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Properties

Label 16-1224e8-1.1-c1e8-0-3
Degree $16$
Conductor $5.038\times 10^{24}$
Sign $1$
Analytic cond. $8.32660\times 10^{7}$
Root an. cond. $3.12629$
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  + 2-s + 4-s − 12·7-s − 8-s − 12·14-s + 16-s − 8·17-s + 16·23-s + 16·25-s − 12·28-s + 24·31-s − 32-s − 8·34-s + 16·46-s − 4·47-s + 48·49-s + 16·50-s + 12·56-s + 24·62-s + 64-s − 8·68-s + 36·71-s + 8·73-s + 24·79-s − 8·89-s + 16·92-s − 4·94-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 4.53·7-s − 0.353·8-s − 3.20·14-s + 1/4·16-s − 1.94·17-s + 3.33·23-s + 16/5·25-s − 2.26·28-s + 4.31·31-s − 0.176·32-s − 1.37·34-s + 2.35·46-s − 0.583·47-s + 48/7·49-s + 2.26·50-s + 1.60·56-s + 3.04·62-s + 1/8·64-s − 0.970·68-s + 4.27·71-s + 0.936·73-s + 2.70·79-s − 0.847·89-s + 1.66·92-s − 0.412·94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{16} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(8.32660\times 10^{7}\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{16} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.298550005\)
\(L(\frac12)\) \(\approx\) \(5.298550005\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T + p T^{3} - p^{2} T^{4} + p^{2} T^{5} - p^{3} T^{7} + p^{4} T^{8} \)
3 \( 1 \)
17 \( ( 1 + T )^{8} \)
good5 \( 1 - 16 T^{2} + 28 p T^{4} - 832 T^{6} + 4326 T^{8} - 832 p^{2} T^{10} + 28 p^{5} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
7 \( ( 1 + 6 T + 30 T^{2} + 108 T^{3} + 306 T^{4} + 108 p T^{5} + 30 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 2 T - 20 T^{2} + 8 T^{3} + 318 T^{4} + 8 p T^{5} - 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )( 1 + 2 T - 20 T^{2} - 8 T^{3} + 318 T^{4} - 8 p T^{5} - 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} ) \)
13 \( 1 - 4 p T^{2} + 1172 T^{4} - 16124 T^{6} + 194710 T^{8} - 16124 p^{2} T^{10} + 1172 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
19 \( 1 - 72 T^{2} + 2492 T^{4} - 57784 T^{6} + 1130022 T^{8} - 57784 p^{2} T^{10} + 2492 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 - 8 T + 42 T^{2} - 274 T^{3} + 1818 T^{4} - 274 p T^{5} + 42 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 208 T^{2} + 19532 T^{4} - 1081216 T^{6} + 38589222 T^{8} - 1081216 p^{2} T^{10} + 19532 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 - 12 T + 142 T^{2} - 1022 T^{3} + 6722 T^{4} - 1022 p T^{5} + 142 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 136 T^{2} + 10700 T^{4} - 590792 T^{6} + 24611302 T^{8} - 590792 p^{2} T^{10} + 10700 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 108 T^{2} + 176 T^{3} + 5414 T^{4} + 176 p T^{5} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( 1 - 156 T^{2} + 10820 T^{4} - 476740 T^{6} + 19058262 T^{8} - 476740 p^{2} T^{10} + 10820 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 2 T + 64 T^{2} + 82 T^{3} + 4798 T^{4} + 82 p T^{5} + 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( 1 - 312 T^{2} + 46940 T^{4} - 4401864 T^{6} + 280109350 T^{8} - 4401864 p^{2} T^{10} + 46940 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 172 T^{2} + 20900 T^{4} - 1780276 T^{6} + 121222422 T^{8} - 1780276 p^{2} T^{10} + 20900 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 464 T^{2} + 95564 T^{4} - 11410048 T^{6} + 863785510 T^{8} - 11410048 p^{2} T^{10} + 95564 p^{4} T^{12} - 464 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 424 T^{2} + 82236 T^{4} - 9714776 T^{6} + 778216358 T^{8} - 9714776 p^{2} T^{10} + 82236 p^{4} T^{12} - 424 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 18 T + 338 T^{2} - 3728 T^{3} + 37514 T^{4} - 3728 p T^{5} + 338 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 4 T + 196 T^{2} - 956 T^{3} + 18262 T^{4} - 956 p T^{5} + 196 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 12 T + 314 T^{2} - 2678 T^{3} + 37002 T^{4} - 2678 p T^{5} + 314 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 348 T^{2} + 72644 T^{4} - 9769028 T^{6} + 960783126 T^{8} - 9769028 p^{2} T^{10} + 72644 p^{4} T^{12} - 348 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 4 T + 72 T^{2} - 1040 T^{3} - 5482 T^{4} - 1040 p T^{5} + 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 4 T + 276 T^{2} + 428 T^{3} + 34038 T^{4} + 428 p T^{5} + 276 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.08529236620371252441775699663, −3.91413347949864789909730691387, −3.89670850696967669888883161633, −3.71524099829734897368707090806, −3.49867460546986418603065985608, −3.46013086426460535301612491284, −3.25747428657113696272469454462, −3.08959924036322107673147317595, −3.08540615566510050760350566236, −3.04540371334432396548250639695, −2.80430376654734776449436990998, −2.77776968806371675871839975976, −2.72473966592247212393135909163, −2.71477796110981948197669496853, −2.37695584665376895573436910603, −2.10946896161342768600355726021, −2.00309851616261767414780238234, −1.87293827680636128738072857947, −1.47497139649514274136533100732, −1.38314724309166161412235040496, −0.891587055197963600574124810015, −0.807790654931858911717680244497, −0.794476663834174395489451436831, −0.53119422185242785880146153509, −0.29884304102875100521994951937, 0.29884304102875100521994951937, 0.53119422185242785880146153509, 0.794476663834174395489451436831, 0.807790654931858911717680244497, 0.891587055197963600574124810015, 1.38314724309166161412235040496, 1.47497139649514274136533100732, 1.87293827680636128738072857947, 2.00309851616261767414780238234, 2.10946896161342768600355726021, 2.37695584665376895573436910603, 2.71477796110981948197669496853, 2.72473966592247212393135909163, 2.77776968806371675871839975976, 2.80430376654734776449436990998, 3.04540371334432396548250639695, 3.08540615566510050760350566236, 3.08959924036322107673147317595, 3.25747428657113696272469454462, 3.46013086426460535301612491284, 3.49867460546986418603065985608, 3.71524099829734897368707090806, 3.89670850696967669888883161633, 3.91413347949864789909730691387, 4.08529236620371252441775699663

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

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