Properties

Degree 16
Conductor $ 2^{8} \cdot 11^{8} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 3·3-s + 4·4-s + 5·5-s + 12·6-s − 7-s + 21·9-s + 20·10-s − 155·11-s + 12·12-s + 7·13-s − 4·14-s + 15·15-s + 161·17-s + 84·18-s − 272·19-s + 20·20-s − 3·21-s − 620·22-s + 628·23-s + 129·25-s + 28·26-s − 59·27-s − 4·28-s + 33·29-s + 60·30-s + 323·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.0539·7-s + 7/9·9-s + 0.632·10-s − 4.24·11-s + 0.288·12-s + 0.149·13-s − 0.0763·14-s + 0.258·15-s + 2.29·17-s + 1.09·18-s − 3.28·19-s + 0.223·20-s − 0.0311·21-s − 6.00·22-s + 5.69·23-s + 1.03·25-s + 0.211·26-s − 0.420·27-s − 0.0269·28-s + 0.211·29-s + 0.365·30-s + 1.87·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 11^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{22} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 11^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )$
$L(2)$  $\approx$  $3.56935$
$L(\frac12)$  $\approx$  $3.56935$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11\}$, \(F_p\) is a polynomial of degree 16. If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad2 \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2} \)
11 \( 1 + 155 T + 13111 T^{2} + 68645 p T^{3} + 264456 p^{2} T^{4} + 68645 p^{4} T^{5} + 13111 p^{6} T^{6} + 155 p^{9} T^{7} + p^{12} T^{8} \)
good3 \( 1 - p T - 4 p T^{2} + 158 T^{3} - 1228 T^{4} + 6473 T^{5} + 959 T^{6} - 225344 T^{7} + 1236232 T^{8} - 225344 p^{3} T^{9} + 959 p^{6} T^{10} + 6473 p^{9} T^{11} - 1228 p^{12} T^{12} + 158 p^{15} T^{13} - 4 p^{19} T^{14} - p^{22} T^{15} + p^{24} T^{16} \)
5 \( 1 - p T - 104 T^{2} - 326 p T^{3} + 10006 T^{4} + 56777 p T^{5} + 2835571 T^{6} - 1265534 p^{2} T^{7} - 387953844 T^{8} - 1265534 p^{5} T^{9} + 2835571 p^{6} T^{10} + 56777 p^{10} T^{11} + 10006 p^{12} T^{12} - 326 p^{16} T^{13} - 104 p^{18} T^{14} - p^{22} T^{15} + p^{24} T^{16} \)
7 \( 1 + T + 12 T^{2} - 1104 T^{3} - 87888 T^{4} - 922401 T^{5} + 28062801 T^{6} + 211635072 T^{7} + 13215739812 T^{8} + 211635072 p^{3} T^{9} + 28062801 p^{6} T^{10} - 922401 p^{9} T^{11} - 87888 p^{12} T^{12} - 1104 p^{15} T^{13} + 12 p^{18} T^{14} + p^{21} T^{15} + p^{24} T^{16} \)
13 \( 1 - 7 T + 1930 T^{2} + 79384 T^{3} + 10224596 T^{4} + 326819897 T^{5} + 769971197 p T^{6} + 1235524128730 T^{7} + 47709276128944 T^{8} + 1235524128730 p^{3} T^{9} + 769971197 p^{7} T^{10} + 326819897 p^{9} T^{11} + 10224596 p^{12} T^{12} + 79384 p^{15} T^{13} + 1930 p^{18} T^{14} - 7 p^{21} T^{15} + p^{24} T^{16} \)
17 \( 1 - 161 T + 5020 T^{2} + 211188 T^{3} + 21130156 T^{4} - 1323457999 T^{5} - 265862283281 T^{6} + 778694329080 p T^{7} + 277440196391544 T^{8} + 778694329080 p^{4} T^{9} - 265862283281 p^{6} T^{10} - 1323457999 p^{9} T^{11} + 21130156 p^{12} T^{12} + 211188 p^{15} T^{13} + 5020 p^{18} T^{14} - 161 p^{21} T^{15} + p^{24} T^{16} \)
19 \( 1 + 272 T + 33201 T^{2} + 3039584 T^{3} + 292957015 T^{4} + 21884364992 T^{5} + 909384182595 T^{6} + 32108296725040 T^{7} + 2715838419478696 T^{8} + 32108296725040 p^{3} T^{9} + 909384182595 p^{6} T^{10} + 21884364992 p^{9} T^{11} + 292957015 p^{12} T^{12} + 3039584 p^{15} T^{13} + 33201 p^{18} T^{14} + 272 p^{21} T^{15} + p^{24} T^{16} \)
23 \( ( 1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 8042722 p^{3} T^{5} + 61816 p^{6} T^{6} - 314 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
29 \( 1 - 33 T + 14176 T^{2} + 2091144 T^{3} + 419909220 T^{4} + 81926257017 T^{5} + 7198320043835 T^{6} + 2719607629962300 T^{7} + 63828590693142176 T^{8} + 2719607629962300 p^{3} T^{9} + 7198320043835 p^{6} T^{10} + 81926257017 p^{9} T^{11} + 419909220 p^{12} T^{12} + 2091144 p^{15} T^{13} + 14176 p^{18} T^{14} - 33 p^{21} T^{15} + p^{24} T^{16} \)
31 \( 1 - 323 T - 4098 T^{2} + 5799192 T^{3} + 305184598 T^{4} - 93424183073 T^{5} + 1165032156515 p T^{6} - 3925905573337100 T^{7} - 564631402757797204 T^{8} - 3925905573337100 p^{3} T^{9} + 1165032156515 p^{7} T^{10} - 93424183073 p^{9} T^{11} + 305184598 p^{12} T^{12} + 5799192 p^{15} T^{13} - 4098 p^{18} T^{14} - 323 p^{21} T^{15} + p^{24} T^{16} \)
37 \( 1 - 49 T - 7868 T^{2} + 4260746 T^{3} - 566286138 T^{4} + 1042572020189 T^{5} + 91349107260671 T^{6} - 14416097444880138 T^{7} + 5705343181934634812 T^{8} - 14416097444880138 p^{3} T^{9} + 91349107260671 p^{6} T^{10} + 1042572020189 p^{9} T^{11} - 566286138 p^{12} T^{12} + 4260746 p^{15} T^{13} - 7868 p^{18} T^{14} - 49 p^{21} T^{15} + p^{24} T^{16} \)
41 \( 1 - 361 T - 19140 T^{2} + 12862580 T^{3} + 3853225260 T^{4} - 605378813703 T^{5} - 283363657590617 T^{6} + 99783171614693080 T^{7} - 22290482644766813160 T^{8} + 99783171614693080 p^{3} T^{9} - 283363657590617 p^{6} T^{10} - 605378813703 p^{9} T^{11} + 3853225260 p^{12} T^{12} + 12862580 p^{15} T^{13} - 19140 p^{18} T^{14} - 361 p^{21} T^{15} + p^{24} T^{16} \)
43 \( ( 1 - 721 T + 420117 T^{2} - 154459221 T^{3} + 51447883420 T^{4} - 154459221 p^{3} T^{5} + 420117 p^{6} T^{6} - 721 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
47 \( 1 + 1069 T + 301070 T^{2} - 43368042 T^{3} - 24642745634 T^{4} + 6223420821981 T^{5} + 4635059307315839 T^{6} + 746453682627128500 T^{7} + 29698413240018564184 T^{8} + 746453682627128500 p^{3} T^{9} + 4635059307315839 p^{6} T^{10} + 6223420821981 p^{9} T^{11} - 24642745634 p^{12} T^{12} - 43368042 p^{15} T^{13} + 301070 p^{18} T^{14} + 1069 p^{21} T^{15} + p^{24} T^{16} \)
53 \( 1 + 281 T - 343802 T^{2} - 60864958 T^{3} + 33213391926 T^{4} + 4932564267181 T^{5} + 8874412003121445 T^{6} + 107096647673733916 T^{7} - \)\(25\!\cdots\!56\)\( T^{8} + 107096647673733916 p^{3} T^{9} + 8874412003121445 p^{6} T^{10} + 4932564267181 p^{9} T^{11} + 33213391926 p^{12} T^{12} - 60864958 p^{15} T^{13} - 343802 p^{18} T^{14} + 281 p^{21} T^{15} + p^{24} T^{16} \)
59 \( 1 + 128 T - 87119 T^{2} - 50674984 T^{3} + 39669673335 T^{4} + 496570131448 T^{5} - 4049042036547885 T^{6} - 22790010341236160 T^{7} + \)\(27\!\cdots\!36\)\( T^{8} - 22790010341236160 p^{3} T^{9} - 4049042036547885 p^{6} T^{10} + 496570131448 p^{9} T^{11} + 39669673335 p^{12} T^{12} - 50674984 p^{15} T^{13} - 87119 p^{18} T^{14} + 128 p^{21} T^{15} + p^{24} T^{16} \)
61 \( 1 + 617 T - 60798 T^{2} - 112995618 T^{3} + 22253617738 T^{4} + 1960695737 p^{2} T^{5} - 13214872371636415 T^{6} + 3429112403997460700 T^{7} + \)\(68\!\cdots\!36\)\( T^{8} + 3429112403997460700 p^{3} T^{9} - 13214872371636415 p^{6} T^{10} + 1960695737 p^{11} T^{11} + 22253617738 p^{12} T^{12} - 112995618 p^{15} T^{13} - 60798 p^{18} T^{14} + 617 p^{21} T^{15} + p^{24} T^{16} \)
67 \( ( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 243308709 p^{3} T^{5} + 686735 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
71 \( 1 - 115 T - 138428 T^{2} + 42960000 T^{3} + 77925723838 T^{4} - 71653925274035 T^{5} + 12191505393571699 T^{6} + 3285417296443548450 T^{7} + \)\(11\!\cdots\!80\)\( T^{8} + 3285417296443548450 p^{3} T^{9} + 12191505393571699 p^{6} T^{10} - 71653925274035 p^{9} T^{11} + 77925723838 p^{12} T^{12} + 42960000 p^{15} T^{13} - 138428 p^{18} T^{14} - 115 p^{21} T^{15} + p^{24} T^{16} \)
73 \( 1 + 1487 T + 142728 T^{2} - 14316264 p T^{3} - 569519703768 T^{4} + 366060407928453 T^{5} + 393291607153115139 T^{6} - 68019158899540158384 T^{7} - \)\(19\!\cdots\!28\)\( T^{8} - 68019158899540158384 p^{3} T^{9} + 393291607153115139 p^{6} T^{10} + 366060407928453 p^{9} T^{11} - 569519703768 p^{12} T^{12} - 14316264 p^{16} T^{13} + 142728 p^{18} T^{14} + 1487 p^{21} T^{15} + p^{24} T^{16} \)
79 \( 1 - 71 T - 500222 T^{2} - 73013624 T^{3} + 350790961998 T^{4} + 97550067497959 T^{5} - 161731343227399015 T^{6} + 8534351363848547400 T^{7} + \)\(59\!\cdots\!76\)\( T^{8} + 8534351363848547400 p^{3} T^{9} - 161731343227399015 p^{6} T^{10} + 97550067497959 p^{9} T^{11} + 350790961998 p^{12} T^{12} - 73013624 p^{15} T^{13} - 500222 p^{18} T^{14} - 71 p^{21} T^{15} + p^{24} T^{16} \)
83 \( 1 - 1942 T + 875395 T^{2} + 857035884 T^{3} - 790010610449 T^{4} - 618864428088728 T^{5} + 845503237658086801 T^{6} + \)\(17\!\cdots\!30\)\( T^{7} - \)\(59\!\cdots\!96\)\( T^{8} + \)\(17\!\cdots\!30\)\( p^{3} T^{9} + 845503237658086801 p^{6} T^{10} - 618864428088728 p^{9} T^{11} - 790010610449 p^{12} T^{12} + 857035884 p^{15} T^{13} + 875395 p^{18} T^{14} - 1942 p^{21} T^{15} + p^{24} T^{16} \)
89 \( ( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1914547747 p^{3} T^{5} + 2406895 p^{6} T^{6} + 1101 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
97 \( 1 + 5128 T + 10343177 T^{2} + 9500759764 T^{3} + 1479496697751 T^{4} - 4959875190278012 T^{5} - 1475957592224572125 T^{6} + \)\(96\!\cdots\!12\)\( T^{7} + \)\(15\!\cdots\!64\)\( T^{8} + \)\(96\!\cdots\!12\)\( p^{3} T^{9} - 1475957592224572125 p^{6} T^{10} - 4959875190278012 p^{9} T^{11} + 1479496697751 p^{12} T^{12} + 9500759764 p^{15} T^{13} + 10343177 p^{18} T^{14} + 5128 p^{21} T^{15} + p^{24} T^{16} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.419504813096985549378070924759, −8.379656075122304239408693573735, −8.160973284074851906535174974462, −7.931538838213455941251650101620, −7.61592026690383926944713276178, −7.54063715403432348854756224300, −7.41364396910379173819144166997, −6.91257774629780427196959960341, −6.90775298407155775345447574611, −6.61386950415302146317304446656, −6.06402805807492753657783912086, −6.01124755013911372474612167340, −5.65424161763885758285245969751, −5.52804045030340407986287728335, −4.92848022681168137791735169484, −4.91856253585733280620776097713, −4.87669696456338650979064339367, −4.68985297667322811981123857566, −4.06093736295486618416107366091, −3.90984967770940474486673319475, −2.94895309570397671560633500049, −2.87928150513178740951327685236, −2.86097182176356573253175167642, −2.46453744128779366832478711766, −1.11978324879953669972392833125, 1.11978324879953669972392833125, 2.46453744128779366832478711766, 2.86097182176356573253175167642, 2.87928150513178740951327685236, 2.94895309570397671560633500049, 3.90984967770940474486673319475, 4.06093736295486618416107366091, 4.68985297667322811981123857566, 4.87669696456338650979064339367, 4.91856253585733280620776097713, 4.92848022681168137791735169484, 5.52804045030340407986287728335, 5.65424161763885758285245969751, 6.01124755013911372474612167340, 6.06402805807492753657783912086, 6.61386950415302146317304446656, 6.90775298407155775345447574611, 6.91257774629780427196959960341, 7.41364396910379173819144166997, 7.54063715403432348854756224300, 7.61592026690383926944713276178, 7.931538838213455941251650101620, 8.160973284074851906535174974462, 8.379656075122304239408693573735, 8.419504813096985549378070924759

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

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