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Properties

Label 16-1110e8-1.1-c1e8-0-0
Degree $16$
Conductor $2.305\times 10^{24}$
Sign $1$
Analytic cond. $3.80890\times 10^{7}$
Root an. cond. $2.97714$
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 − 2·5-s − 4·9-s − 10·11-s + 10·16-s + 24·19-s + 8·20-s + 2·25-s − 10·29-s − 38·31-s + 16·36-s + 26·41-s + 40·44-s + 8·45-s + 13·49-s + 20·55-s − 20·59-s − 6·61-s − 20·64-s − 12·71-s − 96·76-s + 16·79-s − 20·80-s + 10·81-s − 64·89-s − 48·95-s + 40·99-s + ⋯
L(s)  = 1  − 2·4-s − 0.894·5-s − 4/3·9-s − 3.01·11-s + 5/2·16-s + 5.50·19-s + 1.78·20-s + 2/5·25-s − 1.85·29-s − 6.82·31-s + 8/3·36-s + 4.06·41-s + 6.03·44-s + 1.19·45-s + 13/7·49-s + 2.69·55-s − 2.60·59-s − 0.768·61-s − 5/2·64-s − 1.42·71-s − 11.0·76-s + 1.80·79-s − 2.23·80-s + 10/9·81-s − 6.78·89-s − 4.92·95-s + 4.02·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8}\)
Sign: $1$
Analytic conductor: \(3.80890\times 10^{7}\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.06335498000\)
\(L(\frac12)\) \(\approx\) \(0.06335498000\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
3 \( ( 1 + T^{2} )^{4} \)
5 \( 1 + 2 T + 2 T^{2} - 6 T^{3} - 46 T^{4} - 6 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
37 \( ( 1 + T^{2} )^{4} \)
good7 \( 1 - 13 T^{2} + 145 T^{4} - 1290 T^{6} + 10466 T^{8} - 1290 p^{2} T^{10} + 145 p^{4} T^{12} - 13 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + 5 T + 3 p T^{2} + 158 T^{3} + 488 T^{4} + 158 p T^{5} + 3 p^{3} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 6 p T^{2} + 2905 T^{4} - 67090 T^{6} + 1048316 T^{8} - 67090 p^{2} T^{10} + 2905 p^{4} T^{12} - 6 p^{7} T^{14} + p^{8} T^{16} \)
17 \( 1 - 61 T^{2} + 109 p T^{4} - 37666 T^{6} + 653190 T^{8} - 37666 p^{2} T^{10} + 109 p^{5} T^{12} - 61 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 12 T + 89 T^{2} - 546 T^{3} + 2788 T^{4} - 546 p T^{5} + 89 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 11 T^{2} + 744 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 5 T + 82 T^{2} + 481 T^{3} + 3058 T^{4} + 481 p T^{5} + 82 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 19 T + 180 T^{2} + 1091 T^{3} + 5910 T^{4} + 1091 p T^{5} + 180 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - 13 T + 156 T^{2} - 1453 T^{3} + 9602 T^{4} - 1453 p T^{5} + 156 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 319 T^{2} + 45506 T^{4} - 3790265 T^{6} + 201569722 T^{8} - 3790265 p^{2} T^{10} + 45506 p^{4} T^{12} - 319 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 168 T^{2} + 16592 T^{4} - 1151736 T^{6} + 60803998 T^{8} - 1151736 p^{2} T^{10} + 16592 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 89 T^{2} + 4313 T^{4} - 124438 T^{6} + 1426822 T^{8} - 124438 p^{2} T^{10} + 4313 p^{4} T^{12} - 89 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 10 T + 254 T^{2} + 1718 T^{3} + 22882 T^{4} + 1718 p T^{5} + 254 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 3 T + 136 T^{2} + 441 T^{3} + 10446 T^{4} + 441 p T^{5} + 136 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 32 T^{2} + 3868 T^{4} - 468768 T^{6} - 19957402 T^{8} - 468768 p^{2} T^{10} + 3868 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 6 T + 34 T^{2} - 366 T^{3} - 5318 T^{4} - 366 p T^{5} + 34 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 306 T^{2} + 52625 T^{4} - 6003822 T^{6} + 508038268 T^{8} - 6003822 p^{2} T^{10} + 52625 p^{4} T^{12} - 306 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 8 T + 146 T^{2} - 472 T^{3} + 9562 T^{4} - 472 p T^{5} + 146 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 146 T^{2} + 25777 T^{4} - 2120010 T^{6} + 232651316 T^{8} - 2120010 p^{2} T^{10} + 25777 p^{4} T^{12} - 146 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 32 T + 695 T^{2} + 9906 T^{3} + 109484 T^{4} + 9906 p T^{5} + 695 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 439 T^{2} + 99722 T^{4} - 15425637 T^{6} + 1741112370 T^{8} - 15425637 p^{2} T^{10} + 99722 p^{4} T^{12} - 439 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.12532904581334507702006683346, −3.88339285770386777788992414707, −3.82925535279596623241851776861, −3.77567590844139524435815484987, −3.72586691924351620731351173842, −3.61650579244009096499461393252, −3.60292839020152355331872349072, −3.37463231593500615219146090466, −3.02189160597283611775731040231, −2.93012395724391499188664157694, −2.92429757324220568683528041095, −2.82313477782778467537032769892, −2.79736409827427722647211706887, −2.55819643038665427589793077870, −2.23718676984734388456551701386, −2.15779988152451439268326632561, −2.04603127975099496166431020128, −1.66217046851468945730540157287, −1.46311790765117242195307524028, −1.21237197701395780407125795517, −1.18888356037158697117871780594, −1.09225905542472047880989066105, −0.53149171999576276133195517229, −0.18010721482245116791057655205, −0.13339224413338955962918997887, 0.13339224413338955962918997887, 0.18010721482245116791057655205, 0.53149171999576276133195517229, 1.09225905542472047880989066105, 1.18888356037158697117871780594, 1.21237197701395780407125795517, 1.46311790765117242195307524028, 1.66217046851468945730540157287, 2.04603127975099496166431020128, 2.15779988152451439268326632561, 2.23718676984734388456551701386, 2.55819643038665427589793077870, 2.79736409827427722647211706887, 2.82313477782778467537032769892, 2.92429757324220568683528041095, 2.93012395724391499188664157694, 3.02189160597283611775731040231, 3.37463231593500615219146090466, 3.60292839020152355331872349072, 3.61650579244009096499461393252, 3.72586691924351620731351173842, 3.77567590844139524435815484987, 3.82925535279596623241851776861, 3.88339285770386777788992414707, 4.12532904581334507702006683346

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

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