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Properties

Label 16-210e8-1.1-c2e8-0-1
Degree $16$
Conductor $3.782\times 10^{18}$
Sign $1$
Analytic cond. $1.14930\times 10^{6}$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·4-s − 12·9-s − 16·11-s + 40·16-s − 20·25-s − 144·29-s − 96·36-s − 48·37-s − 64·43-s − 128·44-s − 12·49-s + 128·53-s + 160·64-s − 192·67-s + 176·71-s − 288·79-s + 90·81-s + 192·99-s − 160·100-s − 192·107-s − 192·109-s + 608·113-s − 1.15e3·116-s − 416·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 2·4-s − 4/3·9-s − 1.45·11-s + 5/2·16-s − 4/5·25-s − 4.96·29-s − 8/3·36-s − 1.29·37-s − 1.48·43-s − 2.90·44-s − 0.244·49-s + 2.41·53-s + 5/2·64-s − 2.86·67-s + 2.47·71-s − 3.64·79-s + 10/9·81-s + 1.93·99-s − 8/5·100-s − 1.79·107-s − 1.76·109-s + 5.38·113-s − 9.93·116-s − 3.43·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.14930\times 10^{6}\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5884852733\)
\(L(\frac12)\) \(\approx\) \(0.5884852733\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p T^{2} )^{4} \)
3 \( ( 1 + p T^{2} )^{4} \)
5 \( ( 1 + p T^{2} )^{4} \)
7 \( 1 + 12 T^{2} - 58 p^{2} T^{4} + 12 p^{4} T^{6} + p^{8} T^{8} \)
good11 \( ( 1 + 8 T + 304 T^{2} + 2120 T^{3} + 45250 T^{4} + 2120 p^{2} T^{5} + 304 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
13 \( 1 - 1088 T^{2} + 557124 T^{4} - 173160640 T^{6} + 35590649030 T^{8} - 173160640 p^{4} T^{10} + 557124 p^{8} T^{12} - 1088 p^{12} T^{14} + p^{16} T^{16} \)
17 \( ( 1 - 716 T^{2} + 266406 T^{4} - 716 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( 1 - 1040 T^{2} + 715428 T^{4} - 373461040 T^{6} + 149925694598 T^{8} - 373461040 p^{4} T^{10} + 715428 p^{8} T^{12} - 1040 p^{12} T^{14} + p^{16} T^{16} \)
23 \( ( 1 + 780 T^{2} + 15360 T^{3} + 320102 T^{4} + 15360 p^{2} T^{5} + 780 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 + 72 T + 4044 T^{2} + 159480 T^{3} + 5105990 T^{4} + 159480 p^{2} T^{5} + 4044 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 - 5072 T^{2} + 12940644 T^{4} - 21077526640 T^{6} + 23985269180870 T^{8} - 21077526640 p^{4} T^{10} + 12940644 p^{8} T^{12} - 5072 p^{12} T^{14} + p^{16} T^{16} \)
37 \( ( 1 + 24 T + 2412 T^{2} - 16728 T^{3} + 1972550 T^{4} - 16728 p^{2} T^{5} + 2412 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( 1 - 11432 T^{2} + 60126684 T^{4} - 189254916760 T^{6} + 389111411703110 T^{8} - 189254916760 p^{4} T^{10} + 60126684 p^{8} T^{12} - 11432 p^{12} T^{14} + p^{16} T^{16} \)
43 \( ( 1 + 32 T + 3396 T^{2} + 140128 T^{3} + 8375270 T^{4} + 140128 p^{2} T^{5} + 3396 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 - 9656 T^{2} + 49532316 T^{4} - 174004173832 T^{6} + 446973571058246 T^{8} - 174004173832 p^{4} T^{10} + 49532316 p^{8} T^{12} - 9656 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 - 64 T + 6168 T^{2} - 270464 T^{3} + 22592978 T^{4} - 270464 p^{2} T^{5} + 6168 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 - 13672 T^{2} + 87440284 T^{4} - 360721973080 T^{6} + 1260995942842630 T^{8} - 360721973080 p^{4} T^{10} + 87440284 p^{8} T^{12} - 13672 p^{12} T^{14} + p^{16} T^{16} \)
61 \( 1 - 5096 T^{2} + 58282524 T^{4} - 205728528856 T^{6} + 1227606280747910 T^{8} - 205728528856 p^{4} T^{10} + 58282524 p^{8} T^{12} - 5096 p^{12} T^{14} + p^{16} T^{16} \)
67 \( ( 1 + 96 T + 14188 T^{2} + 794016 T^{3} + 79706118 T^{4} + 794016 p^{2} T^{5} + 14188 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 88 T + 8304 T^{2} - 522200 T^{3} + 58604450 T^{4} - 522200 p^{2} T^{5} + 8304 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
73 \( 1 - 18048 T^{2} + 179971204 T^{4} - 1281534583680 T^{6} + 7286987478858630 T^{8} - 1281534583680 p^{4} T^{10} + 179971204 p^{8} T^{12} - 18048 p^{12} T^{14} + p^{16} T^{16} \)
79 \( ( 1 + 144 T + 27876 T^{2} + 2532528 T^{3} + 267406406 T^{4} + 2532528 p^{2} T^{5} + 27876 p^{4} T^{6} + 144 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( 1 - 37880 T^{2} + 719278428 T^{4} - 8629223884360 T^{6} + 71063717963313158 T^{8} - 8629223884360 p^{4} T^{10} + 719278428 p^{8} T^{12} - 37880 p^{12} T^{14} + p^{16} T^{16} \)
89 \( 1 - 20024 T^{2} + 352120860 T^{4} - 3746723082376 T^{6} + 35814387700257734 T^{8} - 3746723082376 p^{4} T^{10} + 352120860 p^{8} T^{12} - 20024 p^{12} T^{14} + p^{16} T^{16} \)
97 \( 1 - 40128 T^{2} + 785762308 T^{4} - 10615096489536 T^{6} + 111881094946850310 T^{8} - 10615096489536 p^{4} T^{10} + 785762308 p^{8} T^{12} - 40128 p^{12} T^{14} + p^{16} T^{16} \)
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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

−5.53154024513644878697823257672, −5.43228520906035647168329454286, −5.42134643009890568658092486431, −4.92948366497114540640903005722, −4.63941748870437667970393560319, −4.56510348674573750665686664344, −4.38285785676814908144335365390, −4.36619473384901134335058506337, −3.76158279374081791971704362091, −3.75416009907072505150035035463, −3.65876682380825580706560970191, −3.61742062686905998274088617753, −3.14217801249409807707650257798, −3.13456193556868928046588923907, −2.99971788630997575749674690724, −2.65278981412913512114772624132, −2.51754508473643973126804886287, −2.37583285093329765545618311454, −1.96947072880861016113678569573, −1.88413778006087952795804223226, −1.71275619343256748218941584582, −1.63193564416792539393722530539, −1.08768121457909747741704534403, −0.46378319259128484178661461688, −0.12726194643307457236450208136, 0.12726194643307457236450208136, 0.46378319259128484178661461688, 1.08768121457909747741704534403, 1.63193564416792539393722530539, 1.71275619343256748218941584582, 1.88413778006087952795804223226, 1.96947072880861016113678569573, 2.37583285093329765545618311454, 2.51754508473643973126804886287, 2.65278981412913512114772624132, 2.99971788630997575749674690724, 3.13456193556868928046588923907, 3.14217801249409807707650257798, 3.61742062686905998274088617753, 3.65876682380825580706560970191, 3.75416009907072505150035035463, 3.76158279374081791971704362091, 4.36619473384901134335058506337, 4.38285785676814908144335365390, 4.56510348674573750665686664344, 4.63941748870437667970393560319, 4.92948366497114540640903005722, 5.42134643009890568658092486431, 5.43228520906035647168329454286, 5.53154024513644878697823257672

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

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