Properties

Label 16-325e8-1.1-c1e8-0-1
Degree $16$
Conductor $1.245\times 10^{20}$
Sign $1$
Analytic cond. $2057.23$
Root an. cond. $1.61094$
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  + 2·2-s − 6·3-s + 5·4-s − 12·6-s − 10·7-s + 6·8-s + 14·9-s − 30·12-s − 8·13-s − 20·14-s + 9·16-s − 18·17-s + 28·18-s − 12·19-s + 60·21-s − 6·23-s − 36·24-s − 16·26-s − 12·27-s − 50·28-s + 8·29-s + 6·32-s − 36·34-s + 70·36-s + 2·37-s − 24·38-s + 48·39-s + ⋯
L(s)  = 1  + 1.41·2-s − 3.46·3-s + 5/2·4-s − 4.89·6-s − 3.77·7-s + 2.12·8-s + 14/3·9-s − 8.66·12-s − 2.21·13-s − 5.34·14-s + 9/4·16-s − 4.36·17-s + 6.59·18-s − 2.75·19-s + 13.0·21-s − 1.25·23-s − 7.34·24-s − 3.13·26-s − 2.30·27-s − 9.44·28-s + 1.48·29-s + 1.06·32-s − 6.17·34-s + 35/3·36-s + 0.328·37-s − 3.89·38-s + 7.68·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{16} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(2057.23\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{325} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{16} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1355187643\)
\(L(\frac12)\) \(\approx\) \(0.1355187643\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + 8 T + 16 T^{2} - 40 T^{3} - 290 T^{4} - 40 p T^{5} + 16 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
good2 \( ( 1 - p^{2} T + 5 T^{2} + p T^{3} - 11 T^{4} + p^{2} T^{5} + 5 p^{2} T^{6} - p^{5} T^{7} + p^{4} T^{8} )( 1 + p T + p T^{2} + p T^{3} + T^{4} + p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} ) \)
3 \( 1 + 2 p T + 22 T^{2} + 20 p T^{3} + 43 p T^{4} + 28 p^{2} T^{5} + 466 T^{6} + 94 p^{2} T^{7} + 1516 T^{8} + 94 p^{3} T^{9} + 466 p^{2} T^{10} + 28 p^{5} T^{11} + 43 p^{5} T^{12} + 20 p^{6} T^{13} + 22 p^{6} T^{14} + 2 p^{8} T^{15} + p^{8} T^{16} \)
7 \( 1 + 10 T + 6 p T^{2} + 116 T^{3} + 341 T^{4} + 1020 T^{5} + 2494 T^{6} + 6562 T^{7} + 18804 T^{8} + 6562 p T^{9} + 2494 p^{2} T^{10} + 1020 p^{3} T^{11} + 341 p^{4} T^{12} + 116 p^{5} T^{13} + 6 p^{7} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 + 14 T^{2} + 97 T^{4} - 182 p T^{6} - 236 p^{2} T^{8} - 182 p^{3} T^{10} + 97 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 2 T + 50 T^{2} + 92 T^{3} + 1135 T^{4} + 92 p T^{5} + 50 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )( 1 + 16 T + 128 T^{2} + 688 T^{3} + 3022 T^{4} + 688 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} ) \)
19 \( ( 1 + 6 T + 37 T^{2} + 150 T^{3} + 492 T^{4} + 150 p T^{5} + 37 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 6 T + 66 T^{2} + 324 T^{3} + 1861 T^{4} + 1116 T^{5} + 2502 T^{6} - 8862 p T^{7} - 36276 p T^{8} - 8862 p^{2} T^{9} + 2502 p^{2} T^{10} + 1116 p^{3} T^{11} + 1861 p^{4} T^{12} + 324 p^{5} T^{13} + 66 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 8 T - 34 T^{2} + 528 T^{3} + 353 T^{4} - 19264 T^{5} + 50686 T^{6} + 248280 T^{7} - 2118188 T^{8} + 248280 p T^{9} + 50686 p^{2} T^{10} - 19264 p^{3} T^{11} + 353 p^{4} T^{12} + 528 p^{5} T^{13} - 34 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
31 \( ( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( 1 - 2 T - 90 T^{2} + 332 T^{3} + 4097 T^{4} - 17184 T^{5} - 90854 T^{6} + 352942 T^{7} + 1733340 T^{8} + 352942 p T^{9} - 90854 p^{2} T^{10} - 17184 p^{3} T^{11} + 4097 p^{4} T^{12} + 332 p^{5} T^{13} - 90 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
41 \( ( 1 - 6 T + 93 T^{2} - 486 T^{3} + 5372 T^{4} - 486 p T^{5} + 93 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 18 T + 314 T^{2} - 3708 T^{3} + 42733 T^{4} - 389244 T^{5} + 3413342 T^{6} - 25218810 T^{7} + 179548948 T^{8} - 25218810 p T^{9} + 3413342 p^{2} T^{10} - 389244 p^{3} T^{11} + 42733 p^{4} T^{12} - 3708 p^{5} T^{13} + 314 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \)
47 \( ( 1 - 8 T + 116 T^{2} - 392 T^{3} + 5158 T^{4} - 392 p T^{5} + 116 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( 1 - 352 T^{2} + 56956 T^{4} - 5554336 T^{6} + 357811558 T^{8} - 5554336 p^{2} T^{10} + 56956 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 12 T + 266 T^{2} - 2616 T^{3} + 37117 T^{4} - 331992 T^{5} + 3532226 T^{6} - 27597948 T^{7} + 240376924 T^{8} - 27597948 p T^{9} + 3532226 p^{2} T^{10} - 331992 p^{3} T^{11} + 37117 p^{4} T^{12} - 2616 p^{5} T^{13} + 266 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 28 T + 282 T^{2} + 1880 T^{3} + 26477 T^{4} + 344016 T^{5} + 2759698 T^{6} + 21027916 T^{7} + 175399068 T^{8} + 21027916 p T^{9} + 2759698 p^{2} T^{10} + 344016 p^{3} T^{11} + 26477 p^{4} T^{12} + 1880 p^{5} T^{13} + 282 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 30 T + 302 T^{2} - 2724 T^{3} + 55441 T^{4} - 623940 T^{5} + 3752474 T^{6} - 42317478 T^{7} + 502476364 T^{8} - 42317478 p T^{9} + 3752474 p^{2} T^{10} - 623940 p^{3} T^{11} + 55441 p^{4} T^{12} - 2724 p^{5} T^{13} + 302 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 + 66 T^{2} - 5411 T^{4} - 1080 T^{5} - 5238 T^{6} + 2109672 T^{7} + 49128204 T^{8} + 2109672 p T^{9} - 5238 p^{2} T^{10} - 1080 p^{3} T^{11} - 5411 p^{4} T^{12} + 66 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 - 8 T + 208 T^{2} - 920 T^{3} + 17998 T^{4} - 920 p T^{5} + 208 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 8 T + 184 T^{2} - 1256 T^{3} + 21022 T^{4} - 1256 p T^{5} + 184 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 12 T + 308 T^{2} + 2700 T^{3} + 37158 T^{4} + 2700 p T^{5} + 308 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 + 24 T + 338 T^{2} + 3504 T^{3} + 22477 T^{4} + 143640 T^{5} + 783482 T^{6} + 3203040 T^{7} + 47062828 T^{8} + 3203040 p T^{9} + 783482 p^{2} T^{10} + 143640 p^{3} T^{11} + 22477 p^{4} T^{12} + 3504 p^{5} T^{13} + 338 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 2 T - 294 T^{2} + 1316 T^{3} + 48341 T^{4} - 223416 T^{5} - 5151962 T^{6} + 10980406 T^{7} + 480743076 T^{8} + 10980406 p T^{9} - 5151962 p^{2} T^{10} - 223416 p^{3} T^{11} + 48341 p^{4} T^{12} + 1316 p^{5} T^{13} - 294 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
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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

−5.38340220083211607517104601049, −5.03503245878402742915104211276, −4.88764732345470730006271471301, −4.70235439632025685941638708124, −4.62703860441439314705135119461, −4.60247366845034528954822292871, −4.33745019915534141670267124622, −4.14248105441800728234912013051, −4.13600969841340959498588193098, −3.83039884949639367484794306516, −3.74224231548805103257573395802, −3.63810305490161919611887494604, −3.49297335756398902839334460372, −3.03314746885629165461558401851, −2.64380311575067098626182506105, −2.62271395846107507011782811266, −2.56332928278159905080505475020, −2.48587165078301771644247764813, −2.44067947875183660281909108774, −2.26464725742882068080148381166, −1.97526660976212837833695099119, −1.45662431761944924064373889095, −0.64366625012979612894550011896, −0.48926694816021345980827405187, −0.21855279440746015151208014677, 0.21855279440746015151208014677, 0.48926694816021345980827405187, 0.64366625012979612894550011896, 1.45662431761944924064373889095, 1.97526660976212837833695099119, 2.26464725742882068080148381166, 2.44067947875183660281909108774, 2.48587165078301771644247764813, 2.56332928278159905080505475020, 2.62271395846107507011782811266, 2.64380311575067098626182506105, 3.03314746885629165461558401851, 3.49297335756398902839334460372, 3.63810305490161919611887494604, 3.74224231548805103257573395802, 3.83039884949639367484794306516, 4.13600969841340959498588193098, 4.14248105441800728234912013051, 4.33745019915534141670267124622, 4.60247366845034528954822292871, 4.62703860441439314705135119461, 4.70235439632025685941638708124, 4.88764732345470730006271471301, 5.03503245878402742915104211276, 5.38340220083211607517104601049

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

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