Properties

Label 16-3525e8-1.1-c1e8-0-1
Degree $16$
Conductor $2.384\times 10^{28}$
Sign $1$
Analytic cond. $3.93994\times 10^{11}$
Root an. cond. $5.30539$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 8·3-s − 24·6-s − 8·7-s + 9·8-s + 36·9-s − 8·11-s − 10·13-s + 24·14-s − 9·16-s − 6·17-s − 108·18-s − 2·19-s − 64·21-s + 24·22-s − 10·23-s + 72·24-s + 30·26-s + 120·27-s − 13·29-s − 5·32-s − 64·33-s + 18·34-s − 3·37-s + 6·38-s − 80·39-s − 16·41-s + ⋯
L(s)  = 1  − 2.12·2-s + 4.61·3-s − 9.79·6-s − 3.02·7-s + 3.18·8-s + 12·9-s − 2.41·11-s − 2.77·13-s + 6.41·14-s − 9/4·16-s − 1.45·17-s − 25.4·18-s − 0.458·19-s − 13.9·21-s + 5.11·22-s − 2.08·23-s + 14.6·24-s + 5.88·26-s + 23.0·27-s − 2.41·29-s − 0.883·32-s − 11.1·33-s + 3.08·34-s − 0.493·37-s + 0.973·38-s − 12.8·39-s − 2.49·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{16} \cdot 47^{8}\)
Sign: $1$
Analytic conductor: \(3.93994\times 10^{11}\)
Root analytic conductor: \(5.30539\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3525} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 3^{8} \cdot 5^{16} \cdot 47^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T )^{8} \)
5 \( 1 \)
47 \( ( 1 + T )^{8} \)
good2 \( 1 + 3 T + 9 T^{2} + 9 p T^{3} + 9 p^{2} T^{4} + 59 T^{5} + 25 p^{2} T^{6} + 149 T^{7} + 113 p T^{8} + 149 p T^{9} + 25 p^{4} T^{10} + 59 p^{3} T^{11} + 9 p^{6} T^{12} + 9 p^{6} T^{13} + 9 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 + 8 T + 52 T^{2} + 248 T^{3} + 1034 T^{4} + 3742 T^{5} + 12121 T^{6} + 36307 T^{7} + 99061 T^{8} + 36307 p T^{9} + 12121 p^{2} T^{10} + 3742 p^{3} T^{11} + 1034 p^{4} T^{12} + 248 p^{5} T^{13} + 52 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 + 8 T + 61 T^{2} + 282 T^{3} + 115 p T^{4} + 4098 T^{5} + 14032 T^{6} + 36737 T^{7} + 131254 T^{8} + 36737 p T^{9} + 14032 p^{2} T^{10} + 4098 p^{3} T^{11} + 115 p^{5} T^{12} + 282 p^{5} T^{13} + 61 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 10 T + 102 T^{2} + 674 T^{3} + 4409 T^{4} + 22292 T^{5} + 109708 T^{6} + 445761 T^{7} + 1759339 T^{8} + 445761 p T^{9} + 109708 p^{2} T^{10} + 22292 p^{3} T^{11} + 4409 p^{4} T^{12} + 674 p^{5} T^{13} + 102 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 6 T + 66 T^{2} + 337 T^{3} + 2348 T^{4} + 10419 T^{5} + 56740 T^{6} + 13391 p T^{7} + 1078798 T^{8} + 13391 p^{2} T^{9} + 56740 p^{2} T^{10} + 10419 p^{3} T^{11} + 2348 p^{4} T^{12} + 337 p^{5} T^{13} + 66 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 + 2 T + 59 T^{2} + 233 T^{3} + 2019 T^{4} + 9286 T^{5} + 57260 T^{6} + 223924 T^{7} + 1270509 T^{8} + 223924 p T^{9} + 57260 p^{2} T^{10} + 9286 p^{3} T^{11} + 2019 p^{4} T^{12} + 233 p^{5} T^{13} + 59 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 + 10 T + 91 T^{2} + 627 T^{3} + 4543 T^{4} + 28458 T^{5} + 170754 T^{6} + 869116 T^{7} + 4337188 T^{8} + 869116 p T^{9} + 170754 p^{2} T^{10} + 28458 p^{3} T^{11} + 4543 p^{4} T^{12} + 627 p^{5} T^{13} + 91 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 + 13 T + 124 T^{2} + 827 T^{3} + 5634 T^{4} + 34235 T^{5} + 218140 T^{6} + 1190230 T^{7} + 6671156 T^{8} + 1190230 p T^{9} + 218140 p^{2} T^{10} + 34235 p^{3} T^{11} + 5634 p^{4} T^{12} + 827 p^{5} T^{13} + 124 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 + 80 T^{2} - 287 T^{3} + 4321 T^{4} - 22689 T^{5} + 170100 T^{6} - 1143462 T^{7} + 5512343 T^{8} - 1143462 p T^{9} + 170100 p^{2} T^{10} - 22689 p^{3} T^{11} + 4321 p^{4} T^{12} - 287 p^{5} T^{13} + 80 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 + 3 T + 102 T^{2} + 649 T^{3} + 6473 T^{4} + 51143 T^{5} + 362537 T^{6} + 2393814 T^{7} + 15973392 T^{8} + 2393814 p T^{9} + 362537 p^{2} T^{10} + 51143 p^{3} T^{11} + 6473 p^{4} T^{12} + 649 p^{5} T^{13} + 102 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 16 T + 321 T^{2} + 3577 T^{3} + 43003 T^{4} + 370937 T^{5} + 3331107 T^{6} + 23230063 T^{7} + 167072010 T^{8} + 23230063 p T^{9} + 3331107 p^{2} T^{10} + 370937 p^{3} T^{11} + 43003 p^{4} T^{12} + 3577 p^{5} T^{13} + 321 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 + 25 T + 550 T^{2} + 7915 T^{3} + 101797 T^{4} + 1037754 T^{5} + 9606290 T^{6} + 1735355 p T^{7} + 530338331 T^{8} + 1735355 p^{2} T^{9} + 9606290 p^{2} T^{10} + 1037754 p^{3} T^{11} + 101797 p^{4} T^{12} + 7915 p^{5} T^{13} + 550 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 4 T + 232 T^{2} + 985 T^{3} + 28876 T^{4} + 117797 T^{5} + 2420494 T^{6} + 8998787 T^{7} + 148749654 T^{8} + 8998787 p T^{9} + 2420494 p^{2} T^{10} + 117797 p^{3} T^{11} + 28876 p^{4} T^{12} + 985 p^{5} T^{13} + 232 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 8 T + 169 T^{2} + 865 T^{3} + 18651 T^{4} + 90942 T^{5} + 1542562 T^{6} + 6397462 T^{7} + 101101912 T^{8} + 6397462 p T^{9} + 1542562 p^{2} T^{10} + 90942 p^{3} T^{11} + 18651 p^{4} T^{12} + 865 p^{5} T^{13} + 169 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 15 T + 254 T^{2} - 3105 T^{3} + 40623 T^{4} - 398700 T^{5} + 3930734 T^{6} - 33573867 T^{7} + 289024575 T^{8} - 33573867 p T^{9} + 3930734 p^{2} T^{10} - 398700 p^{3} T^{11} + 40623 p^{4} T^{12} - 3105 p^{5} T^{13} + 254 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 27 T + 564 T^{2} + 8914 T^{3} + 124510 T^{4} + 1474352 T^{5} + 15779767 T^{6} + 149485590 T^{7} + 1298028401 T^{8} + 149485590 p T^{9} + 15779767 p^{2} T^{10} + 1474352 p^{3} T^{11} + 124510 p^{4} T^{12} + 8914 p^{5} T^{13} + 564 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 14 T + 317 T^{2} - 2547 T^{3} + 34559 T^{4} - 128008 T^{5} + 22988 p T^{6} + 3352700 T^{7} + 56012760 T^{8} + 3352700 p T^{9} + 22988 p^{3} T^{10} - 128008 p^{3} T^{11} + 34559 p^{4} T^{12} - 2547 p^{5} T^{13} + 317 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 28 T + 684 T^{2} + 10734 T^{3} + 156267 T^{4} + 1788319 T^{5} + 19785479 T^{6} + 184380946 T^{7} + 1699586792 T^{8} + 184380946 p T^{9} + 19785479 p^{2} T^{10} + 1788319 p^{3} T^{11} + 156267 p^{4} T^{12} + 10734 p^{5} T^{13} + 684 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 7 T + 105 T^{2} + 234 T^{3} - 26 p T^{4} + 57397 T^{5} + 484944 T^{6} - 3225775 T^{7} + 108421600 T^{8} - 3225775 p T^{9} + 484944 p^{2} T^{10} + 57397 p^{3} T^{11} - 26 p^{5} T^{12} + 234 p^{5} T^{13} + 105 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 60 T + 1855 T^{2} + 38358 T^{3} + 595730 T^{4} + 7448435 T^{5} + 79541972 T^{6} + 768628920 T^{7} + 7088425040 T^{8} + 768628920 p T^{9} + 79541972 p^{2} T^{10} + 7448435 p^{3} T^{11} + 595730 p^{4} T^{12} + 38358 p^{5} T^{13} + 1855 p^{6} T^{14} + 60 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 + 34 T + 898 T^{2} + 15553 T^{3} + 233339 T^{4} + 2779838 T^{5} + 31019514 T^{6} + 302671774 T^{7} + 2981633366 T^{8} + 302671774 p T^{9} + 31019514 p^{2} T^{10} + 2779838 p^{3} T^{11} + 233339 p^{4} T^{12} + 15553 p^{5} T^{13} + 898 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 7 T + 516 T^{2} + 3468 T^{3} + 131014 T^{4} + 804980 T^{5} + 21467103 T^{6} + 115773622 T^{7} + 2464027597 T^{8} + 115773622 p T^{9} + 21467103 p^{2} T^{10} + 804980 p^{3} T^{11} + 131014 p^{4} T^{12} + 3468 p^{5} T^{13} + 516 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.81216757145875168860377187675, −3.80633321373275284692218020861, −3.59868009686670003615862631449, −3.56554348111583456114898137555, −3.44223257485394838157142732881, −3.31189474350332799348506403916, −3.19296087155105010772192928363, −3.12020347096287027280422928091, −3.07253277576104968253709304821, −2.88265102350872203576384543294, −2.86021293419878178897549180192, −2.81798701677671439179171381901, −2.55138374380944435493321749900, −2.45409091950486997231737017383, −2.33393397010475608769293471420, −2.22911314832556469154857253700, −2.20896956125419026396511463892, −2.18580449455154231732463787712, −1.84534808731926428958766115825, −1.59687128992922751286071825607, −1.53095065260380383555879054465, −1.44590110017290191107953706622, −1.37048080437691646416580379375, −1.20220945587164917344787781257, −1.18794120054339960493656086797, 0, 0, 0, 0, 0, 0, 0, 0, 1.18794120054339960493656086797, 1.20220945587164917344787781257, 1.37048080437691646416580379375, 1.44590110017290191107953706622, 1.53095065260380383555879054465, 1.59687128992922751286071825607, 1.84534808731926428958766115825, 2.18580449455154231732463787712, 2.20896956125419026396511463892, 2.22911314832556469154857253700, 2.33393397010475608769293471420, 2.45409091950486997231737017383, 2.55138374380944435493321749900, 2.81798701677671439179171381901, 2.86021293419878178897549180192, 2.88265102350872203576384543294, 3.07253277576104968253709304821, 3.12020347096287027280422928091, 3.19296087155105010772192928363, 3.31189474350332799348506403916, 3.44223257485394838157142732881, 3.56554348111583456114898137555, 3.59868009686670003615862631449, 3.80633321373275284692218020861, 3.81216757145875168860377187675

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

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