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Properties

Label 16-1680e8-1.1-c1e8-0-8
Degree $16$
Conductor $6.346\times 10^{25}$
Sign $1$
Analytic cond. $1.04879\times 10^{9}$
Root an. cond. $3.66263$
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  − 4·7-s − 8·11-s − 16·13-s + 12·17-s + 8·19-s − 16·23-s − 2·25-s − 28·37-s − 24·47-s + 10·49-s − 8·53-s − 8·59-s − 8·71-s − 28·73-s + 32·77-s − 2·81-s + 16·83-s − 64·89-s + 64·91-s − 28·97-s − 28·103-s − 32·107-s − 24·113-s − 48·119-s + 24·121-s + 16·125-s + 127-s + ⋯
L(s)  = 1  − 1.51·7-s − 2.41·11-s − 4.43·13-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 + 10/7·49-s − 1.09·53-s − 1.04·59-s − 0.949·71-s − 3.27·73-s + 3.64·77-s − 2/9·81-s + 1.75·83-s − 6.78·89-s + 6.70·91-s − 2.84·97-s − 2.75·103-s − 3.09·107-s − 2.25·113-s − 4.40·119-s + 2.18·121-s + 1.43·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.04879\times 10^{9}\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2727051605\)
\(L(\frac12)\) \(\approx\) \(0.2727051605\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{4} )^{2} \)
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 \( 1 + 4 T + 6 T^{2} - 4 p T^{3} - 18 p T^{4} - 4 p^{2} T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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

−4.01528857812071394433852880176, −3.88943373526910179860821724533, −3.80725820343353860168440631051, −3.49175464631941596263157418858, −3.22809907952366777174876708812, −3.13769404579031524302401550847, −3.03140518261646428050293783838, −3.00735100584843107552104027571, −2.92925138558121036575306605209, −2.92600711070620696486207981969, −2.84851308372743953299040352949, −2.66935343945688264052363546363, −2.48071460316109801297825996751, −2.33582470660664648162683188774, −1.92590521224968597106288514904, −1.91036186847974624461036165131, −1.73571555516774424111658384677, −1.67575965923047136454112414314, −1.53925055628880890876541183767, −1.40131728843969068155276865295, −1.32673205022033832266734675956, −0.53620889626874959294760425656, −0.40352713608824583044500650284, −0.24956037898509668044881231487, −0.19452056800123924840806825159, 0.19452056800123924840806825159, 0.24956037898509668044881231487, 0.40352713608824583044500650284, 0.53620889626874959294760425656, 1.32673205022033832266734675956, 1.40131728843969068155276865295, 1.53925055628880890876541183767, 1.67575965923047136454112414314, 1.73571555516774424111658384677, 1.91036186847974624461036165131, 1.92590521224968597106288514904, 2.33582470660664648162683188774, 2.48071460316109801297825996751, 2.66935343945688264052363546363, 2.84851308372743953299040352949, 2.92600711070620696486207981969, 2.92925138558121036575306605209, 3.00735100584843107552104027571, 3.03140518261646428050293783838, 3.13769404579031524302401550847, 3.22809907952366777174876708812, 3.49175464631941596263157418858, 3.80725820343353860168440631051, 3.88943373526910179860821724533, 4.01528857812071394433852880176

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

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