Properties

Label 16-7448e8-1.1-c1e8-0-1
Degree $16$
Conductor $9.469\times 10^{30}$
Sign $1$
Analytic cond. $1.56507\times 10^{14}$
Root an. cond. $7.71184$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 9·9-s + 9·11-s + 4·17-s + 8·19-s + 25·23-s − 12·25-s + 27-s + 6·29-s + 13·37-s − 16·41-s + 17·43-s + 9·45-s − 24·47-s + 2·53-s − 9·55-s + 2·59-s − 13·61-s + 2·67-s + 10·71-s + 5·73-s + 16·79-s + 33·81-s − 43·83-s − 4·85-s + 8·89-s − 8·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 3·9-s + 2.71·11-s + 0.970·17-s + 1.83·19-s + 5.21·23-s − 2.39·25-s + 0.192·27-s + 1.11·29-s + 2.13·37-s − 2.49·41-s + 2.59·43-s + 1.34·45-s − 3.50·47-s + 0.274·53-s − 1.21·55-s + 0.260·59-s − 1.66·61-s + 0.244·67-s + 1.18·71-s + 0.585·73-s + 1.80·79-s + 11/3·81-s − 4.71·83-s − 0.433·85-s + 0.847·89-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 7^{16} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(1.56507\times 10^{14}\)
Root analytic conductor: \(7.71184\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7448} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 7^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(54.76683685\)
\(L(\frac12)\) \(\approx\) \(54.76683685\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( ( 1 - T )^{8} \)
good3 \( 1 + p^{2} T^{2} - T^{3} + 16 p T^{4} - 11 T^{5} + 203 T^{6} - 46 T^{7} + 694 T^{8} - 46 p T^{9} + 203 p^{2} T^{10} - 11 p^{3} T^{11} + 16 p^{5} T^{12} - p^{5} T^{13} + p^{8} T^{14} + p^{8} T^{16} \)
5 \( 1 + T + 13 T^{2} + 26 T^{3} + 112 T^{4} + 248 T^{5} + 851 T^{6} + 1537 T^{7} + 5006 T^{8} + 1537 p T^{9} + 851 p^{2} T^{10} + 248 p^{3} T^{11} + 112 p^{4} T^{12} + 26 p^{5} T^{13} + 13 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 - 9 T + 82 T^{2} - 459 T^{3} + 2417 T^{4} - 9780 T^{5} + 38622 T^{6} - 128886 T^{7} + 453764 T^{8} - 128886 p T^{9} + 38622 p^{2} T^{10} - 9780 p^{3} T^{11} + 2417 p^{4} T^{12} - 459 p^{5} T^{13} + 82 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 63 T^{2} + T^{3} + 1844 T^{4} + 265 T^{5} + 34701 T^{6} + 9002 T^{7} + 498638 T^{8} + 9002 p T^{9} + 34701 p^{2} T^{10} + 265 p^{3} T^{11} + 1844 p^{4} T^{12} + p^{5} T^{13} + 63 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 4 T + 58 T^{2} - 185 T^{3} + 121 p T^{4} - 6234 T^{5} + 53760 T^{6} - 139113 T^{7} + 1017128 T^{8} - 139113 p T^{9} + 53760 p^{2} T^{10} - 6234 p^{3} T^{11} + 121 p^{5} T^{12} - 185 p^{5} T^{13} + 58 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 25 T + 413 T^{2} - 4867 T^{3} + 46668 T^{4} - 369316 T^{5} + 2515264 T^{6} - 14747354 T^{7} + 75826491 T^{8} - 14747354 p T^{9} + 2515264 p^{2} T^{10} - 369316 p^{3} T^{11} + 46668 p^{4} T^{12} - 4867 p^{5} T^{13} + 413 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 6 T + 127 T^{2} - 889 T^{3} + 8908 T^{4} - 58681 T^{5} + 443085 T^{6} - 2388316 T^{7} + 15502590 T^{8} - 2388316 p T^{9} + 443085 p^{2} T^{10} - 58681 p^{3} T^{11} + 8908 p^{4} T^{12} - 889 p^{5} T^{13} + 127 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 + 148 T^{2} - 116 T^{3} + 10252 T^{4} - 17596 T^{5} + 457996 T^{6} - 1076104 T^{7} + 15671846 T^{8} - 1076104 p T^{9} + 457996 p^{2} T^{10} - 17596 p^{3} T^{11} + 10252 p^{4} T^{12} - 116 p^{5} T^{13} + 148 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 13 T + 207 T^{2} - 1750 T^{3} + 16402 T^{4} - 112836 T^{5} + 816117 T^{6} - 5071053 T^{7} + 32170002 T^{8} - 5071053 p T^{9} + 816117 p^{2} T^{10} - 112836 p^{3} T^{11} + 16402 p^{4} T^{12} - 1750 p^{5} T^{13} + 207 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \)
41 \( ( 1 + 2 T + p T^{2} )^{8} \)
43 \( 1 - 17 T + 355 T^{2} - 4036 T^{3} + 49646 T^{4} - 429858 T^{5} + 3936957 T^{6} - 642057 p T^{7} + 204548050 T^{8} - 642057 p^{2} T^{9} + 3936957 p^{2} T^{10} - 429858 p^{3} T^{11} + 49646 p^{4} T^{12} - 4036 p^{5} T^{13} + 355 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 24 T + 392 T^{2} + 4364 T^{3} + 37602 T^{4} + 233364 T^{5} + 1036208 T^{6} + 2240436 T^{7} + 2928963 T^{8} + 2240436 p T^{9} + 1036208 p^{2} T^{10} + 233364 p^{3} T^{11} + 37602 p^{4} T^{12} + 4364 p^{5} T^{13} + 392 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 2 T + 129 T^{2} + 37 T^{3} + 10574 T^{4} - 3559 T^{5} + 772851 T^{6} - 539736 T^{7} + 43318986 T^{8} - 539736 p T^{9} + 772851 p^{2} T^{10} - 3559 p^{3} T^{11} + 10574 p^{4} T^{12} + 37 p^{5} T^{13} + 129 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 2 T + 161 T^{2} - 671 T^{3} + 14184 T^{4} - 82025 T^{5} + 987807 T^{6} - 6008672 T^{7} + 60662110 T^{8} - 6008672 p T^{9} + 987807 p^{2} T^{10} - 82025 p^{3} T^{11} + 14184 p^{4} T^{12} - 671 p^{5} T^{13} + 161 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 13 T + 330 T^{2} + 3221 T^{3} + 51309 T^{4} + 421256 T^{5} + 5202698 T^{6} + 36815298 T^{7} + 375128276 T^{8} + 36815298 p T^{9} + 5202698 p^{2} T^{10} + 421256 p^{3} T^{11} + 51309 p^{4} T^{12} + 3221 p^{5} T^{13} + 330 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 2 T + 453 T^{2} - 869 T^{3} + 94066 T^{4} - 164739 T^{5} + 11710995 T^{6} - 17822484 T^{7} + 956952666 T^{8} - 17822484 p T^{9} + 11710995 p^{2} T^{10} - 164739 p^{3} T^{11} + 94066 p^{4} T^{12} - 869 p^{5} T^{13} + 453 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 10 T + 328 T^{2} - 2494 T^{3} + 51564 T^{4} - 327342 T^{5} + 78728 p T^{6} - 31264890 T^{7} + 457121126 T^{8} - 31264890 p T^{9} + 78728 p^{3} T^{10} - 327342 p^{3} T^{11} + 51564 p^{4} T^{12} - 2494 p^{5} T^{13} + 328 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 5 T + 227 T^{2} - 759 T^{3} + 19426 T^{4} + 26030 T^{5} + 597516 T^{6} + 13022320 T^{7} - 270615 T^{8} + 13022320 p T^{9} + 597516 p^{2} T^{10} + 26030 p^{3} T^{11} + 19426 p^{4} T^{12} - 759 p^{5} T^{13} + 227 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 16 T + 412 T^{2} - 5444 T^{3} + 81524 T^{4} - 930572 T^{5} + 10484580 T^{6} - 103560600 T^{7} + 965277206 T^{8} - 103560600 p T^{9} + 10484580 p^{2} T^{10} - 930572 p^{3} T^{11} + 81524 p^{4} T^{12} - 5444 p^{5} T^{13} + 412 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 43 T + 1290 T^{2} + 27837 T^{3} + 495581 T^{4} + 7318764 T^{5} + 93657586 T^{6} + 1035827238 T^{7} + 10099433844 T^{8} + 1035827238 p T^{9} + 93657586 p^{2} T^{10} + 7318764 p^{3} T^{11} + 495581 p^{4} T^{12} + 27837 p^{5} T^{13} + 1290 p^{6} T^{14} + 43 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 8 T + 448 T^{2} - 3580 T^{3} + 98092 T^{4} - 710908 T^{5} + 13956896 T^{6} - 88249184 T^{7} + 16134934 p T^{8} - 88249184 p T^{9} + 13956896 p^{2} T^{10} - 710908 p^{3} T^{11} + 98092 p^{4} T^{12} - 3580 p^{5} T^{13} + 448 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 12 T + 608 T^{2} + 5812 T^{3} + 170620 T^{4} + 1355452 T^{5} + 29452384 T^{6} + 196855172 T^{7} + 3436368006 T^{8} + 196855172 p T^{9} + 29452384 p^{2} T^{10} + 1355452 p^{3} T^{11} + 170620 p^{4} T^{12} + 5812 p^{5} T^{13} + 608 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.29207584995394540115068442489, −2.97025566599721680851085903937, −2.95743450417319163042219427380, −2.93966823292409946157197193714, −2.80398561467016136394838585001, −2.75882413259309349605950079167, −2.72069388028561751606906667875, −2.60359025475657129036670078771, −2.50910114169373286090819234576, −2.05781482670116854294927745620, −1.92100403643489553231850075471, −1.91874588274641452236736494683, −1.88039250670180818372252089599, −1.77318043839249126537234618105, −1.65922171927053373976948418559, −1.45500311099123761259675854012, −1.29095645664281311727707889651, −1.12376992463193954143160693599, −1.09802029650983048767998412188, −0.979391882788534858588754955441, −0.56757633580984542772623305889, −0.53934515358848748387159892336, −0.53325613418273381647312438425, −0.50755109545045332146904247087, −0.42456785024167406924971147747, 0.42456785024167406924971147747, 0.50755109545045332146904247087, 0.53325613418273381647312438425, 0.53934515358848748387159892336, 0.56757633580984542772623305889, 0.979391882788534858588754955441, 1.09802029650983048767998412188, 1.12376992463193954143160693599, 1.29095645664281311727707889651, 1.45500311099123761259675854012, 1.65922171927053373976948418559, 1.77318043839249126537234618105, 1.88039250670180818372252089599, 1.91874588274641452236736494683, 1.92100403643489553231850075471, 2.05781482670116854294927745620, 2.50910114169373286090819234576, 2.60359025475657129036670078771, 2.72069388028561751606906667875, 2.75882413259309349605950079167, 2.80398561467016136394838585001, 2.93966823292409946157197193714, 2.95743450417319163042219427380, 2.97025566599721680851085903937, 3.29207584995394540115068442489

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

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