Properties

Label 16-2352e8-1.1-c1e8-0-13
Degree $16$
Conductor $9.365\times 10^{26}$
Sign $1$
Analytic cond. $1.54780\times 10^{10}$
Root an. cond. $4.33368$
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  + 4·3-s + 6·9-s − 24·23-s − 4·25-s + 16·29-s − 16·31-s − 8·47-s − 8·53-s + 24·59-s + 48·61-s + 48·67-s − 96·69-s + 48·73-s − 16·75-s + 24·79-s − 15·81-s + 64·87-s + 48·89-s − 64·93-s + 24·101-s + 24·107-s − 8·109-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s − 5.00·23-s − 4/5·25-s + 2.97·29-s − 2.87·31-s − 1.16·47-s − 1.09·53-s + 3.12·59-s + 6.14·61-s + 5.86·67-s − 11.5·69-s + 5.61·73-s − 1.84·75-s + 2.70·79-s − 5/3·81-s + 6.86·87-s + 5.08·89-s − 6.63·93-s + 2.38·101-s + 2.32·107-s − 0.766·109-s − 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{8} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.54780\times 10^{10}\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{8} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(37.44144708\)
\(L(\frac12)\) \(\approx\) \(37.44144708\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - T + T^{2} )^{4} \)
7 \( 1 \)
good5 \( 1 + 4 T^{2} + 12 T^{4} - 24 T^{5} + 8 T^{6} + 96 T^{7} - 49 T^{8} + 96 p T^{9} + 8 p^{2} T^{10} - 24 p^{3} T^{11} + 12 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 + 24 T^{2} + 294 T^{4} + 624 T^{5} + 1728 T^{6} + 13536 T^{7} + 3635 T^{8} + 13536 p T^{9} + 1728 p^{2} T^{10} + 624 p^{3} T^{11} + 294 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - 32 T^{2} + 592 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( 1 + 36 T^{2} + 732 T^{4} + 3192 T^{5} + 8904 T^{6} + 94752 T^{7} + 19967 T^{8} + 94752 p T^{9} + 8904 p^{2} T^{10} + 3192 p^{3} T^{11} + 732 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \)
19 \( 1 - 36 T^{2} - 192 T^{3} + 706 T^{4} + 5280 T^{5} + 13968 T^{6} - 74688 T^{7} - 425181 T^{8} - 74688 p T^{9} + 13968 p^{2} T^{10} + 5280 p^{3} T^{11} + 706 p^{4} T^{12} - 192 p^{5} T^{13} - 36 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 + 24 T + 336 T^{2} + 3456 T^{3} + 28614 T^{4} + 8712 p T^{5} + 1230336 T^{6} + 294456 p T^{7} + 33940163 T^{8} + 294456 p^{2} T^{9} + 1230336 p^{2} T^{10} + 8712 p^{4} T^{11} + 28614 p^{4} T^{12} + 3456 p^{5} T^{13} + 336 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 - 8 T + 112 T^{2} - 664 T^{3} + 4842 T^{4} - 664 p T^{5} + 112 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 16 T + 76 T^{2} + 224 T^{3} + 2738 T^{4} + 10960 T^{5} - 63664 T^{6} - 402352 T^{7} - 682637 T^{8} - 402352 p T^{9} - 63664 p^{2} T^{10} + 10960 p^{3} T^{11} + 2738 p^{4} T^{12} + 224 p^{5} T^{13} + 76 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 48 T^{2} + 384 T^{3} + 526 T^{4} - 16320 T^{5} + 82944 T^{6} + 340608 T^{7} - 3231021 T^{8} + 340608 p T^{9} + 82944 p^{2} T^{10} - 16320 p^{3} T^{11} + 526 p^{4} T^{12} + 384 p^{5} T^{13} - 48 p^{6} T^{14} + p^{8} T^{16} \)
41 \( 1 - 248 T^{2} + 29208 T^{4} - 2129944 T^{6} + 2560930 p T^{8} - 2129944 p^{2} T^{10} + 29208 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 + 24 T^{2} - 324 T^{4} + 63336 T^{6} + 5311334 T^{8} + 63336 p^{2} T^{10} - 324 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 + 8 T - 108 T^{2} - 432 T^{3} + 10994 T^{4} + 16248 T^{5} - 716368 T^{6} - 478072 T^{7} + 33653619 T^{8} - 478072 p T^{9} - 716368 p^{2} T^{10} + 16248 p^{3} T^{11} + 10994 p^{4} T^{12} - 432 p^{5} T^{13} - 108 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 8 T - 60 T^{2} - 1488 T^{3} - 2854 T^{4} + 86520 T^{5} + 678032 T^{6} - 2016616 T^{7} - 44462109 T^{8} - 2016616 p T^{9} + 678032 p^{2} T^{10} + 86520 p^{3} T^{11} - 2854 p^{4} T^{12} - 1488 p^{5} T^{13} - 60 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 24 T + 260 T^{2} - 1968 T^{3} + 10594 T^{4} - 3240 T^{5} - 802192 T^{6} + 11964744 T^{7} - 110029565 T^{8} + 11964744 p T^{9} - 802192 p^{2} T^{10} - 3240 p^{3} T^{11} + 10594 p^{4} T^{12} - 1968 p^{5} T^{13} + 260 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 48 T + 1248 T^{2} - 23040 T^{3} + 335856 T^{4} - 4102848 T^{5} + 43415616 T^{6} - 404880144 T^{7} + 3352897967 T^{8} - 404880144 p T^{9} + 43415616 p^{2} T^{10} - 4102848 p^{3} T^{11} + 335856 p^{4} T^{12} - 23040 p^{5} T^{13} + 1248 p^{6} T^{14} - 48 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 48 T + 1292 T^{2} - 25152 T^{3} + 391386 T^{4} - 5111088 T^{5} + 57558448 T^{6} - 567562704 T^{7} + 4939433987 T^{8} - 567562704 p T^{9} + 57558448 p^{2} T^{10} - 5111088 p^{3} T^{11} + 391386 p^{4} T^{12} - 25152 p^{5} T^{13} + 1292 p^{6} T^{14} - 48 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 208 T^{2} + 27604 T^{4} - 2815216 T^{6} + 229790374 T^{8} - 2815216 p^{2} T^{10} + 27604 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 48 T + 1280 T^{2} - 24576 T^{3} + 373824 T^{4} - 4762272 T^{5} + 52785664 T^{6} - 521722704 T^{7} + 4670755679 T^{8} - 521722704 p T^{9} + 52785664 p^{2} T^{10} - 4762272 p^{3} T^{11} + 373824 p^{4} T^{12} - 24576 p^{5} T^{13} + 1280 p^{6} T^{14} - 48 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 24 T + 452 T^{2} - 6240 T^{3} + 73866 T^{4} - 803640 T^{5} + 8125840 T^{6} - 79110984 T^{7} + 724752659 T^{8} - 79110984 p T^{9} + 8125840 p^{2} T^{10} - 803640 p^{3} T^{11} + 73866 p^{4} T^{12} - 6240 p^{5} T^{13} + 452 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 + 76 T^{2} + 192 T^{3} + 11094 T^{4} + 192 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( 1 - 48 T + 1092 T^{2} - 15552 T^{3} + 140124 T^{4} - 513240 T^{5} - 6718968 T^{6} + 157471440 T^{7} - 1835542177 T^{8} + 157471440 p T^{9} - 6718968 p^{2} T^{10} - 513240 p^{3} T^{11} + 140124 p^{4} T^{12} - 15552 p^{5} T^{13} + 1092 p^{6} T^{14} - 48 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 192 T^{2} + 44928 T^{4} - 5229504 T^{6} + 668134658 T^{8} - 5229504 p^{2} T^{10} + 44928 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.70256229899562092186933323684, −3.63542042908262378568289225709, −3.49074815902218048554112600374, −3.42022022519722923137013599859, −3.37860678360342502143612346726, −3.17048534572636319076338147557, −3.10951951748663715111733188713, −3.10200734398998696188461307192, −2.46118163830090786301185223897, −2.44199574243798501088633332371, −2.40199786335123758772719899022, −2.28697815360706736910329634593, −2.27600206776195834413878474046, −2.27045567211827485949823530820, −2.04105714449523361009413947797, −1.91292312349485684265856584721, −1.90567000229695492126029684291, −1.90192837977972710649925783370, −1.36722351135435809534342521408, −1.22152352125500906209760036755, −0.954423420583798568274496786683, −0.66918692419737620894198176933, −0.57453679438056411293271795807, −0.56742443791190630739745746497, −0.41050940949349008742264155803, 0.41050940949349008742264155803, 0.56742443791190630739745746497, 0.57453679438056411293271795807, 0.66918692419737620894198176933, 0.954423420583798568274496786683, 1.22152352125500906209760036755, 1.36722351135435809534342521408, 1.90192837977972710649925783370, 1.90567000229695492126029684291, 1.91292312349485684265856584721, 2.04105714449523361009413947797, 2.27045567211827485949823530820, 2.27600206776195834413878474046, 2.28697815360706736910329634593, 2.40199786335123758772719899022, 2.44199574243798501088633332371, 2.46118163830090786301185223897, 3.10200734398998696188461307192, 3.10951951748663715111733188713, 3.17048534572636319076338147557, 3.37860678360342502143612346726, 3.42022022519722923137013599859, 3.49074815902218048554112600374, 3.63542042908262378568289225709, 3.70256229899562092186933323684

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

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