.167×10238.167\times 10^{23}
Sign 11
Analytic cond. 1.34975×1071.34975\times 10^{7}
Root an. cond. 2.790232.79023
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  + 8·4-s + 24·16-s − 8·19-s − 48·41-s − 4·49-s − 48·59-s − 24·61-s − 16·71-s − 64·76-s − 2·81-s + 24·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 384·164-s + 167-s + 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 4·4-s + 6·16-s − 1.83·19-s − 7.49·41-s − 4/7·49-s − 6.24·59-s − 3.07·61-s − 1.89·71-s − 7.34·76-s − 2/9·81-s + 2.29·109-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 − 29.9·164-s + 0.0773·167-s + 2.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

Λ(s)=((38⋅516⋅138)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((38⋅516⋅138)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 38⋅516⋅1383^{8} \cdot 5^{16} \cdot 13^{8}
Sign: 11
Analytic conductor: 1.34975×1071.34975\times 10^{7}
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 38⋅516⋅138, ( :[1/2]8), 1)(16,\ 3^{8} \cdot 5^{16} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.32913993150.3291399315
L(12)L(\frac12) ≈\approx 0.32913993150.3291399315
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}
ppFp(T)F_p(T)
bad3 (1+T4)2 ( 1 + T^{4} )^{2}
5 1 1
13 (1−18T2+p2T4)2 ( 1 - 18 T^{2} + p^{2} T^{4} )^{2}
good2 (1−pT2+p2T4)4 ( 1 - p T^{2} + p^{2} T^{4} )^{4}
7 (1+2T2+11T4+2p2T6+p4T8)2 ( 1 + 2 T^{2} + 11 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2}
11 (1−p2T4+p4T8)2 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2}
17 1−738T4+260611T8−738p4T12+p8T16 1 - 738 T^{4} + 260611 T^{8} - 738 p^{4} T^{12} + p^{8} T^{16}
19 (1+4T+8T2−92T3−706T4−92pT5+8p2T6+4p3T7+p4T8)2 ( 1 + 4 T + 8 T^{2} - 92 T^{3} - 706 T^{4} - 92 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
23 1−82T4−151437T8−82p4T12+p8T16 1 - 82 T^{4} - 151437 T^{8} - 82 p^{4} T^{12} + p^{8} T^{16}
29 (1−10T+pT2)4(1+10T+pT2)4 ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4}
31 (1+p2T4)4 ( 1 + p^{2} T^{4} )^{4}
37 (1+90T2+3971T4+90p2T6+p4T8)2 ( 1 + 90 T^{2} + 3971 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8} )^{2}
41 (1+24T+288T2+2448T3+17087T4+2448pT5+288p2T6+24p3T7+p4T8)2 ( 1 + 24 T + 288 T^{2} + 2448 T^{3} + 17087 T^{4} + 2448 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2}
43 (1−42T2+p2T4)2(1+42T2+p2T4)2 ( 1 - 42 T^{2} + p^{2} T^{4} )^{2}( 1 + 42 T^{2} + p^{2} T^{4} )^{2}
47 (1+64T2+2274T4+64p2T6+p4T8)2 ( 1 + 64 T^{2} + 2274 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2}
53 1+1598T4−1400637T8+1598p4T12+p8T16 1 + 1598 T^{4} - 1400637 T^{8} + 1598 p^{4} T^{12} + p^{8} T^{16}
59 (1+12T+72T2+12pT3+p2T4)4 ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4}
61 (1+6T+43T2+6pT3+p2T4)4 ( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4}
67 (1−164T2+14294T4−164p2T6+p4T8)2 ( 1 - 164 T^{2} + 14294 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2}
71 (1+8T+32T2−160T3−7481T4−160pT5+32p2T6+8p3T7+p4T8)2 ( 1 + 8 T + 32 T^{2} - 160 T^{3} - 7481 T^{4} - 160 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
73 (1−pT2)8 ( 1 - p T^{2} )^{8}
79 (1+11T2+p2T4)4 ( 1 + 11 T^{2} + p^{2} T^{4} )^{4}
83 (1+208T2+21426T4+208p2T6+p4T8)2 ( 1 + 208 T^{2} + 21426 T^{4} + 208 p^{2} T^{6} + p^{4} T^{8} )^{2}
89 (1+12047T4+p4T8)2 ( 1 + 12047 T^{4} + p^{4} T^{8} )^{2}
97 (1−362T2+51491T4−362p2T6+p4T8)2 ( 1 - 362 T^{2} + 51491 T^{4} - 362 p^{2} T^{6} + p^{4} T^{8} )^{2}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.40751536378015578937536694933, −4.26057583502248076429575993198, −3.93389289515505025387269162016, −3.84320049625471115333413552002, −3.58372266970314975514213571921, −3.55142847862123418462709648409, −3.36024782776231887111903217202, −3.20994416659107175825842625686, −3.17044178120927572523927738524, −2.97648994488580056963403815144, −2.94313949999248358892698566527, −2.79177242598209592354363272080, −2.62727269809712788902464924981, −2.61263839225020512700691530282, −2.15941482382976557238570574292, −1.96659787742113864749354811769, −1.92492565319593449004144839771, −1.85138091249805560905234243434, −1.82497702744534381826520074396, −1.70288741715018417600935607709, −1.45890451130054961601956984256, −1.21415500593990526747619915748, −1.18643012438184014092550364318, −0.16820852232885409235583198051, −0.12386141711356014105886337824, 0.12386141711356014105886337824, 0.16820852232885409235583198051, 1.18643012438184014092550364318, 1.21415500593990526747619915748, 1.45890451130054961601956984256, 1.70288741715018417600935607709, 1.82497702744534381826520074396, 1.85138091249805560905234243434, 1.92492565319593449004144839771, 1.96659787742113864749354811769, 2.15941482382976557238570574292, 2.61263839225020512700691530282, 2.62727269809712788902464924981, 2.79177242598209592354363272080, 2.94313949999248358892698566527, 2.97648994488580056963403815144, 3.17044178120927572523927738524, 3.20994416659107175825842625686, 3.36024782776231887111903217202, 3.55142847862123418462709648409, 3.58372266970314975514213571921, 3.84320049625471115333413552002, 3.93389289515505025387269162016, 4.26057583502248076429575993198, 4.40751536378015578937536694933

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.