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Properties

Label 16-1176e8-1.1-c1e8-0-0
Degree $16$
Conductor $3.658\times 10^{24}$
Sign $1$
Analytic cond. $6.04610\times 10^{7}$
Root an. cond. $3.06437$
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-s − 2·8-s − 4·9-s − 16-s − 4·17-s + 12·23-s + 8·25-s − 8·31-s − 8·32-s − 4·36-s + 4·41-s + 5·64-s − 4·68-s − 28·71-s + 8·72-s + 8·73-s − 40·79-s + 10·81-s − 20·89-s + 12·92-s − 40·97-s + 8·100-s − 8·103-s − 8·113-s + 24·121-s − 8·124-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.707·8-s − 4/3·9-s − 1/4·16-s − 0.970·17-s + 2.50·23-s + 8/5·25-s − 1.43·31-s − 1.41·32-s − 2/3·36-s + 0.624·41-s + 5/8·64-s − 0.485·68-s − 3.32·71-s + 0.942·72-s + 0.936·73-s − 4.50·79-s + 10/9·81-s − 2.11·89-s + 1.25·92-s − 4.06·97-s + 4/5·100-s − 0.788·103-s − 0.752·113-s + 2.18·121-s − 0.718·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(6.04610\times 10^{7}\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{8} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.209784416\)
\(L(\frac12)\) \(\approx\) \(4.209784416\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + p T^{3} + p T^{4} + p^{2} T^{5} - p^{2} T^{6} + p^{4} T^{8} \)
3 \( ( 1 + T^{2} )^{4} \)
7 \( 1 \)
good5 \( 1 - 8 T^{2} + 16 T^{4} - 168 T^{6} + 1694 T^{8} - 168 p^{2} T^{10} + 16 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 24 T^{2} + 592 T^{4} - 8312 T^{6} + 114206 T^{8} - 8312 p^{2} T^{10} + 592 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 48 T^{2} + 1340 T^{4} - 26704 T^{6} + 396198 T^{8} - 26704 p^{2} T^{10} + 1340 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 2 T + 38 T^{2} + 70 T^{3} + 706 T^{4} + 70 p T^{5} + 38 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 64 T^{2} + 2012 T^{4} - 42432 T^{6} + 793446 T^{8} - 42432 p^{2} T^{10} + 2012 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 - 6 T + 74 T^{2} - 334 T^{3} + 2282 T^{4} - 334 p T^{5} + 74 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 16 T^{2} + 2876 T^{4} - 34928 T^{6} + 3439654 T^{8} - 34928 p^{2} T^{10} + 2876 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 + 4 T + 80 T^{2} + 244 T^{3} + 3294 T^{4} + 244 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 192 T^{2} + 18716 T^{4} - 1175872 T^{6} + 51538086 T^{8} - 1175872 p^{2} T^{10} + 18716 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 2 T + 134 T^{2} - 214 T^{3} + 7618 T^{4} - 214 p T^{5} + 134 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 168 T^{2} + 15676 T^{4} - 1026520 T^{6} + 50753126 T^{8} - 1026520 p^{2} T^{10} + 15676 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 44 T^{2} + 128 T^{3} + 3302 T^{4} + 128 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( 1 - 336 T^{2} + 52604 T^{4} - 5027376 T^{6} + 321653350 T^{8} - 5027376 p^{2} T^{10} + 52604 p^{4} T^{12} - 336 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \)
61 \( 1 - 272 T^{2} + 35516 T^{4} - 3030896 T^{6} + 201654822 T^{8} - 3030896 p^{2} T^{10} + 35516 p^{4} T^{12} - 272 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 344 T^{2} + 59996 T^{4} - 6783272 T^{6} + 536949606 T^{8} - 6783272 p^{2} T^{10} + 59996 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 14 T + 194 T^{2} + 1686 T^{3} + 14330 T^{4} + 1686 p T^{5} + 194 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 4 T + 92 T^{2} + 580 T^{3} + 166 T^{4} + 580 p T^{5} + 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 20 T + 340 T^{2} + 3972 T^{3} + 38678 T^{4} + 3972 p T^{5} + 340 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 248 T^{2} + 532 p T^{4} - 5204488 T^{6} + 499369126 T^{8} - 5204488 p^{2} T^{10} + 532 p^{5} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 10 T + 198 T^{2} + 494 T^{3} + 13122 T^{4} + 494 p T^{5} + 198 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 20 T + 300 T^{2} + 2572 T^{3} + 25574 T^{4} + 2572 p T^{5} + 300 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.16730612445960539540793907595, −4.06861034779744461689935191126, −4.01198626763122450767780633303, −3.81803855499865508007472176814, −3.47984868721251698315256884844, −3.46881399497171996588338833510, −3.29870560066338510348074400032, −3.15987746437749513216674021682, −3.11346337696149302092315294531, −2.89823713561261682044088141664, −2.76587161961034791897129857279, −2.74077844229113321164435014942, −2.64845264138563585425585672333, −2.53646152289368337652540636060, −2.48349391284503846407558891941, −1.90225289481434728338024218342, −1.79750994881580529603354006217, −1.76118727631777295518365836685, −1.61884589458139723971548233748, −1.57693403205061851227987675865, −1.25000245294838359676855441983, −0.928406191025036971333334412505, −0.57379120543433173168641922071, −0.45062681538664243966747178911, −0.33651751094094145339208542915, 0.33651751094094145339208542915, 0.45062681538664243966747178911, 0.57379120543433173168641922071, 0.928406191025036971333334412505, 1.25000245294838359676855441983, 1.57693403205061851227987675865, 1.61884589458139723971548233748, 1.76118727631777295518365836685, 1.79750994881580529603354006217, 1.90225289481434728338024218342, 2.48349391284503846407558891941, 2.53646152289368337652540636060, 2.64845264138563585425585672333, 2.74077844229113321164435014942, 2.76587161961034791897129857279, 2.89823713561261682044088141664, 3.11346337696149302092315294531, 3.15987746437749513216674021682, 3.29870560066338510348074400032, 3.46881399497171996588338833510, 3.47984868721251698315256884844, 3.81803855499865508007472176814, 4.01198626763122450767780633303, 4.06861034779744461689935191126, 4.16730612445960539540793907595

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

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