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Properties

Label 16-1050e8-1.1-c1e8-0-3
Degree $16$
Conductor $1.477\times 10^{24}$
Sign $1$
Analytic cond. $2.44191\times 10^{7}$
Root an. cond. $2.89556$
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  − 8·7-s + 8·11-s − 16·13-s − 2·16-s + 12·17-s + 8·19-s − 16·23-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 − 2·81-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 + 127-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 + 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 − 2/9·81-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 + 0.0887·127-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.44191\times 10^{7}\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4559185051\)
\(L(\frac12)\) \(\approx\) \(0.4559185051\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T^{4} )^{2} \)
5 \( 1 \)
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

−4.33790459834400034469267619851, −4.16042190689855710638449864095, −3.86303797138459812329995246133, −3.70409785308301018124933650044, −3.65063925614762982953861715813, −3.64400546510106452233912374404, −3.51882546823795202024416384590, −3.35186439831131940526085502824, −3.30801352958486327869612611125, −2.89182949558700399096470552078, −2.81277015641787847517485526631, −2.71625484974733886104627207637, −2.67350645091101412253687160915, −2.53649211284595339544935462096, −2.42297354547907190292559516304, −2.12565583021432757236372087290, −2.04407021213137014139165170020, −2.03009664733081522437852671870, −1.41105568614945595356961934807, −1.40748571826862331958179969322, −1.23661185582416206368245566268, −0.815141783717198515190489480399, −0.69798031453266553057746502995, −0.62150572730848687507966209927, −0.097250887657807710419426111156, 0.097250887657807710419426111156, 0.62150572730848687507966209927, 0.69798031453266553057746502995, 0.815141783717198515190489480399, 1.23661185582416206368245566268, 1.40748571826862331958179969322, 1.41105568614945595356961934807, 2.03009664733081522437852671870, 2.04407021213137014139165170020, 2.12565583021432757236372087290, 2.42297354547907190292559516304, 2.53649211284595339544935462096, 2.67350645091101412253687160915, 2.71625484974733886104627207637, 2.81277015641787847517485526631, 2.89182949558700399096470552078, 3.30801352958486327869612611125, 3.35186439831131940526085502824, 3.51882546823795202024416384590, 3.64400546510106452233912374404, 3.65063925614762982953861715813, 3.70409785308301018124933650044, 3.86303797138459812329995246133, 4.16042190689855710638449864095, 4.33790459834400034469267619851

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

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