Properties

Label 16-1250e8-1.1-c1e8-0-1
Degree $16$
Conductor $5.960\times 10^{24}$
Sign $1$
Analytic cond. $9.85137\times 10^{7}$
Root an. cond. $3.15931$
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  − 4·4-s + 5·9-s − 4·11-s + 10·16-s + 10·19-s − 30·29-s − 24·31-s − 20·36-s + 26·41-s + 16·44-s + 30·49-s + 16·61-s − 20·64-s − 44·71-s − 40·76-s + 20·79-s − 6·81-s + 10·89-s − 20·99-s + 16·101-s − 30·109-s + 120·116-s − 54·121-s + 96·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 2·4-s + 5/3·9-s − 1.20·11-s + 5/2·16-s + 2.29·19-s − 5.57·29-s − 4.31·31-s − 3.33·36-s + 4.06·41-s + 2.41·44-s + 30/7·49-s + 2.04·61-s − 5/2·64-s − 5.22·71-s − 4.58·76-s + 2.25·79-s − 2/3·81-s + 1.05·89-s − 2.01·99-s + 1.59·101-s − 2.87·109-s + 11.1·116-s − 4.90·121-s + 8.62·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(9.85137\times 10^{7}\)
Root analytic conductor: \(3.15931\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.530584298\)
\(L(\frac12)\) \(\approx\) \(1.530584298\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
5 \( 1 \)
good3 \( 1 - 5 T^{2} + 31 T^{4} - 95 T^{6} + 376 T^{8} - 95 p^{2} T^{10} + 31 p^{4} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 - 30 T^{2} + 481 T^{4} - 5230 T^{6} + 42116 T^{8} - 5230 p^{2} T^{10} + 481 p^{4} T^{12} - 30 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + 2 T + 3 p T^{2} + 54 T^{3} + 500 T^{4} + 54 p T^{5} + 3 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 35 T^{2} + 926 T^{4} - 16240 T^{6} + 245521 T^{8} - 16240 p^{2} T^{10} + 926 p^{4} T^{12} - 35 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 70 T^{2} + 2711 T^{4} - 72380 T^{6} + 1426621 T^{8} - 72380 p^{2} T^{10} + 2711 p^{4} T^{12} - 70 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 5 T + 41 T^{2} - 265 T^{3} + 916 T^{4} - 265 p T^{5} + 41 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 110 T^{2} + 5441 T^{4} - 170590 T^{6} + 4207876 T^{8} - 170590 p^{2} T^{10} + 5441 p^{4} T^{12} - 110 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 15 T + 186 T^{2} + 1410 T^{3} + 9111 T^{4} + 1410 p T^{5} + 186 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 12 T + 123 T^{2} + 684 T^{3} + 4440 T^{4} + 684 p T^{5} + 123 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 190 T^{2} + 17751 T^{4} - 1088900 T^{6} + 47481221 T^{8} - 1088900 p^{2} T^{10} + 17751 p^{4} T^{12} - 190 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 13 T + 198 T^{2} - 1576 T^{3} + 12785 T^{4} - 1576 p T^{5} + 198 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 230 T^{2} + 25001 T^{4} - 1727030 T^{6} + 85837476 T^{8} - 1727030 p^{2} T^{10} + 25001 p^{4} T^{12} - 230 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 350 T^{2} + 54721 T^{4} - 4989870 T^{6} + 290207396 T^{8} - 4989870 p^{2} T^{10} + 54721 p^{4} T^{12} - 350 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 205 T^{2} + 22131 T^{4} - 1650815 T^{6} + 95978576 T^{8} - 1650815 p^{2} T^{10} + 22131 p^{4} T^{12} - 205 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 96 T^{2} + 560 T^{3} + 4046 T^{4} + 560 p T^{5} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 8 T + 143 T^{2} - 506 T^{3} + 8295 T^{4} - 506 p T^{5} + 143 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 190 T^{2} + 14641 T^{4} - 602350 T^{6} + 24146836 T^{8} - 602350 p^{2} T^{10} + 14641 p^{4} T^{12} - 190 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 22 T + 413 T^{2} + 4854 T^{3} + 48500 T^{4} + 4854 p T^{5} + 413 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 305 T^{2} + 40391 T^{4} - 3168935 T^{6} + 213856176 T^{8} - 3168935 p^{2} T^{10} + 40391 p^{4} T^{12} - 305 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 10 T + 296 T^{2} - 2130 T^{3} + 33966 T^{4} - 2130 p T^{5} + 296 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 350 T^{2} + 68241 T^{4} - 8936350 T^{6} + 861968756 T^{8} - 8936350 p^{2} T^{10} + 68241 p^{4} T^{12} - 350 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 5 T + 191 T^{2} + 165 T^{3} + 15056 T^{4} + 165 p T^{5} + 191 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 265 T^{2} + 32593 T^{4} - 265 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−4.14559232662747553593798641940, −3.95143065630618760544603787901, −3.93125732392760524096456050091, −3.75727419712785892919034017607, −3.72491935608904419135264301627, −3.52840428302495339065615874657, −3.46920833661267858839790320077, −3.42251820685545613392792334096, −3.12285897341955242627930926105, −3.07251700946408139375180077542, −2.75771456122971824493645057370, −2.48715653810262041909912497459, −2.45670601016537798690751687996, −2.41502187586504972719759957706, −2.38035760877541155612847155274, −1.99618097458223584295107897201, −1.77431918345264034419577831693, −1.65083638701624411416418529535, −1.51380370547656573594467768799, −1.42686658328713912079255982451, −1.12762956010657924566065798618, −0.982699038392073290433343711749, −0.62609648221076953288456069148, −0.35839607037691932125531672962, −0.24836444855857479739218596089, 0.24836444855857479739218596089, 0.35839607037691932125531672962, 0.62609648221076953288456069148, 0.982699038392073290433343711749, 1.12762956010657924566065798618, 1.42686658328713912079255982451, 1.51380370547656573594467768799, 1.65083638701624411416418529535, 1.77431918345264034419577831693, 1.99618097458223584295107897201, 2.38035760877541155612847155274, 2.41502187586504972719759957706, 2.45670601016537798690751687996, 2.48715653810262041909912497459, 2.75771456122971824493645057370, 3.07251700946408139375180077542, 3.12285897341955242627930926105, 3.42251820685545613392792334096, 3.46920833661267858839790320077, 3.52840428302495339065615874657, 3.72491935608904419135264301627, 3.75727419712785892919034017607, 3.93125732392760524096456050091, 3.95143065630618760544603787901, 4.14559232662747553593798641940

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

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