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Properties

Label 16-630e8-1.1-c1e8-0-7
Degree $16$
Conductor $2.482\times 10^{22}$
Sign $1$
Analytic cond. $410148.$
Root an. cond. $2.24289$
Motivic weight $1$
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  + 8·7-s − 8·11-s + 16·13-s − 2·16-s + 12·17-s + 8·19-s − 16·23-s − 2·25-s − 28·37-s + 24·47-s + 30·49-s + 8·53-s + 8·59-s − 8·71-s + 28·73-s − 64·77-s − 16·83-s − 64·89-s + 128·91-s + 28·97-s − 28·103-s − 32·107-s − 16·112-s + 24·113-s + 96·119-s + 24·121-s + 16·125-s + ⋯
L(s)  = 1  + 3.02·7-s − 2.41·11-s + 4.43·13-s − 1/2·16-s + 2.91·17-s + 1.83·19-s − 3.33·23-s − 2/5·25-s − 4.60·37-s + 3.50·47-s + 30/7·49-s + 1.09·53-s + 1.04·59-s − 0.949·71-s + 3.27·73-s − 7.29·77-s − 1.75·83-s − 6.78·89-s + 13.4·91-s + 2.84·97-s − 2.75·103-s − 3.09·107-s − 1.51·112-s + 2.25·113-s + 8.80·119-s + 2.18·121-s + 1.43·125-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(410148.\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.093738115\)
\(L(\frac12)\) \(\approx\) \(6.093738115\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( 1 \)
5 \( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 16 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \)
7 \( 1 - 8 T + 34 T^{2} - 16 p T^{3} + 46 p T^{4} - 16 p^{2} T^{5} + 34 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
good11 \( ( 1 + 4 T + 12 T^{2} - 4 T^{3} + 18 T^{4} - 4 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 562 T^{4} - 136 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 - 12 T + 72 T^{2} - 388 T^{3} + 1696 T^{4} - 4732 T^{5} + 9944 T^{6} - 2420 T^{7} - 98754 T^{8} - 2420 p T^{9} + 9944 p^{2} T^{10} - 4732 p^{3} T^{11} + 1696 p^{4} T^{12} - 388 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
19 \( ( 1 - 4 T + 62 T^{2} - 212 T^{3} + 1666 T^{4} - 212 p T^{5} + 62 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 16 T + 128 T^{2} + 656 T^{3} + 3300 T^{4} + 21520 T^{5} + 137088 T^{6} + 691344 T^{7} + 3238918 T^{8} + 691344 p T^{9} + 137088 p^{2} T^{10} + 21520 p^{3} T^{11} + 3300 p^{4} T^{12} + 656 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 4 T^{2} + 200 T^{4} - 5548 T^{6} + 1325934 T^{8} - 5548 p^{2} T^{10} + 200 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 - 148 T^{2} + 11192 T^{4} - 565212 T^{6} + 20507310 T^{8} - 565212 p^{2} T^{10} + 11192 p^{4} T^{12} - 148 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 + 28 T + 392 T^{2} + 4180 T^{3} + 38912 T^{4} + 311708 T^{5} + 2210520 T^{6} + 14742164 T^{7} + 92944158 T^{8} + 14742164 p T^{9} + 2210520 p^{2} T^{10} + 311708 p^{3} T^{11} + 38912 p^{4} T^{12} + 4180 p^{5} T^{13} + 392 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 176 T^{2} + 13468 T^{4} - 643920 T^{6} + 26131718 T^{8} - 643920 p^{2} T^{10} + 13468 p^{4} T^{12} - 176 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 32 T^{3} - 6588 T^{4} - 1632 T^{5} + 512 T^{6} + 117440 T^{7} + 17663206 T^{8} + 117440 p T^{9} + 512 p^{2} T^{10} - 1632 p^{3} T^{11} - 6588 p^{4} T^{12} - 32 p^{5} T^{13} + p^{8} T^{16} \)
47 \( 1 - 24 T + 288 T^{2} - 2728 T^{3} + 26308 T^{4} - 243208 T^{5} + 1981280 T^{6} - 14819768 T^{7} + 104930310 T^{8} - 14819768 p T^{9} + 1981280 p^{2} T^{10} - 243208 p^{3} T^{11} + 26308 p^{4} T^{12} - 2728 p^{5} T^{13} + 288 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 8 T + 32 T^{2} - 264 T^{3} + 5828 T^{4} - 46136 T^{5} + 217440 T^{6} - 2204664 T^{7} + 21970150 T^{8} - 2204664 p T^{9} + 217440 p^{2} T^{10} - 46136 p^{3} T^{11} + 5828 p^{4} T^{12} - 264 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 - 4 T + 70 T^{2} - 116 T^{3} + 802 T^{4} - 116 p T^{5} + 70 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( 1 - 264 T^{2} + 37532 T^{4} - 3615096 T^{6} + 255247078 T^{8} - 3615096 p^{2} T^{10} + 37532 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 64 T^{3} + 9348 T^{4} + 10560 T^{5} + 2048 T^{6} + 428416 T^{7} + 51599014 T^{8} + 428416 p T^{9} + 2048 p^{2} T^{10} + 10560 p^{3} T^{11} + 9348 p^{4} T^{12} - 64 p^{5} T^{13} + p^{8} T^{16} \)
71 \( ( 1 + 4 T + 158 T^{2} - 220 T^{3} + 10018 T^{4} - 220 p T^{5} + 158 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 28 T + 392 T^{2} - 4756 T^{3} + 46736 T^{4} - 302508 T^{5} + 1459480 T^{6} - 3207460 T^{7} - 21483554 T^{8} - 3207460 p T^{9} + 1459480 p^{2} T^{10} - 302508 p^{3} T^{11} + 46736 p^{4} T^{12} - 4756 p^{5} T^{13} + 392 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 272 T^{2} + 39644 T^{4} - 4069360 T^{6} + 348070342 T^{8} - 4069360 p^{2} T^{10} + 39644 p^{4} T^{12} - 272 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 + 16 T + 128 T^{2} + 976 T^{3} + 16964 T^{4} + 235664 T^{5} + 2075520 T^{6} + 16972368 T^{7} + 135669670 T^{8} + 16972368 p T^{9} + 2075520 p^{2} T^{10} + 235664 p^{3} T^{11} + 16964 p^{4} T^{12} + 976 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 + 32 T + 628 T^{2} + 8320 T^{3} + 88630 T^{4} + 8320 p T^{5} + 628 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 28 T + 392 T^{2} - 4900 T^{3} + 61488 T^{4} - 704956 T^{5} + 7640472 T^{6} - 89011524 T^{7} + 981394654 T^{8} - 89011524 p T^{9} + 7640472 p^{2} T^{10} - 704956 p^{3} T^{11} + 61488 p^{4} T^{12} - 4900 p^{5} T^{13} + 392 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.55520674215189696224129501859, −4.33301799750897758156348347869, −4.25803894740846275230514249605, −4.22770498119929956341995241052, −4.07249899712044638069370938166, −3.88174229477348259726616740533, −3.72269013117269494187693888516, −3.70116108394110836753813354004, −3.68515771865377441957743017960, −3.38930301047317147429043280738, −3.25859550172253433226256542885, −2.96998978507533333978662578613, −2.90655607202619786549175727890, −2.73061556599519458384176666158, −2.51886434183815301342362868654, −2.27294203162021120141116723756, −2.24837830757679292323102637361, −1.79529297059397454398445285610, −1.69419999305939079951124337237, −1.66405441867546583516164266230, −1.35120167931502393584662066971, −1.33130153545064257602933956375, −1.07362964584401041388161648022, −0.840559680775090187968352606714, −0.29522668069004027955878630690, 0.29522668069004027955878630690, 0.840559680775090187968352606714, 1.07362964584401041388161648022, 1.33130153545064257602933956375, 1.35120167931502393584662066971, 1.66405441867546583516164266230, 1.69419999305939079951124337237, 1.79529297059397454398445285610, 2.24837830757679292323102637361, 2.27294203162021120141116723756, 2.51886434183815301342362868654, 2.73061556599519458384176666158, 2.90655607202619786549175727890, 2.96998978507533333978662578613, 3.25859550172253433226256542885, 3.38930301047317147429043280738, 3.68515771865377441957743017960, 3.70116108394110836753813354004, 3.72269013117269494187693888516, 3.88174229477348259726616740533, 4.07249899712044638069370938166, 4.22770498119929956341995241052, 4.25803894740846275230514249605, 4.33301799750897758156348347869, 4.55520674215189696224129501859

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

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