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Properties

Label 16-5e32-1.1-c1e8-0-0
Degree $16$
Conductor $2.328\times 10^{22}$
Sign $1$
Analytic cond. $384819.$
Root an. cond. $2.23397$
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  − 5·4-s − 5·9-s + 16·11-s + 19·16-s − 20·19-s + 6·31-s + 25·36-s − 4·41-s − 80·44-s + 50·49-s − 30·59-s − 34·61-s − 45·64-s + 6·71-s + 100·76-s + 20·79-s + 9·81-s − 40·89-s − 80·99-s − 24·101-s + 20·109-s + 166·121-s − 30·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 5/2·4-s − 5/3·9-s + 4.82·11-s + 19/4·16-s − 4.58·19-s + 1.07·31-s + 25/6·36-s − 0.624·41-s − 12.0·44-s + 50/7·49-s − 3.90·59-s − 4.35·61-s − 5.62·64-s + 0.712·71-s + 11.4·76-s + 2.25·79-s + 81-s − 4.23·89-s − 8.04·99-s − 2.38·101-s + 1.91·109-s + 15.0·121-s − 2.69·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(5^{32}\)
Sign: $1$
Analytic conductor: \(384819.\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1886164008\)
\(L(\frac12)\) \(\approx\) \(0.1886164008\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + 5 T^{2} + 3 p T^{4} - 5 p^{2} T^{6} - 79 T^{8} - 5 p^{4} T^{10} + 3 p^{5} T^{12} + 5 p^{6} T^{14} + p^{8} T^{16} \)
3 \( 1 + 5 T^{2} + 16 T^{4} + 35 T^{6} + 31 T^{8} + 35 p^{2} T^{10} + 16 p^{4} T^{12} + 5 p^{6} T^{14} + p^{8} T^{16} \)
7 \( ( 1 - 25 T^{2} + 253 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 8 T + 13 T^{2} + 74 T^{3} - 435 T^{4} + 74 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 10 T^{2} + 291 T^{4} - 2300 T^{6} + 64541 T^{8} - 2300 p^{2} T^{10} + 291 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 10 T^{2} + 651 T^{4} - 560 T^{6} + 183941 T^{8} - 560 p^{2} T^{10} + 651 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 70 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 55 T^{2} + 72 p T^{4} - 50465 T^{6} + 1421471 T^{8} - 50465 p^{2} T^{10} + 72 p^{5} T^{12} - 55 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 19 T^{2} - 120 T^{3} + 721 T^{4} - 120 p T^{5} - 19 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 159 p T^{5} - 22 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 + 45 T^{2} + 3096 T^{4} + 174355 T^{6} + 5372031 T^{8} + 174355 p^{2} T^{10} + 3096 p^{4} T^{12} + 45 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 2 T - 17 T^{2} + 214 T^{3} + 2025 T^{4} + 214 p T^{5} - 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 145 T^{2} + 8853 T^{4} - 145 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( 1 + 95 T^{2} + 6351 T^{4} + 361285 T^{6} + 20346536 T^{8} + 361285 p^{2} T^{10} + 6351 p^{4} T^{12} + 95 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 105 T^{2} + 4656 T^{4} - 13105 T^{6} - 11269449 T^{8} - 13105 p^{2} T^{10} + 4656 p^{4} T^{12} + 105 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 15 T + 31 T^{2} - 15 p T^{3} - 10424 T^{4} - 15 p^{2} T^{5} + 31 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 17 T + 78 T^{2} + 289 T^{3} + 3755 T^{4} + 289 p T^{5} + 78 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 150 T^{2} + 4851 T^{4} - 664040 T^{6} - 82538859 T^{8} - 664040 p^{2} T^{10} + 4851 p^{4} T^{12} + 150 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 3 T - 37 T^{2} + 549 T^{3} + 1480 T^{4} + 549 p T^{5} - 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 65 T^{2} - 1104 T^{4} - 418145 T^{6} - 21296209 T^{8} - 418145 p^{2} T^{10} - 1104 p^{4} T^{12} + 65 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 10 T + 21 T^{2} - 770 T^{3} + 12791 T^{4} - 770 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 105 T^{2} + 15096 T^{4} + 1832615 T^{6} + 123748551 T^{8} + 1832615 p^{2} T^{10} + 15096 p^{4} T^{12} + 105 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 20 T + 151 T^{2} + 1600 T^{3} + 21441 T^{4} + 1600 p T^{5} + 151 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + 175 T^{2} + 23931 T^{4} + 2891765 T^{6} + 282813776 T^{8} + 2891765 p^{2} T^{10} + 23931 p^{4} T^{12} + 175 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.54757771259265046858850554958, −4.46559433853447565775391666186, −4.36832528999113139985474841193, −4.13210361310465520356830187030, −4.03189173079629639045283845111, −4.00569653509453866624447546655, −3.99729116852239846503369236000, −3.92272647336488407525183372518, −3.46688237155887287518498989637, −3.27347254163573227871448285235, −3.25911920481400876539747965211, −3.22688244565535515838487044744, −3.11282067300901489233530388102, −2.82986446107966829873626040067, −2.44433168153135731208698323175, −2.27542243445622023579124077154, −2.14967868734203532320845479041, −2.04818440716311450594993671294, −1.87902239337358694727753033058, −1.44348615768788869020512657310, −1.26343213274245579410755474701, −1.14873653393707835910480627989, −0.919674443844184187233151908836, −0.62549793148592428382273598844, −0.089680074265771067128043224633, 0.089680074265771067128043224633, 0.62549793148592428382273598844, 0.919674443844184187233151908836, 1.14873653393707835910480627989, 1.26343213274245579410755474701, 1.44348615768788869020512657310, 1.87902239337358694727753033058, 2.04818440716311450594993671294, 2.14967868734203532320845479041, 2.27542243445622023579124077154, 2.44433168153135731208698323175, 2.82986446107966829873626040067, 3.11282067300901489233530388102, 3.22688244565535515838487044744, 3.25911920481400876539747965211, 3.27347254163573227871448285235, 3.46688237155887287518498989637, 3.92272647336488407525183372518, 3.99729116852239846503369236000, 4.00569653509453866624447546655, 4.03189173079629639045283845111, 4.13210361310465520356830187030, 4.36832528999113139985474841193, 4.46559433853447565775391666186, 4.54757771259265046858850554958

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

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