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Properties

Label 16-1734e8-1.1-c1e8-0-0
Degree $16$
Conductor $8.173\times 10^{25}$
Sign $1$
Analytic cond. $1.35085\times 10^{9}$
Root an. cond. $3.72102$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 8·5-s − 8·7-s + 10·16-s − 32·20-s + 8·23-s + 32·25-s + 32·28-s − 24·29-s − 8·31-s − 64·35-s − 40·37-s − 32·41-s + 32·49-s − 8·61-s − 20·64-s − 48·67-s − 24·71-s + 32·73-s − 24·79-s + 80·80-s − 2·81-s − 32·92-s − 16·97-s − 128·100-s + 32·101-s − 32·103-s + ⋯
L(s)  = 1  − 2·4-s + 3.57·5-s − 3.02·7-s + 5/2·16-s − 7.15·20-s + 1.66·23-s + 32/5·25-s + 6.04·28-s − 4.45·29-s − 1.43·31-s − 10.8·35-s − 6.57·37-s − 4.99·41-s + 32/7·49-s − 1.02·61-s − 5/2·64-s − 5.86·67-s − 2.84·71-s + 3.74·73-s − 2.70·79-s + 8.94·80-s − 2/9·81-s − 3.33·92-s − 1.62·97-s − 12.7·100-s + 3.18·101-s − 3.15·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 17^{16}\)
Sign: $1$
Analytic conductor: \(1.35085\times 10^{9}\)
Root analytic conductor: \(3.72102\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 17^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.01703644015\)
\(L(\frac12)\) \(\approx\) \(0.01703644015\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
3 \( ( 1 + T^{4} )^{2} \)
17 \( 1 \)
good5 \( 1 - 8 T + 32 T^{2} - 104 T^{3} + 328 T^{4} - 936 T^{5} + 96 p^{2} T^{6} - 1192 p T^{7} + 14034 T^{8} - 1192 p^{2} T^{9} + 96 p^{4} T^{10} - 936 p^{3} T^{11} + 328 p^{4} T^{12} - 104 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
7 \( ( 1 + 4 T + 8 T^{2} + 20 T^{3} + 46 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( 1 + 64 T^{3} + 36 T^{4} - 576 T^{5} + 2048 T^{6} + 2432 T^{7} - 34394 T^{8} + 2432 p T^{9} + 2048 p^{2} T^{10} - 576 p^{3} T^{11} + 36 p^{4} T^{12} + 64 p^{5} T^{13} + p^{8} T^{16} \)
13 \( ( 1 + 40 T^{2} + 16 T^{3} + 698 T^{4} + 16 p T^{5} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( 1 - 56 T^{2} + 1964 T^{4} - 2776 p T^{6} + 1155334 T^{8} - 2776 p^{3} T^{10} + 1964 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 - 8 T + 32 T^{2} - 232 T^{3} + 1404 T^{4} - 3272 T^{5} + 8160 T^{6} - 27240 T^{7} + 28486 T^{8} - 27240 p T^{9} + 8160 p^{2} T^{10} - 3272 p^{3} T^{11} + 1404 p^{4} T^{12} - 232 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16616 T^{4} + 89640 T^{5} + 421344 T^{6} + 1890072 T^{7} + 9201426 T^{8} + 1890072 p T^{9} + 421344 p^{2} T^{10} + 89640 p^{3} T^{11} + 16616 p^{4} T^{12} + 2472 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
31 \( ( 1 + 4 T + 8 T^{2} + 68 T^{3} + 382 T^{4} + 68 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 + 40 T + 800 T^{2} + 10744 T^{3} + 109672 T^{4} + 916792 T^{5} + 6650848 T^{6} + 43941864 T^{7} + 273602898 T^{8} + 43941864 p T^{9} + 6650848 p^{2} T^{10} + 916792 p^{3} T^{11} + 109672 p^{4} T^{12} + 10744 p^{5} T^{13} + 800 p^{6} T^{14} + 40 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 32 T + 512 T^{2} + 5872 T^{3} + 59488 T^{4} + 557552 T^{5} + 4624000 T^{6} + 33576096 T^{7} + 222260418 T^{8} + 33576096 p T^{9} + 4624000 p^{2} T^{10} + 557552 p^{3} T^{11} + 59488 p^{4} T^{12} + 5872 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 88 T^{2} + 9212 T^{4} - 470568 T^{6} + 26633958 T^{8} - 470568 p^{2} T^{10} + 9212 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 132 T^{2} - 224 T^{3} + 7742 T^{4} - 224 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( 1 - 144 T^{2} + 9460 T^{4} - 576944 T^{6} + 35120774 T^{8} - 576944 p^{2} T^{10} + 9460 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 248 T^{2} + 34540 T^{4} - 3214536 T^{6} + 220136774 T^{8} - 3214536 p^{2} T^{10} + 34540 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 + 8 T + 32 T^{2} + 296 T^{3} + 2504 T^{4} + 22056 T^{5} + 140128 T^{6} + 2201480 T^{7} + 34278034 T^{8} + 2201480 p T^{9} + 140128 p^{2} T^{10} + 22056 p^{3} T^{11} + 2504 p^{4} T^{12} + 296 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
67 \( ( 1 + 24 T + 332 T^{2} + 3640 T^{3} + 33198 T^{4} + 3640 p T^{5} + 332 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( 1 + 24 T + 288 T^{2} + 2616 T^{3} + 26108 T^{4} + 294744 T^{5} + 2976480 T^{6} + 25958328 T^{7} + 218272966 T^{8} + 25958328 p T^{9} + 2976480 p^{2} T^{10} + 294744 p^{3} T^{11} + 26108 p^{4} T^{12} + 2616 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 32 T + 512 T^{2} - 6336 T^{3} + 58304 T^{4} - 325312 T^{5} + 630784 T^{6} + 12568800 T^{7} - 187932350 T^{8} + 12568800 p T^{9} + 630784 p^{2} T^{10} - 325312 p^{3} T^{11} + 58304 p^{4} T^{12} - 6336 p^{5} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 24 T + 288 T^{2} + 1912 T^{3} + 636 T^{4} - 143400 T^{5} - 1796896 T^{6} - 157432 p T^{7} - 13754 p^{2} T^{8} - 157432 p^{2} T^{9} - 1796896 p^{2} T^{10} - 143400 p^{3} T^{11} + 636 p^{4} T^{12} + 1912 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 504 T^{2} + 121388 T^{4} - 18062408 T^{6} + 1807461894 T^{8} - 18062408 p^{2} T^{10} + 121388 p^{4} T^{12} - 504 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 176 T^{2} + 160 T^{3} + 15554 T^{4} + 160 p T^{5} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( 1 + 16 T + 128 T^{2} + 528 T^{3} - 12736 T^{4} - 151024 T^{5} - 646784 T^{6} + 6063120 T^{7} + 170809090 T^{8} + 6063120 p T^{9} - 646784 p^{2} T^{10} - 151024 p^{3} T^{11} - 12736 p^{4} T^{12} + 528 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.88880694240836283892884098831, −3.67004392170533298168316426193, −3.61575143176249607568985534804, −3.53807638993943074957111444416, −3.42522245318451022933827650439, −3.22898922141671060664005133169, −3.22554399873238721692897656746, −3.16766072584194488489946328084, −3.00808381652038882853036335698, −2.82580874740341343832592648968, −2.68834130853381329337405517199, −2.64860203517705300749194335317, −2.34004668511974536073765049737, −2.02120242625190702141979301820, −1.86175493074233579017788579410, −1.84626322914872330022685035021, −1.76746720999170936840628341511, −1.69813511188983178428674872185, −1.48177632551125004475580801261, −1.47393832403312983248751417133, −1.29226695299841537362575258443, −0.864979095599951551659807698236, −0.40589272294363697501154616563, −0.10956575061313375285417781756, −0.07239098625659021723325852602, 0.07239098625659021723325852602, 0.10956575061313375285417781756, 0.40589272294363697501154616563, 0.864979095599951551659807698236, 1.29226695299841537362575258443, 1.47393832403312983248751417133, 1.48177632551125004475580801261, 1.69813511188983178428674872185, 1.76746720999170936840628341511, 1.84626322914872330022685035021, 1.86175493074233579017788579410, 2.02120242625190702141979301820, 2.34004668511974536073765049737, 2.64860203517705300749194335317, 2.68834130853381329337405517199, 2.82580874740341343832592648968, 3.00808381652038882853036335698, 3.16766072584194488489946328084, 3.22554399873238721692897656746, 3.22898922141671060664005133169, 3.42522245318451022933827650439, 3.53807638993943074957111444416, 3.61575143176249607568985534804, 3.67004392170533298168316426193, 3.88880694240836283892884098831

Graph of the $Z$-function along the critical line

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