Properties

Label 16-4025e8-1.1-c1e8-0-1
Degree $16$
Conductor $6.889\times 10^{28}$
Sign $1$
Analytic cond. $1.13852\times 10^{12}$
Root an. cond. $5.66919$
Motivic weight $1$
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  + 2-s + 4·3-s − 5·4-s + 4·6-s − 8·7-s − 6·8-s − 4·9-s + 3·11-s − 20·12-s + 5·13-s − 8·14-s + 9·16-s + 5·17-s − 4·18-s − 2·19-s − 32·21-s + 3·22-s + 8·23-s − 24·24-s + 5·26-s − 39·27-s + 40·28-s − 9·29-s − 3·31-s + 16·32-s + 12·33-s + 5·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.30·3-s − 5/2·4-s + 1.63·6-s − 3.02·7-s − 2.12·8-s − 4/3·9-s + 0.904·11-s − 5.77·12-s + 1.38·13-s − 2.13·14-s + 9/4·16-s + 1.21·17-s − 0.942·18-s − 0.458·19-s − 6.98·21-s + 0.639·22-s + 1.66·23-s − 4.89·24-s + 0.980·26-s − 7.50·27-s + 7.55·28-s − 1.67·29-s − 0.538·31-s + 2.82·32-s + 2.08·33-s + 0.857·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{16} \cdot 7^{8} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(1.13852\times 10^{12}\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{16} \cdot 7^{8} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.05199059\)
\(L(\frac12)\) \(\approx\) \(18.05199059\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( ( 1 + T )^{8} \)
23 \( ( 1 - T )^{8} \)
good2 \( 1 - T + 3 p T^{2} - 5 T^{3} + 5 p^{2} T^{4} - p^{4} T^{5} + 7 p^{3} T^{6} - 45 T^{7} + 131 T^{8} - 45 p T^{9} + 7 p^{5} T^{10} - p^{7} T^{11} + 5 p^{6} T^{12} - 5 p^{5} T^{13} + 3 p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
3 \( 1 - 4 T + 20 T^{2} - 19 p T^{3} + 59 p T^{4} - 398 T^{5} + 317 p T^{6} - 1768 T^{7} + 3437 T^{8} - 1768 p T^{9} + 317 p^{3} T^{10} - 398 p^{3} T^{11} + 59 p^{5} T^{12} - 19 p^{6} T^{13} + 20 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 - 3 T + 58 T^{2} - 188 T^{3} + 155 p T^{4} - 5286 T^{5} + 32172 T^{6} - 89557 T^{7} + 420105 T^{8} - 89557 p T^{9} + 32172 p^{2} T^{10} - 5286 p^{3} T^{11} + 155 p^{5} T^{12} - 188 p^{5} T^{13} + 58 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 - 5 T + 74 T^{2} - 262 T^{3} + 178 p T^{4} - 457 p T^{5} + 43418 T^{6} - 86890 T^{7} + 614789 T^{8} - 86890 p T^{9} + 43418 p^{2} T^{10} - 457 p^{4} T^{11} + 178 p^{5} T^{12} - 262 p^{5} T^{13} + 74 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 5 T + 93 T^{2} - 477 T^{3} + 4414 T^{4} - 20794 T^{5} + 133012 T^{6} - 544109 T^{7} + 2721975 T^{8} - 544109 p T^{9} + 133012 p^{2} T^{10} - 20794 p^{3} T^{11} + 4414 p^{4} T^{12} - 477 p^{5} T^{13} + 93 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 + 2 T + 123 T^{2} + 222 T^{3} + 7083 T^{4} + 11277 T^{5} + 247444 T^{6} + 337556 T^{7} + 5716293 T^{8} + 337556 p T^{9} + 247444 p^{2} T^{10} + 11277 p^{3} T^{11} + 7083 p^{4} T^{12} + 222 p^{5} T^{13} + 123 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 + 9 T + 177 T^{2} + 37 p T^{3} + 12317 T^{4} + 54195 T^{5} + 500732 T^{6} + 1763562 T^{7} + 15551717 T^{8} + 1763562 p T^{9} + 500732 p^{2} T^{10} + 54195 p^{3} T^{11} + 12317 p^{4} T^{12} + 37 p^{6} T^{13} + 177 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 + 3 T + 202 T^{2} + 494 T^{3} + 18723 T^{4} + 37846 T^{5} + 1052980 T^{6} + 1772343 T^{7} + 39487773 T^{8} + 1772343 p T^{9} + 1052980 p^{2} T^{10} + 37846 p^{3} T^{11} + 18723 p^{4} T^{12} + 494 p^{5} T^{13} + 202 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 6 T + 128 T^{2} - 483 T^{3} + 6815 T^{4} - 18492 T^{5} + 279952 T^{6} - 724717 T^{7} + 11064919 T^{8} - 724717 p T^{9} + 279952 p^{2} T^{10} - 18492 p^{3} T^{11} + 6815 p^{4} T^{12} - 483 p^{5} T^{13} + 128 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 7 T + 146 T^{2} + 290 T^{3} + 6657 T^{4} - 14616 T^{5} + 316450 T^{6} - 398833 T^{7} + 18250885 T^{8} - 398833 p T^{9} + 316450 p^{2} T^{10} - 14616 p^{3} T^{11} + 6657 p^{4} T^{12} + 290 p^{5} T^{13} + 146 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 8 T + 186 T^{2} - 811 T^{3} + 12532 T^{4} - 3752 T^{5} + 314858 T^{6} + 2838902 T^{7} + 3830323 T^{8} + 2838902 p T^{9} + 314858 p^{2} T^{10} - 3752 p^{3} T^{11} + 12532 p^{4} T^{12} - 811 p^{5} T^{13} + 186 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 22 T + 477 T^{2} - 6490 T^{3} + 84170 T^{4} - 846631 T^{5} + 8082338 T^{6} - 63730055 T^{7} + 476894657 T^{8} - 63730055 p T^{9} + 8082338 p^{2} T^{10} - 846631 p^{3} T^{11} + 84170 p^{4} T^{12} - 6490 p^{5} T^{13} + 477 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 21 T + 545 T^{2} - 7508 T^{3} + 111457 T^{4} - 1140983 T^{5} + 12195509 T^{6} - 97831066 T^{7} + 813054453 T^{8} - 97831066 p T^{9} + 12195509 p^{2} T^{10} - 1140983 p^{3} T^{11} + 111457 p^{4} T^{12} - 7508 p^{5} T^{13} + 545 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 14 T + 265 T^{2} - 2791 T^{3} + 35713 T^{4} - 324717 T^{5} + 3355461 T^{6} - 26197372 T^{7} + 229201335 T^{8} - 26197372 p T^{9} + 3355461 p^{2} T^{10} - 324717 p^{3} T^{11} + 35713 p^{4} T^{12} - 2791 p^{5} T^{13} + 265 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 8 T + 404 T^{2} - 2889 T^{3} + 76073 T^{4} - 473002 T^{5} + 8636309 T^{6} - 45493588 T^{7} + 643227315 T^{8} - 45493588 p T^{9} + 8636309 p^{2} T^{10} - 473002 p^{3} T^{11} + 76073 p^{4} T^{12} - 2889 p^{5} T^{13} + 404 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 21 T + 559 T^{2} - 7909 T^{3} + 124053 T^{4} - 1337889 T^{5} + 15557612 T^{6} - 135484528 T^{7} + 1266969303 T^{8} - 135484528 p T^{9} + 15557612 p^{2} T^{10} - 1337889 p^{3} T^{11} + 124053 p^{4} T^{12} - 7909 p^{5} T^{13} + 559 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 11 T + 393 T^{2} - 3817 T^{3} + 76733 T^{4} - 635255 T^{5} + 9436196 T^{6} - 66559266 T^{7} + 796900451 T^{8} - 66559266 p T^{9} + 9436196 p^{2} T^{10} - 635255 p^{3} T^{11} + 76733 p^{4} T^{12} - 3817 p^{5} T^{13} + 393 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 26 T + 716 T^{2} - 12063 T^{3} + 196639 T^{4} - 2464044 T^{5} + 29416957 T^{6} - 289161782 T^{7} + 2699414091 T^{8} - 289161782 p T^{9} + 29416957 p^{2} T^{10} - 2464044 p^{3} T^{11} + 196639 p^{4} T^{12} - 12063 p^{5} T^{13} + 716 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 16 T + 468 T^{2} + 5518 T^{3} + 90791 T^{4} + 852531 T^{5} + 10480272 T^{6} + 84705860 T^{7} + 905161519 T^{8} + 84705860 p T^{9} + 10480272 p^{2} T^{10} + 852531 p^{3} T^{11} + 90791 p^{4} T^{12} + 5518 p^{5} T^{13} + 468 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 20 T + 605 T^{2} - 9107 T^{3} + 155840 T^{4} - 1898117 T^{5} + 23665964 T^{6} - 239462516 T^{7} + 2378527291 T^{8} - 239462516 p T^{9} + 23665964 p^{2} T^{10} - 1898117 p^{3} T^{11} + 155840 p^{4} T^{12} - 9107 p^{5} T^{13} + 605 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 15 T + 358 T^{2} - 5282 T^{3} + 75100 T^{4} - 930075 T^{5} + 10738476 T^{6} - 109583110 T^{7} + 1120131273 T^{8} - 109583110 p T^{9} + 10738476 p^{2} T^{10} - 930075 p^{3} T^{11} + 75100 p^{4} T^{12} - 5282 p^{5} T^{13} + 358 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - T + 546 T^{2} + 252 T^{3} + 136930 T^{4} + 233989 T^{5} + 21403884 T^{6} + 49846696 T^{7} + 2395310017 T^{8} + 49846696 p T^{9} + 21403884 p^{2} T^{10} + 233989 p^{3} T^{11} + 136930 p^{4} T^{12} + 252 p^{5} T^{13} + 546 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.45981289200928368402441051749, −3.32823697449614473929321478180, −3.28224674539582691234203868985, −3.16398486642984790577909858379, −3.13878990911522863703253576687, −3.09378277930784995207356914910, −2.90566722237345607926932785647, −2.72481955136895767687617800647, −2.57754111005712293689574996417, −2.40114761580678700916016132739, −2.35763087165908383970584449379, −2.31007484878664853550698621017, −2.26984568610796584626902493164, −2.08009096433889468457006938076, −2.06655016037952633097703197159, −1.68720940746349895277201632762, −1.48552290225663474933487958010, −1.46644643420939015334913032106, −0.918257654980547558916138210331, −0.75027390374802439012494657002, −0.71194409535736708582045698988, −0.68338040569004458944626418174, −0.62591820482858108190174143341, −0.46651248674646385016142024975, −0.32448755616436489634783123980, 0.32448755616436489634783123980, 0.46651248674646385016142024975, 0.62591820482858108190174143341, 0.68338040569004458944626418174, 0.71194409535736708582045698988, 0.75027390374802439012494657002, 0.918257654980547558916138210331, 1.46644643420939015334913032106, 1.48552290225663474933487958010, 1.68720940746349895277201632762, 2.06655016037952633097703197159, 2.08009096433889468457006938076, 2.26984568610796584626902493164, 2.31007484878664853550698621017, 2.35763087165908383970584449379, 2.40114761580678700916016132739, 2.57754111005712293689574996417, 2.72481955136895767687617800647, 2.90566722237345607926932785647, 3.09378277930784995207356914910, 3.13878990911522863703253576687, 3.16398486642984790577909858379, 3.28224674539582691234203868985, 3.32823697449614473929321478180, 3.45981289200928368402441051749

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

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