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Properties

Label 16-3150e8-1.1-c1e8-0-15
Degree $16$
Conductor $9.694\times 10^{27}$
Sign $1$
Analytic cond. $1.60214\times 10^{11}$
Root an. cond. $5.01526$
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  + 2·4-s + 24·11-s + 16-s − 12·23-s − 24·37-s + 32·41-s + 16·43-s + 48·44-s + 8·47-s − 24·53-s − 24·59-s − 2·64-s + 24·67-s − 24·79-s + 16·83-s − 16·89-s − 24·92-s − 8·101-s + 24·107-s + 24·109-s + 272·121-s + 127-s + 131-s + 137-s + 139-s − 48·148-s + 149-s + ⋯
L(s)  = 1  + 4-s + 7.23·11-s + 1/4·16-s − 2.50·23-s − 3.94·37-s + 4.99·41-s + 2.43·43-s + 7.23·44-s + 1.16·47-s − 3.29·53-s − 3.12·59-s − 1/4·64-s + 2.93·67-s − 2.70·79-s + 1.75·83-s − 1.69·89-s − 2.50·92-s − 0.796·101-s + 2.32·107-s + 2.29·109-s + 24.7·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.94·148-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.60214\times 10^{11}\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(22.35915683\)
\(L(\frac12)\) \(\approx\) \(22.35915683\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{2} \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - 94 T^{4} + p^{4} T^{8} \)
good11 \( 1 - 24 T + 304 T^{2} - 2688 T^{3} + 18481 T^{4} - 104232 T^{5} + 496816 T^{6} - 2036016 T^{7} + 7239280 T^{8} - 2036016 p T^{9} + 496816 p^{2} T^{10} - 104232 p^{3} T^{11} + 18481 p^{4} T^{12} - 2688 p^{5} T^{13} + 304 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 - 80 T^{2} + 3006 T^{4} - 69568 T^{6} + 1087139 T^{8} - 69568 p^{2} T^{10} + 3006 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 28 T^{2} - 192 T^{3} + 26 p T^{4} + 4320 T^{5} + 15824 T^{6} - 58560 T^{7} - 349805 T^{8} - 58560 p T^{9} + 15824 p^{2} T^{10} + 4320 p^{3} T^{11} + 26 p^{5} T^{12} - 192 p^{5} T^{13} - 28 p^{6} T^{14} + p^{8} T^{16} \)
19 \( 1 + 58 T^{2} + 99 p T^{4} - 1872 T^{5} + 44906 T^{6} - 71184 T^{7} + 869156 T^{8} - 71184 p T^{9} + 44906 p^{2} T^{10} - 1872 p^{3} T^{11} + 99 p^{5} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 + 6 T + 57 T^{2} + 270 T^{3} + 1772 T^{4} + 270 p T^{5} + 57 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 36 T^{2} + 470 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( 1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 237200 p^{2} T^{10} + 5706 p^{4} T^{12} + 100 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 + 24 T + 216 T^{2} + 1680 T^{3} + 21025 T^{4} + 178920 T^{5} + 986040 T^{6} + 7528032 T^{7} + 58723920 T^{8} + 7528032 p T^{9} + 986040 p^{2} T^{10} + 178920 p^{3} T^{11} + 21025 p^{4} T^{12} + 1680 p^{5} T^{13} + 216 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
41 \( ( 1 - 16 T + 232 T^{2} - 2000 T^{3} + 15591 T^{4} - 2000 p T^{5} + 232 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 8 T + 180 T^{2} - 1000 T^{3} + 11750 T^{4} - 1000 p T^{5} + 180 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 8 T - 78 T^{2} + 240 T^{3} + 6305 T^{4} + 3648 T^{5} - 344398 T^{6} + 95176 T^{7} + 11426244 T^{8} + 95176 p T^{9} - 344398 p^{2} T^{10} + 3648 p^{3} T^{11} + 6305 p^{4} T^{12} + 240 p^{5} T^{13} - 78 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 24 T + 362 T^{2} + 4080 T^{3} + 37881 T^{4} + 336960 T^{5} + 2978650 T^{6} + 25456152 T^{7} + 197950532 T^{8} + 25456152 p T^{9} + 2978650 p^{2} T^{10} + 336960 p^{3} T^{11} + 37881 p^{4} T^{12} + 4080 p^{5} T^{13} + 362 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 24 T + 224 T^{2} + 1392 T^{3} + 9358 T^{4} + 16056 T^{5} - 613888 T^{6} - 6806952 T^{7} - 48734957 T^{8} - 6806952 p T^{9} - 613888 p^{2} T^{10} + 16056 p^{3} T^{11} + 9358 p^{4} T^{12} + 1392 p^{5} T^{13} + 224 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 4411200 p T^{9} + 293552 p^{2} T^{10} + 30240 p^{3} T^{11} + 6186 p^{4} T^{12} + 124 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 24 T + 332 T^{2} - 3696 T^{3} + 30922 T^{4} - 206568 T^{5} + 963056 T^{6} - 1103736 T^{7} - 11820413 T^{8} - 1103736 p T^{9} + 963056 p^{2} T^{10} - 206568 p^{3} T^{11} + 30922 p^{4} T^{12} - 3696 p^{5} T^{13} + 332 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 4371368 p^{2} T^{10} + 41532 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 + 156 T^{2} + 12202 T^{4} + 230256 T^{6} - 10850829 T^{8} + 230256 p^{2} T^{10} + 12202 p^{4} T^{12} + 156 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 + 24 T + 252 T^{2} + 1584 T^{3} + 5050 T^{4} + 17832 T^{5} - 542160 T^{6} - 18801768 T^{7} - 226869549 T^{8} - 18801768 p T^{9} - 542160 p^{2} T^{10} + 17832 p^{3} T^{11} + 5050 p^{4} T^{12} + 1584 p^{5} T^{13} + 252 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 - 8 T + 148 T^{2} - 1192 T^{3} + 18438 T^{4} - 1192 p T^{5} + 148 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 + 16 T - 48 T^{2} - 1824 T^{3} + 2078 T^{4} + 135504 T^{5} + 38912 T^{6} - 7970384 T^{7} - 72309597 T^{8} - 7970384 p T^{9} + 38912 p^{2} T^{10} + 135504 p^{3} T^{11} + 2078 p^{4} T^{12} - 1824 p^{5} T^{13} - 48 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 24 T^{2} + 19036 T^{4} - 653400 T^{6} + 157960902 T^{8} - 653400 p^{2} T^{10} + 19036 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.62359060013966794349398271639, −3.49661915373879796242190744178, −3.44298153663581433000624997825, −3.33402712191915878305625853260, −3.18715458234126220354981940192, −3.09837895885299246185249207642, −2.99288672112097463071876877885, −2.67824660660693284817720585860, −2.62850871805637710856031411042, −2.45130708644196121041914425772, −2.20894627774537117853757703480, −2.17610158209895862146298291427, −2.15986903264985027112467001917, −1.99912987424262084346938670304, −1.90723850324376831958447251358, −1.63796282448156504885092077764, −1.44616642469027448073438007462, −1.31659106807686206814217734119, −1.28990333862767017115949858069, −1.25077375353777933346491739195, −1.14033892577187205122942301716, −1.01768218163845986120464682904, −0.62639867694176628327600978419, −0.38830614694181141914515757695, −0.24787602332205104803329279802, 0.24787602332205104803329279802, 0.38830614694181141914515757695, 0.62639867694176628327600978419, 1.01768218163845986120464682904, 1.14033892577187205122942301716, 1.25077375353777933346491739195, 1.28990333862767017115949858069, 1.31659106807686206814217734119, 1.44616642469027448073438007462, 1.63796282448156504885092077764, 1.90723850324376831958447251358, 1.99912987424262084346938670304, 2.15986903264985027112467001917, 2.17610158209895862146298291427, 2.20894627774537117853757703480, 2.45130708644196121041914425772, 2.62850871805637710856031411042, 2.67824660660693284817720585860, 2.99288672112097463071876877885, 3.09837895885299246185249207642, 3.18715458234126220354981940192, 3.33402712191915878305625853260, 3.44298153663581433000624997825, 3.49661915373879796242190744178, 3.62359060013966794349398271639

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

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