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Properties

Label 16-405e8-1.1-c2e8-0-10
Degree 1616
Conductor 7.238×10207.238\times 10^{20}
Sign 11
Analytic cond. 2.19948×1082.19948\times 10^{8}
Root an. cond. 3.321963.32196
Motivic weight 22
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  + 20·7-s − 40·13-s − 23·16-s + 40·25-s − 32·31-s + 80·37-s − 40·43-s + 200·49-s + 232·61-s − 280·67-s + 440·73-s − 800·91-s + 20·97-s + 140·103-s − 460·112-s − 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 800·169-s + 173-s + ⋯
L(s)  = 1  + 20/7·7-s − 3.07·13-s − 1.43·16-s + 8/5·25-s − 1.03·31-s + 2.16·37-s − 0.930·43-s + 4.08·49-s + 3.80·61-s − 4.17·67-s + 6.02·73-s − 8.79·91-s + 0.206·97-s + 1.35·103-s − 4.10·112-s − 0.132·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4.73·169-s + 0.00578·173-s + ⋯

Functional equation

Λ(s)=((332⋅58)s/2 ΓC(s)8 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((332⋅58)s/2 ΓC(s+1)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 332⋅583^{32} \cdot 5^{8}
Sign: 11
Analytic conductor: 2.19948×1082.19948\times 10^{8}
Root analytic conductor: 3.321963.32196
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 332⋅58, ( :[1]8), 1)(16,\ 3^{32} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 17.2298253117.22982531
L(12)L(\frac12) ≈\approx 17.2298253117.22982531
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}
ppFp(T)F_p(T)
bad3 1 1
5 1−8pT2+39p2T4−8p5T6+p8T8 1 - 8 p T^{2} + 39 p^{2} T^{4} - 8 p^{5} T^{6} + p^{8} T^{8}
good2 1+23T4+273T8+23p8T12+p16T16 1 + 23 T^{4} + 273 T^{8} + 23 p^{8} T^{12} + p^{16} T^{16}
7 (1−10T+50T2+480T3−4801T4+480p2T5+50p4T6−10p6T7+p8T8)2 ( 1 - 10 T + 50 T^{2} + 480 T^{3} - 4801 T^{4} + 480 p^{2} T^{5} + 50 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} )^{2}
11 (1+8T2−14577T4+8p4T6+p8T8)2 ( 1 + 8 T^{2} - 14577 T^{4} + 8 p^{4} T^{6} + p^{8} T^{8} )^{2}
13 (1+20T+200T2−2760T3−56161T4−2760p2T5+200p4T6+20p6T7+p8T8)2 ( 1 + 20 T + 200 T^{2} - 2760 T^{3} - 56161 T^{4} - 2760 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2}
17 (1+144322T4+p8T8)2 ( 1 + 144322 T^{4} + p^{8} T^{8} )^{2}
19 (1−398T2+p4T4)4 ( 1 - 398 T^{2} + p^{4} T^{4} )^{4}
23 1−517762T4+189766503363T8−517762p8T12+p16T16 1 - 517762 T^{4} + 189766503363 T^{8} - 517762 p^{8} T^{12} + p^{16} T^{16}
29 (1−568T2−384657T4−568p4T6+p8T8)2 ( 1 - 568 T^{2} - 384657 T^{4} - 568 p^{4} T^{6} + p^{8} T^{8} )^{2}
31 (1+8T−897T2+8p2T3+p4T4)4 ( 1 + 8 T - 897 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{4}
37 (1−20T+200T2−20p2T3+p4T4)4 ( 1 - 20 T + 200 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{4}
41 (1−2362T2+2753283T4−2362p4T6+p8T8)2 ( 1 - 2362 T^{2} + 2753283 T^{4} - 2362 p^{4} T^{6} + p^{8} T^{8} )^{2}
43 (1+20T+200T2−69960T3−4118401T4−69960p2T5+200p4T6+20p6T7+p8T8)2 ( 1 + 20 T + 200 T^{2} - 69960 T^{3} - 4118401 T^{4} - 69960 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2}
47 1+8681918T4+51564413496963T8+8681918p8T12+p16T16 1 + 8681918 T^{4} + 51564413496963 T^{8} + 8681918 p^{8} T^{12} + p^{16} T^{16}
53 (1+3037282T4+p8T8)2 ( 1 + 3037282 T^{4} + p^{8} T^{8} )^{2}
59 (1+4712T2+10085583T4+4712p4T6+p8T8)2 ( 1 + 4712 T^{2} + 10085583 T^{4} + 4712 p^{4} T^{6} + p^{8} T^{8} )^{2}
61 (1−58T−357T2−58p2T3+p4T4)4 ( 1 - 58 T - 357 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{4}
67 (1+140T+9800T2+115080T3−12095521T4+115080p2T5+9800p4T6+140p6T7+p8T8)2 ( 1 + 140 T + 9800 T^{2} + 115080 T^{3} - 12095521 T^{4} + 115080 p^{2} T^{5} + 9800 p^{4} T^{6} + 140 p^{6} T^{7} + p^{8} T^{8} )^{2}
71 (1+6082T2+p4T4)4 ( 1 + 6082 T^{2} + p^{4} T^{4} )^{4}
73 (1−110T+6050T2−110p2T3+p4T4)4 ( 1 - 110 T + 6050 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} )^{4}
79 (1+12338T2+113276163T4+12338p4T6+p8T8)2 ( 1 + 12338 T^{2} + 113276163 T^{4} + 12338 p^{4} T^{6} + p^{8} T^{8} )^{2}
83 1+30948638T4−1294474038083997T8+30948638p8T12+p16T16 1 + 30948638 T^{4} - 1294474038083997 T^{8} + 30948638 p^{8} T^{12} + p^{16} T^{16}
89 (1−pT)8(1+pT)8 ( 1 - p T )^{8}( 1 + p T )^{8}
97 (1−10T+50T2+187680T3−89467681T4+187680p2T5+50p4T6−10p6T7+p8T8)2 ( 1 - 10 T + 50 T^{2} + 187680 T^{3} - 89467681 T^{4} + 187680 p^{2} T^{5} + 50 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} 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

−4.64897182936310250161074030089, −4.54641298182923400719600449977, −4.46968934274527551023347130824, −4.46681186747022836630959840992, −4.18783359091639900176481467873, −4.10581530603600100404992290225, −4.01277910219000292246981813985, −3.93024760374787806435654056782, −3.30051113190840078345788710464, −3.28799782025716302389956245667, −3.19958990164059556225284551991, −3.04842711188112782948636152664, −2.74751683014254781328319201531, −2.73108853449954190467969906399, −2.41338314048273461268797627623, −2.17756299101604149336355291148, −2.05536650468224523762537964178, −2.00053678896078430869242214178, −1.84061328904944644722237138066, −1.60851625053961948931938487570, −1.51740588257524858104384594683, −0.72917276988044757311693842144, −0.62725646656248607109211676289, −0.62317473013916487506518725824, −0.54715903863425491032257209176, 0.54715903863425491032257209176, 0.62317473013916487506518725824, 0.62725646656248607109211676289, 0.72917276988044757311693842144, 1.51740588257524858104384594683, 1.60851625053961948931938487570, 1.84061328904944644722237138066, 2.00053678896078430869242214178, 2.05536650468224523762537964178, 2.17756299101604149336355291148, 2.41338314048273461268797627623, 2.73108853449954190467969906399, 2.74751683014254781328319201531, 3.04842711188112782948636152664, 3.19958990164059556225284551991, 3.28799782025716302389956245667, 3.30051113190840078345788710464, 3.93024760374787806435654056782, 4.01277910219000292246981813985, 4.10581530603600100404992290225, 4.18783359091639900176481467873, 4.46681186747022836630959840992, 4.46968934274527551023347130824, 4.54641298182923400719600449977, 4.64897182936310250161074030089

Graph of the ZZ-function along the critical line

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