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Properties

Label 16-3150e8-1.1-c1e8-0-7
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  + 4·2-s + 6·4-s − 24·11-s − 16·13-s − 15·16-s + 24·17-s − 96·22-s − 8·23-s − 64·26-s − 24·32-s + 96·34-s − 32·41-s − 144·44-s − 32·46-s + 12·47-s − 96·52-s + 4·53-s − 24·59-s − 6·64-s − 48·67-s + 144·68-s + 24·73-s + 24·79-s − 128·82-s − 16·89-s − 48·92-s + 48·94-s + ⋯
L(s)  = 1  + 2.82·2-s + 3·4-s − 7.23·11-s − 4.43·13-s − 3.75·16-s + 5.82·17-s − 20.4·22-s − 1.66·23-s − 12.5·26-s − 4.24·32-s + 16.4·34-s − 4.99·41-s − 21.7·44-s − 4.71·46-s + 1.75·47-s − 13.3·52-s + 0.549·53-s − 3.12·59-s − 3/4·64-s − 5.86·67-s + 17.4·68-s + 2.80·73-s + 2.70·79-s − 14.1·82-s − 1.69·89-s − 5.00·92-s + 4.95·94-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

\(L(1)\) \(\approx\) \(0.5396842196\)
\(L(\frac12)\) \(\approx\) \(0.5396842196\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{4} \)
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 \( ( 1 + 8 T + 72 T^{2} + 328 T^{3} + 1535 T^{4} + 328 p T^{5} + 72 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 - 24 T + 316 T^{2} - 2976 T^{3} + 22330 T^{4} - 141240 T^{5} + 776752 T^{6} - 3781944 T^{7} + 16463059 T^{8} - 3781944 p T^{9} + 776752 p^{2} T^{10} - 141240 p^{3} T^{11} + 22330 p^{4} T^{12} - 2976 p^{5} T^{13} + 316 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
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 \( ( 1 + 4 T - 31 T^{2} + 4 T^{3} + 1312 T^{4} + 4 p T^{5} - 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 + 72 T^{2} + 1441 T^{4} - 2520 T^{5} + 74088 T^{6} - 269496 T^{7} + 5102928 T^{8} - 269496 p T^{9} + 74088 p^{2} T^{10} - 2520 p^{3} T^{11} + 1441 p^{4} T^{12} + 72 p^{6} T^{14} + p^{8} T^{16} \)
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 \( 1 - 296 T^{2} + 39900 T^{4} - 3207640 T^{6} + 168538406 T^{8} - 3207640 p^{2} T^{10} + 39900 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 12 T + 182 T^{2} - 1608 T^{3} + 14001 T^{4} - 79344 T^{5} + 532582 T^{6} - 2135988 T^{7} + 16324772 T^{8} - 2135988 p T^{9} + 532582 p^{2} T^{10} - 79344 p^{3} T^{11} + 14001 p^{4} T^{12} - 1608 p^{5} T^{13} + 182 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 4 T - 66 T^{2} + 984 T^{3} - 631 T^{4} - 50712 T^{5} + 363998 T^{6} + 1081556 T^{7} - 22439580 T^{8} + 1081556 p T^{9} + 363998 p^{2} T^{10} - 50712 p^{3} T^{11} - 631 p^{4} T^{12} + 984 p^{5} T^{13} - 66 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
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 \( 1 + 48 T + 1108 T^{2} + 16320 T^{3} + 166858 T^{4} + 1140048 T^{5} + 3573520 T^{6} - 22726896 T^{7} - 369529661 T^{8} - 22726896 p T^{9} + 3573520 p^{2} T^{10} + 1140048 p^{3} T^{11} + 166858 p^{4} T^{12} + 16320 p^{5} T^{13} + 1108 p^{6} T^{14} + 48 p^{7} T^{15} + p^{8} T^{16} \)
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 \( ( 1 - 12 T - 6 T^{2} - 48 T^{3} + 6659 T^{4} - 48 p T^{5} - 6 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 - 232 T^{2} + 39708 T^{4} - 4492952 T^{6} + 427617638 T^{8} - 4492952 p^{2} T^{10} + 39708 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \)
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 \( ( 1 - 24 T + 276 T^{2} - 1608 T^{3} + 10022 T^{4} - 1608 p T^{5} + 276 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−3.41024905765645980818057340139, −3.36583894945112643829736141209, −3.35065730318655083143488994015, −3.32630804528499381012007099660, −3.10720839661079125702849744909, −3.06986145795823201279516495328, −2.96831919612418465866214186228, −2.88630636271534328618609698983, −2.81143098735057610283676889671, −2.57844140518747871574934421314, −2.49174462520679842273035628936, −2.35398186055841636489501675538, −2.25692920006159304473315372718, −2.21349939972424243821909203790, −2.07269406179304581366334927281, −1.92441419481194787165205795114, −1.82299298221926028083235423501, −1.40986879871683962213407304302, −1.34004454611283292087291896969, −1.24006243124168893905803777250, −0.853752704663457373343975829840, −0.56864169516755822384818064907, −0.36280820123741468153197208753, −0.20166980826989213963798523100, −0.12405743037466701370456116708, 0.12405743037466701370456116708, 0.20166980826989213963798523100, 0.36280820123741468153197208753, 0.56864169516755822384818064907, 0.853752704663457373343975829840, 1.24006243124168893905803777250, 1.34004454611283292087291896969, 1.40986879871683962213407304302, 1.82299298221926028083235423501, 1.92441419481194787165205795114, 2.07269406179304581366334927281, 2.21349939972424243821909203790, 2.25692920006159304473315372718, 2.35398186055841636489501675538, 2.49174462520679842273035628936, 2.57844140518747871574934421314, 2.81143098735057610283676889671, 2.88630636271534328618609698983, 2.96831919612418465866214186228, 3.06986145795823201279516495328, 3.10720839661079125702849744909, 3.32630804528499381012007099660, 3.35065730318655083143488994015, 3.36583894945112643829736141209, 3.41024905765645980818057340139

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

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