Properties

Degree 16
Conductor $ 2^{24} $
Sign $1$
Motivic weight 9
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 18·2-s − 52·4-s + 4.80e3·7-s + 1.75e3·8-s + 5.90e4·9-s − 8.64e4·14-s + 5.93e4·16-s − 1.02e5·17-s − 1.06e6·18-s + 3.41e6·23-s + 6.60e6·25-s − 2.49e5·28-s + 8.03e5·31-s − 3.81e6·32-s + 1.83e6·34-s − 3.07e6·36-s − 2.18e6·41-s − 6.14e7·46-s + 7.43e6·47-s − 1.37e8·49-s − 1.18e8·50-s + 8.40e6·56-s − 1.44e7·62-s + 2.83e8·63-s − 5.83e6·64-s + 5.30e6·68-s + 5.60e8·71-s + ⋯
L(s)  = 1  − 0.795·2-s − 0.101·4-s + 0.755·7-s + 0.151·8-s + 2.99·9-s − 0.601·14-s + 0.226·16-s − 0.296·17-s − 2.38·18-s + 2.54·23-s + 3.37·25-s − 0.0767·28-s + 0.156·31-s − 0.643·32-s + 0.235·34-s − 0.304·36-s − 0.120·41-s − 2.02·46-s + 0.222·47-s − 3.41·49-s − 2.68·50-s + 0.114·56-s − 0.124·62-s + 2.26·63-s − 0.0434·64-s + 0.0300·68-s + 2.61·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16777216 ^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16777216 ^{s/2} \, \Gamma_{\C}(s+9/2)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(16777216\)    =    \(2^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  induced by $\chi_{8} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 16777216,\ (\ :[9/2]^{8}),\ 1)$
$L(5)$  $\approx$  $5.25952$
$L(\frac12)$  $\approx$  $5.25952$
$L(\frac{11}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 16. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 + 9 p T + 47 p^{3} T^{2} + 93 p^{6} T^{3} + 35 p^{10} T^{4} + 93 p^{15} T^{5} + 47 p^{21} T^{6} + 9 p^{28} T^{7} + p^{36} T^{8} \)
good3 \( 1 - 59048 T^{2} + 69594932 p^{3} T^{4} - 6090909784 p^{8} T^{6} + 1071668796386710 p^{6} T^{8} - 6090909784 p^{26} T^{10} + 69594932 p^{39} T^{12} - 59048 p^{54} T^{14} + p^{72} T^{16} \)
5 \( 1 - 6600808 T^{2} + 17461449664316 T^{4} - 969001730842818264 p^{2} T^{6} + \)\(49\!\cdots\!78\)\( p^{4} T^{8} - 969001730842818264 p^{20} T^{10} + 17461449664316 p^{36} T^{12} - 6600808 p^{54} T^{14} + p^{72} T^{16} \)
7 \( ( 1 - 2400 T + 11068292 p T^{2} + 255830496 p^{3} T^{3} + 8426386960458 p^{3} T^{4} + 255830496 p^{12} T^{5} + 11068292 p^{19} T^{6} - 2400 p^{27} T^{7} + p^{36} T^{8} )^{2} \)
11 \( 1 - 10626901608 T^{2} + 58781092281363321020 T^{4} - \)\(22\!\cdots\!52\)\( T^{6} + \)\(60\!\cdots\!38\)\( T^{8} - \)\(22\!\cdots\!52\)\( p^{18} T^{10} + 58781092281363321020 p^{36} T^{12} - 10626901608 p^{54} T^{14} + p^{72} T^{16} \)
13 \( 1 - 52177437864 T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(19\!\cdots\!44\)\( T^{6} + \)\(22\!\cdots\!38\)\( T^{8} - \)\(19\!\cdots\!44\)\( p^{18} T^{10} + \)\(12\!\cdots\!60\)\( p^{36} T^{12} - 52177437864 p^{54} T^{14} + p^{72} T^{16} \)
17 \( ( 1 + 3000 p T + 277845292252 T^{2} + 28596206694990600 T^{3} + \)\(14\!\cdots\!62\)\( p^{2} T^{4} + 28596206694990600 p^{9} T^{5} + 277845292252 p^{18} T^{6} + 3000 p^{28} T^{7} + p^{36} T^{8} )^{2} \)
19 \( 1 - 1485325196328 T^{2} + \)\(12\!\cdots\!88\)\( T^{4} - \)\(65\!\cdots\!60\)\( T^{6} + \)\(25\!\cdots\!58\)\( T^{8} - \)\(65\!\cdots\!60\)\( p^{18} T^{10} + \)\(12\!\cdots\!88\)\( p^{36} T^{12} - 1485325196328 p^{54} T^{14} + p^{72} T^{16} \)
23 \( ( 1 - 1706016 T + 3178422376156 T^{2} - 5879850573514848 p T^{3} + \)\(67\!\cdots\!42\)\( T^{4} - 5879850573514848 p^{10} T^{5} + 3178422376156 p^{18} T^{6} - 1706016 p^{27} T^{7} + p^{36} T^{8} )^{2} \)
29 \( 1 - 78392306727720 T^{2} + \)\(28\!\cdots\!40\)\( T^{4} - \)\(67\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!38\)\( T^{8} - \)\(67\!\cdots\!00\)\( p^{18} T^{10} + \)\(28\!\cdots\!40\)\( p^{36} T^{12} - 78392306727720 p^{54} T^{14} + p^{72} T^{16} \)
31 \( ( 1 - 401792 T + 86180537957500 T^{2} - 37247601810974485376 T^{3} + \)\(32\!\cdots\!74\)\( T^{4} - 37247601810974485376 p^{9} T^{5} + 86180537957500 p^{18} T^{6} - 401792 p^{27} T^{7} + p^{36} T^{8} )^{2} \)
37 \( 1 - 713593321977192 T^{2} + \)\(24\!\cdots\!96\)\( T^{4} - \)\(53\!\cdots\!04\)\( T^{6} + \)\(82\!\cdots\!86\)\( T^{8} - \)\(53\!\cdots\!04\)\( p^{18} T^{10} + \)\(24\!\cdots\!96\)\( p^{36} T^{12} - 713593321977192 p^{54} T^{14} + p^{72} T^{16} \)
41 \( ( 1 + 1090392 T + 1059071398836988 T^{2} + \)\(88\!\cdots\!32\)\( T^{3} + \)\(48\!\cdots\!74\)\( T^{4} + \)\(88\!\cdots\!32\)\( p^{9} T^{5} + 1059071398836988 p^{18} T^{6} + 1090392 p^{27} T^{7} + p^{36} T^{8} )^{2} \)
43 \( 1 - 1649449028392296 T^{2} + \)\(12\!\cdots\!08\)\( T^{4} - \)\(66\!\cdots\!36\)\( T^{6} + \)\(34\!\cdots\!62\)\( T^{8} - \)\(66\!\cdots\!36\)\( p^{18} T^{10} + \)\(12\!\cdots\!08\)\( p^{36} T^{12} - 1649449028392296 p^{54} T^{14} + p^{72} T^{16} \)
47 \( ( 1 - 3716160 T + 1896060860372156 T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!38\)\( T^{4} - \)\(18\!\cdots\!60\)\( p^{9} T^{5} + 1896060860372156 p^{18} T^{6} - 3716160 p^{27} T^{7} + p^{36} T^{8} )^{2} \)
53 \( 1 - 8766485591758824 T^{2} + \)\(43\!\cdots\!44\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(33\!\cdots\!70\)\( T^{8} - \)\(11\!\cdots\!80\)\( p^{18} T^{10} + \)\(43\!\cdots\!44\)\( p^{36} T^{12} - 8766485591758824 p^{54} T^{14} + p^{72} T^{16} \)
59 \( 1 - 65444821078817512 T^{2} + \)\(19\!\cdots\!36\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{6} + \)\(34\!\cdots\!90\)\( T^{8} - \)\(32\!\cdots\!40\)\( p^{18} T^{10} + \)\(19\!\cdots\!36\)\( p^{36} T^{12} - 65444821078817512 p^{54} T^{14} + p^{72} T^{16} \)
61 \( 1 - 33480681785208872 T^{2} + \)\(58\!\cdots\!36\)\( T^{4} - \)\(62\!\cdots\!40\)\( T^{6} + \)\(64\!\cdots\!50\)\( T^{8} - \)\(62\!\cdots\!40\)\( p^{18} T^{10} + \)\(58\!\cdots\!36\)\( p^{36} T^{12} - 33480681785208872 p^{54} T^{14} + p^{72} T^{16} \)
67 \( 1 - 104404589351487656 T^{2} + \)\(63\!\cdots\!84\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(82\!\cdots\!50\)\( T^{8} - \)\(26\!\cdots\!20\)\( p^{18} T^{10} + \)\(63\!\cdots\!84\)\( p^{36} T^{12} - 104404589351487656 p^{54} T^{14} + p^{72} T^{16} \)
71 \( ( 1 - 280117344 T + 145831167184353692 T^{2} - \)\(30\!\cdots\!96\)\( T^{3} + \)\(97\!\cdots\!34\)\( T^{4} - \)\(30\!\cdots\!96\)\( p^{9} T^{5} + 145831167184353692 p^{18} T^{6} - 280117344 p^{27} T^{7} + p^{36} T^{8} )^{2} \)
73 \( ( 1 + 261993560 T + 145417231313639804 T^{2} + \)\(33\!\cdots\!76\)\( T^{3} + \)\(99\!\cdots\!54\)\( T^{4} + \)\(33\!\cdots\!76\)\( p^{9} T^{5} + 145417231313639804 p^{18} T^{6} + 261993560 p^{27} T^{7} + p^{36} T^{8} )^{2} \)
79 \( ( 1 + 124471872 T + 432348011474041148 T^{2} + \)\(40\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!42\)\( T^{4} + \)\(40\!\cdots\!20\)\( p^{9} T^{5} + 432348011474041148 p^{18} T^{6} + 124471872 p^{27} T^{7} + p^{36} T^{8} )^{2} \)
83 \( 1 - 620814363205633576 T^{2} + \)\(25\!\cdots\!08\)\( T^{4} - \)\(71\!\cdots\!76\)\( T^{6} + \)\(15\!\cdots\!22\)\( T^{8} - \)\(71\!\cdots\!76\)\( p^{18} T^{10} + \)\(25\!\cdots\!08\)\( p^{36} T^{12} - 620814363205633576 p^{54} T^{14} + p^{72} T^{16} \)
89 \( ( 1 - 372413928 T + 500734440582085948 T^{2} - \)\(19\!\cdots\!20\)\( p T^{3} + \)\(28\!\cdots\!22\)\( T^{4} - \)\(19\!\cdots\!20\)\( p^{10} T^{5} + 500734440582085948 p^{18} T^{6} - 372413928 p^{27} T^{7} + p^{36} T^{8} )^{2} \)
97 \( ( 1 + 4966392 T + 2046417225380869532 T^{2} - \)\(22\!\cdots\!04\)\( T^{3} + \)\(19\!\cdots\!82\)\( T^{4} - \)\(22\!\cdots\!04\)\( p^{9} T^{5} + 2046417225380869532 p^{18} T^{6} + 4966392 p^{27} T^{7} + p^{36} T^{8} )^{2} \)
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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

−9.066119657328896677898637071310, −8.938761873834235855602559250976, −8.539255450859165808688795095002, −8.216668033323703049741391489157, −7.899257537411288051599426808184, −7.73811334198341082897571137608, −7.42601265088147744082964880155, −6.89286082018099205740364667346, −6.82589886433999997973353112969, −6.71858881119435499044570943558, −6.65862218504348266853261555250, −5.73111269654623798831699249891, −5.47507876430754981413015897677, −4.91200434140504340858827066423, −4.70123618930792420651478544594, −4.51827021190430141688253419569, −4.45881051517567841731270577497, −3.53901561045568655763416667963, −3.19765859938353322157558525430, −2.96656224618974369709787245977, −1.99593194959048974090888365981, −1.70993775660762988295087346178, −1.12264873584756363001482525246, −1.02553271768728285703819471725, −0.52850205214568098847237753075, 0.52850205214568098847237753075, 1.02553271768728285703819471725, 1.12264873584756363001482525246, 1.70993775660762988295087346178, 1.99593194959048974090888365981, 2.96656224618974369709787245977, 3.19765859938353322157558525430, 3.53901561045568655763416667963, 4.45881051517567841731270577497, 4.51827021190430141688253419569, 4.70123618930792420651478544594, 4.91200434140504340858827066423, 5.47507876430754981413015897677, 5.73111269654623798831699249891, 6.65862218504348266853261555250, 6.71858881119435499044570943558, 6.82589886433999997973353112969, 6.89286082018099205740364667346, 7.42601265088147744082964880155, 7.73811334198341082897571137608, 7.899257537411288051599426808184, 8.216668033323703049741391489157, 8.539255450859165808688795095002, 8.938761873834235855602559250976, 9.066119657328896677898637071310

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

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