Properties

Label 16-684e8-1.1-c2e8-0-1
Degree $16$
Conductor $4.791\times 10^{22}$
Sign $1$
Analytic cond. $1.45589\times 10^{10}$
Root an. cond. $4.31713$
Motivic weight $2$
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  − 8·7-s − 60·13-s + 32·19-s + 32·25-s − 20·43-s − 332·49-s + 124·61-s − 228·67-s + 100·73-s − 396·79-s + 480·91-s + 216·97-s − 360·109-s − 304·121-s + 127-s + 131-s − 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.41e3·169-s + 173-s − 256·175-s + ⋯
L(s)  = 1  − 8/7·7-s − 4.61·13-s + 1.68·19-s + 1.27·25-s − 0.465·43-s − 6.77·49-s + 2.03·61-s − 3.40·67-s + 1.36·73-s − 5.01·79-s + 5.27·91-s + 2.22·97-s − 3.30·109-s − 2.51·121-s + 0.00787·127-s + 0.00763·131-s − 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 8.39·169-s + 0.00578·173-s − 1.46·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(1.45589\times 10^{10}\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{684} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 19^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.077289657\)
\(L(\frac12)\) \(\approx\) \(1.077289657\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( ( 1 - 16 T + 30 p T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
good5 \( 1 - 32 T^{2} - 266 T^{4} - 256 p T^{6} + 25339 p^{2} T^{8} - 256 p^{5} T^{10} - 266 p^{8} T^{12} - 32 p^{12} T^{14} + p^{16} T^{16} \)
7 \( ( 1 + 2 T + 93 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
11 \( ( 1 + 152 T^{2} + 31002 T^{4} + 152 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 30 T + 641 T^{2} + 10230 T^{3} + 138420 T^{4} + 10230 p^{2} T^{5} + 641 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( 1 - 764 T^{2} + 305386 T^{4} - 85008752 T^{6} + 21729825043 T^{8} - 85008752 p^{4} T^{10} + 305386 p^{8} T^{12} - 764 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 2048 T^{2} + 2586262 T^{4} - 2147041280 T^{6} + 1333749862819 T^{8} - 2147041280 p^{4} T^{10} + 2586262 p^{8} T^{12} - 2048 p^{12} T^{14} + p^{16} T^{16} \)
29 \( 1 + 2188 T^{2} + 2487850 T^{4} + 1936231216 T^{6} + 1401428791411 T^{8} + 1936231216 p^{4} T^{10} + 2487850 p^{8} T^{12} + 2188 p^{12} T^{14} + p^{16} T^{16} \)
31 \( ( 1 - 2458 T^{2} + 2920083 T^{4} - 2458 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
37 \( ( 1 - 4174 T^{2} + 8096115 T^{4} - 4174 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
41 \( 1 + 1828 T^{2} - 2025590 T^{4} - 519788144 T^{6} + 11379113771731 T^{8} - 519788144 p^{4} T^{10} - 2025590 p^{8} T^{12} + 1828 p^{12} T^{14} + p^{16} T^{16} \)
43 \( ( 1 + 10 T - 2897 T^{2} - 7010 T^{3} + 5378308 T^{4} - 7010 p^{2} T^{5} - 2897 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 - 2564 T^{2} + 4043146 T^{4} + 394332944 p T^{6} - 22401570893 p^{2} T^{8} + 394332944 p^{5} T^{10} + 4043146 p^{8} T^{12} - 2564 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 + 9376 T^{2} + 50804470 T^{4} + 199933298944 T^{6} + 620867097807619 T^{8} + 199933298944 p^{4} T^{10} + 50804470 p^{8} T^{12} + 9376 p^{12} T^{14} + p^{16} T^{16} \)
59 \( 1 + 5080 T^{2} + 9386662 T^{4} - 39700118720 T^{6} - 189928106526077 T^{8} - 39700118720 p^{4} T^{10} + 9386662 p^{8} T^{12} + 5080 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 - 62 T + 2377 T^{2} + 370450 T^{3} - 25997276 T^{4} + 370450 p^{2} T^{5} + 2377 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( ( 1 + 114 T + 7895 T^{2} + 406182 T^{3} + 11990196 T^{4} + 406182 p^{2} T^{5} + 7895 p^{4} T^{6} + 114 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( 1 + 9580 T^{2} + 43273162 T^{4} - 22226787920 T^{6} - 625382034889517 T^{8} - 22226787920 p^{4} T^{10} + 43273162 p^{8} T^{12} + 9580 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 - 50 T - 8759 T^{2} - 30050 T^{3} + 85044340 T^{4} - 30050 p^{2} T^{5} - 8759 p^{4} T^{6} - 50 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( ( 1 + 198 T + 28799 T^{2} + 3114738 T^{3} + 290071668 T^{4} + 3114738 p^{2} T^{5} + 28799 p^{4} T^{6} + 198 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( ( 1 + 4508 T^{2} - 13851258 T^{4} + 4508 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( 1 + 6208 T^{2} - 20623850 T^{4} - 411723052544 T^{6} - 2854354537375709 T^{8} - 411723052544 p^{4} T^{10} - 20623850 p^{8} T^{12} + 6208 p^{12} T^{14} + p^{16} T^{16} \)
97 \( ( 1 - 54 T + 10381 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−4.35846468468732824129422781917, −4.29911801644164864028920385924, −4.06829966728666885950444712568, −3.97418350951627161616868894611, −3.69620568110846372210615494531, −3.49500197853600739938361653630, −3.46190443158437838916187159338, −3.25864949806396695890905100104, −2.98445756952354130170692505616, −2.90266625916954492714415457786, −2.89741487280811206721813905389, −2.88648179682267841225486944030, −2.69155936118757774017153476117, −2.46806691719167902033152938229, −2.43164709314542113442842171321, −1.94074133881091418927663379065, −1.88711066358892767490727634836, −1.58042429262620762102350260251, −1.53717491798908231660460952428, −1.53707831408483976973715107165, −1.18044273800897214490308390394, −0.69847906787698184686371475923, −0.46004419599633723835319142159, −0.26182243302771700753875927416, −0.22086700658372112064774587270, 0.22086700658372112064774587270, 0.26182243302771700753875927416, 0.46004419599633723835319142159, 0.69847906787698184686371475923, 1.18044273800897214490308390394, 1.53707831408483976973715107165, 1.53717491798908231660460952428, 1.58042429262620762102350260251, 1.88711066358892767490727634836, 1.94074133881091418927663379065, 2.43164709314542113442842171321, 2.46806691719167902033152938229, 2.69155936118757774017153476117, 2.88648179682267841225486944030, 2.89741487280811206721813905389, 2.90266625916954492714415457786, 2.98445756952354130170692505616, 3.25864949806396695890905100104, 3.46190443158437838916187159338, 3.49500197853600739938361653630, 3.69620568110846372210615494531, 3.97418350951627161616868894611, 4.06829966728666885950444712568, 4.29911801644164864028920385924, 4.35846468468732824129422781917

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

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