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Properties

Label 16-960e8-1.1-c2e8-0-13
Degree $16$
Conductor $7.214\times 10^{23}$
Sign $1$
Analytic cond. $2.19204\times 10^{11}$
Root an. cond. $5.11449$
Motivic weight $2$
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  − 12·9-s − 16·13-s + 20·25-s − 64·29-s + 112·37-s − 16·41-s + 168·49-s − 352·53-s + 176·61-s − 240·73-s + 90·81-s + 80·89-s + 432·97-s + 224·101-s − 368·109-s − 288·113-s + 192·117-s + 552·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 4/3·9-s − 1.23·13-s + 4/5·25-s − 2.20·29-s + 3.02·37-s − 0.390·41-s + 24/7·49-s − 6.64·53-s + 2.88·61-s − 3.28·73-s + 10/9·81-s + 0.898·89-s + 4.45·97-s + 2.21·101-s − 3.37·109-s − 2.54·113-s + 1.64·117-s + 4.56·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(2.19204\times 10^{11}\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(6.235778914\)
\(L(\frac12)\) \(\approx\) \(6.235778914\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + p T^{2} )^{4} \)
5 \( ( 1 - p T^{2} )^{4} \)
good7 \( 1 - 24 p T^{2} + 13404 T^{4} - 690840 T^{6} + 31402310 T^{8} - 690840 p^{4} T^{10} + 13404 p^{8} T^{12} - 24 p^{13} T^{14} + p^{16} T^{16} \)
11 \( ( 1 - 276 T^{2} + 48006 T^{4} - 276 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 8 T + 204 T^{2} - 1736 T^{3} - 634 T^{4} - 1736 p^{2} T^{5} + 204 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( ( 1 + 732 T^{2} + 3840 T^{3} + 247238 T^{4} + 3840 p^{2} T^{5} + 732 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( 1 - 1192 T^{2} + 901404 T^{4} - 482591000 T^{6} + 198221377670 T^{8} - 482591000 p^{4} T^{10} + 901404 p^{8} T^{12} - 1192 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 616 T^{2} + 755676 T^{4} - 531969752 T^{6} + 267063306566 T^{8} - 531969752 p^{4} T^{10} + 755676 p^{8} T^{12} - 616 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 + 32 T + 1212 T^{2} + 46048 T^{3} + 1958438 T^{4} + 46048 p^{2} T^{5} + 1212 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 - 2280 T^{2} + 2108508 T^{4} - 623021400 T^{6} - 295925282362 T^{8} - 623021400 p^{4} T^{10} + 2108508 p^{8} T^{12} - 2280 p^{12} T^{14} + p^{16} T^{16} \)
37 \( ( 1 - 56 T + 3948 T^{2} - 174856 T^{3} + 6816518 T^{4} - 174856 p^{2} T^{5} + 3948 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( ( 1 + 8 T + 4924 T^{2} + 33080 T^{3} + 10990150 T^{4} + 33080 p^{2} T^{5} + 4924 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 - 3976 T^{2} + 15992796 T^{4} - 34977997112 T^{6} + 80511749221766 T^{8} - 34977997112 p^{4} T^{10} + 15992796 p^{8} T^{12} - 3976 p^{12} T^{14} + p^{16} T^{16} \)
47 \( 1 - 9640 T^{2} + 45901788 T^{4} - 144992767640 T^{6} + 353863561499078 T^{8} - 144992767640 p^{4} T^{10} + 45901788 p^{8} T^{12} - 9640 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 + 176 T + 396 p T^{2} + 1644496 T^{3} + 101651558 T^{4} + 1644496 p^{2} T^{5} + 396 p^{5} T^{6} + 176 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 - 22952 T^{2} + 243302044 T^{4} - 1557392711960 T^{6} + 6588978912358150 T^{8} - 1557392711960 p^{4} T^{10} + 243302044 p^{8} T^{12} - 22952 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 - 88 T + 12348 T^{2} - 708776 T^{3} + 62059430 T^{4} - 708776 p^{2} T^{5} + 12348 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 - 19848 T^{2} + 189118044 T^{4} - 1228457910840 T^{6} + 6186195617725190 T^{8} - 1228457910840 p^{4} T^{10} + 189118044 p^{8} T^{12} - 19848 p^{12} T^{14} + p^{16} T^{16} \)
71 \( 1 - 26376 T^{2} + 353772444 T^{4} - 3049343603256 T^{6} + 18245696307579590 T^{8} - 3049343603256 p^{4} T^{10} + 353772444 p^{8} T^{12} - 26376 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 + 120 T + 19740 T^{2} + 1592520 T^{3} + 158554502 T^{4} + 1592520 p^{2} T^{5} + 19740 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 8040 T^{2} + 72964188 T^{4} - 471847486680 T^{6} + 4278949597529798 T^{8} - 471847486680 p^{4} T^{10} + 72964188 p^{8} T^{12} - 8040 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 - 18184 T^{2} + 266916444 T^{4} - 2437903506104 T^{6} + 19925145362261510 T^{8} - 2437903506104 p^{4} T^{10} + 266916444 p^{8} T^{12} - 18184 p^{12} T^{14} + p^{16} T^{16} \)
89 \( ( 1 - 40 T + 11100 T^{2} - 192920 T^{3} + 121014662 T^{4} - 192920 p^{2} T^{5} + 11100 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 216 T + 43516 T^{2} - 4942440 T^{3} + 582543750 T^{4} - 4942440 p^{2} T^{5} + 43516 p^{4} T^{6} - 216 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−3.96608428300161901765833540380, −3.89437039912116536280406962352, −3.78221894755826770551670402629, −3.73936792643024632912806634484, −3.45354241644570808814705670596, −3.40322607350227051921871235410, −3.36384702391692608210620722252, −3.00006790253194522526937148526, −2.93073745730350223265265847765, −2.86427786003574437718585228359, −2.80339986852901628314326050096, −2.50470021392958276982795259266, −2.35635112212887860498338534658, −2.28411183160345118039927774506, −2.17049585924333193810913307847, −1.95737981992521532780769062645, −1.83089860032269415088676942279, −1.48015874928725388434319849530, −1.43455627878793867837476283835, −1.19448600151075283678428800655, −1.08343738117363926729833908671, −0.59971928685199392444214869114, −0.43934344230141729695294494565, −0.42564272124677550874536050716, −0.28656827504248652817376524546, 0.28656827504248652817376524546, 0.42564272124677550874536050716, 0.43934344230141729695294494565, 0.59971928685199392444214869114, 1.08343738117363926729833908671, 1.19448600151075283678428800655, 1.43455627878793867837476283835, 1.48015874928725388434319849530, 1.83089860032269415088676942279, 1.95737981992521532780769062645, 2.17049585924333193810913307847, 2.28411183160345118039927774506, 2.35635112212887860498338534658, 2.50470021392958276982795259266, 2.80339986852901628314326050096, 2.86427786003574437718585228359, 2.93073745730350223265265847765, 3.00006790253194522526937148526, 3.36384702391692608210620722252, 3.40322607350227051921871235410, 3.45354241644570808814705670596, 3.73936792643024632912806634484, 3.78221894755826770551670402629, 3.89437039912116536280406962352, 3.96608428300161901765833540380

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

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