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Properties

Label 16-510e8-1.1-c1e8-0-6
Degree $16$
Conductor $4.577\times 10^{21}$
Sign $1$
Analytic cond. $75644.6$
Root an. cond. $2.01801$
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  − 8·2-s + 36·4-s − 8·5-s − 4·7-s − 120·8-s + 64·10-s − 16·11-s + 32·14-s + 330·16-s − 4·17-s − 288·20-s + 128·22-s + 16·23-s + 24·25-s − 144·28-s + 8·29-s + 16·31-s − 792·32-s + 32·34-s + 32·35-s + 960·40-s − 4·41-s − 24·43-s − 576·44-s − 128·46-s + 8·49-s − 192·50-s + ⋯
L(s)  = 1  − 5.65·2-s + 18·4-s − 3.57·5-s − 1.51·7-s − 42.4·8-s + 20.2·10-s − 4.82·11-s + 8.55·14-s + 82.5·16-s − 0.970·17-s − 64.3·20-s + 27.2·22-s + 3.33·23-s + 24/5·25-s − 27.2·28-s + 1.48·29-s + 2.87·31-s − 140.·32-s + 5.48·34-s + 5.40·35-s + 151.·40-s − 0.624·41-s − 3.65·43-s − 86.8·44-s − 18.8·46-s + 8/7·49-s − 27.1·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(75644.6\)
Root analytic conductor: \(2.01801\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.02888730173\)
\(L(\frac12)\) \(\approx\) \(0.02888730173\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{8} \)
3 \( ( 1 + T^{4} )^{2} \)
5 \( ( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
17 \( 1 + 4 T - 24 T^{2} - 4 p T^{3} + 254 T^{4} - 4 p^{2} T^{5} - 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
good7 \( 1 + 4 T + 8 T^{2} + 60 T^{3} + 152 T^{4} + 44 T^{5} + 760 T^{6} + 1332 T^{7} - 4322 T^{8} + 1332 p T^{9} + 760 p^{2} T^{10} + 44 p^{3} T^{11} + 152 p^{4} T^{12} + 60 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
11 \( ( 1 + 8 T + 32 T^{2} + 120 T^{3} + 434 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 28 T^{2} + 512 T^{4} - 5060 T^{6} + 59758 T^{8} - 5060 p^{2} T^{10} + 512 p^{4} T^{12} - 28 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \)
23 \( 1 - 16 T + 128 T^{2} - 800 T^{3} + 4428 T^{4} - 20992 T^{5} + 89088 T^{6} - 332912 T^{7} + 1309094 T^{8} - 332912 p T^{9} + 89088 p^{2} T^{10} - 20992 p^{3} T^{11} + 4428 p^{4} T^{12} - 800 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 8 T + 32 T^{2} - 200 T^{3} + 1540 T^{4} - 8760 T^{5} + 40800 T^{6} - 320312 T^{7} + 2487462 T^{8} - 320312 p T^{9} + 40800 p^{2} T^{10} - 8760 p^{3} T^{11} + 1540 p^{4} T^{12} - 200 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 16 T + 128 T^{2} - 704 T^{3} + 2700 T^{4} - 7136 T^{5} + 16384 T^{6} + 22736 T^{7} - 482842 T^{8} + 22736 p T^{9} + 16384 p^{2} T^{10} - 7136 p^{3} T^{11} + 2700 p^{4} T^{12} - 704 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 512 T^{3} + 932 T^{4} - 512 T^{5} + 131072 T^{6} - 588800 T^{7} + 232998 T^{8} - 588800 p T^{9} + 131072 p^{2} T^{10} - 512 p^{3} T^{11} + 932 p^{4} T^{12} - 512 p^{5} T^{13} + p^{8} T^{16} \)
41 \( 1 + 4 T + 8 T^{2} + 100 T^{3} + 3544 T^{4} + 14028 T^{5} + 32760 T^{6} + 485260 T^{7} + 6916254 T^{8} + 485260 p T^{9} + 32760 p^{2} T^{10} + 14028 p^{3} T^{11} + 3544 p^{4} T^{12} + 100 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
43 \( ( 1 + 12 T + 170 T^{2} + 1380 T^{3} + 10990 T^{4} + 1380 p T^{5} + 170 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 112 T^{2} + 9852 T^{4} - 508304 T^{6} + 27287174 T^{8} - 508304 p^{2} T^{10} + 9852 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 8 T + 116 T^{2} - 504 T^{3} + 5510 T^{4} - 504 p T^{5} + 116 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 188 T^{2} + 20048 T^{4} - 1577284 T^{6} + 100869454 T^{8} - 1577284 p^{2} T^{10} + 20048 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 16 T + 128 T^{2} - 1456 T^{3} + 20708 T^{4} - 170000 T^{5} + 1129344 T^{6} - 11137712 T^{7} + 108970086 T^{8} - 11137712 p T^{9} + 1129344 p^{2} T^{10} - 170000 p^{3} T^{11} + 20708 p^{4} T^{12} - 1456 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 388 T^{2} + 71072 T^{4} - 8169116 T^{6} + 649676974 T^{8} - 8169116 p^{2} T^{10} + 71072 p^{4} T^{12} - 388 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 + 44 T + 968 T^{2} + 14932 T^{3} + 191816 T^{4} + 2223460 T^{5} + 23636664 T^{6} + 229699388 T^{7} + 2031298110 T^{8} + 229699388 p T^{9} + 23636664 p^{2} T^{10} + 2223460 p^{3} T^{11} + 191816 p^{4} T^{12} + 14932 p^{5} T^{13} + 968 p^{6} T^{14} + 44 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 20 T + 200 T^{2} - 2516 T^{3} + 17672 T^{4} + 34596 T^{5} - 1061192 T^{6} + 18597700 T^{7} - 244687682 T^{8} + 18597700 p T^{9} - 1061192 p^{2} T^{10} + 34596 p^{3} T^{11} + 17672 p^{4} T^{12} - 2516 p^{5} T^{13} + 200 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 16 T + 128 T^{2} - 1600 T^{3} + 30476 T^{4} - 301152 T^{5} + 2197504 T^{6} - 25390384 T^{7} + 292020070 T^{8} - 25390384 p T^{9} + 2197504 p^{2} T^{10} - 301152 p^{3} T^{11} + 30476 p^{4} T^{12} - 1600 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 + 8 T + 236 T^{2} + 1032 T^{3} + 23798 T^{4} + 1032 p T^{5} + 236 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 8 T + 260 T^{2} - 1368 T^{3} + 29270 T^{4} - 1368 p T^{5} + 260 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 20 T + 200 T^{2} - 596 T^{3} + 2888 T^{4} - 58620 T^{5} + 772408 T^{6} + 8765284 T^{7} - 150044546 T^{8} + 8765284 p T^{9} + 772408 p^{2} T^{10} - 58620 p^{3} T^{11} + 2888 p^{4} T^{12} - 596 p^{5} T^{13} + 200 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.86214529280837629646906969152, −4.85883307130217694555912807373, −4.50128617759122019698213157937, −4.30360643056310099908544442994, −4.17491458853913874231612560655, −3.95062536831331482804806350457, −3.79468011199524384998011514751, −3.43026090716625514138509292143, −3.41189226062541138122423565936, −3.10266398879273983654919096075, −3.09950656590568625502875301879, −3.08144967959927567952729118442, −2.91057569697486066306950587063, −2.84044962742645273074089584039, −2.55776252381125777888756595138, −2.33852728929492422224641559174, −2.24416272008334361264962180815, −2.03810961389930604907478385792, −1.88340419383704290654059014671, −1.44129580371357153121722111387, −1.15512846734810589397472321114, −0.65591623340699743035283799911, −0.56622709642715172326987810973, −0.42648286112857963254234252025, −0.38470285535381666299787094766, 0.38470285535381666299787094766, 0.42648286112857963254234252025, 0.56622709642715172326987810973, 0.65591623340699743035283799911, 1.15512846734810589397472321114, 1.44129580371357153121722111387, 1.88340419383704290654059014671, 2.03810961389930604907478385792, 2.24416272008334361264962180815, 2.33852728929492422224641559174, 2.55776252381125777888756595138, 2.84044962742645273074089584039, 2.91057569697486066306950587063, 3.08144967959927567952729118442, 3.09950656590568625502875301879, 3.10266398879273983654919096075, 3.41189226062541138122423565936, 3.43026090716625514138509292143, 3.79468011199524384998011514751, 3.95062536831331482804806350457, 4.17491458853913874231612560655, 4.30360643056310099908544442994, 4.50128617759122019698213157937, 4.85883307130217694555912807373, 4.86214529280837629646906969152

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

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