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Properties

Label 8-2e16-1.1-c14e4-0-0
Degree $8$
Conductor $65536$
Sign $1$
Analytic cond. $156591.$
Root an. cond. $4.46011$
Motivic weight $14$
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  − 8.70e4·5-s − 4.34e5·9-s + 6.68e7·13-s + 1.31e7·17-s − 1.68e10·25-s − 1.75e10·29-s − 1.39e11·37-s + 6.62e11·41-s + 3.77e10·45-s + 1.53e12·49-s − 2.97e12·53-s − 6.62e12·61-s − 5.81e12·65-s − 1.20e13·73-s + 2.62e13·81-s − 1.14e12·85-s + 1.06e14·89-s + 9.57e13·97-s − 4.60e14·101-s − 7.43e14·109-s + 3.48e14·113-s − 2.90e13·117-s + 1.33e15·121-s + 2.08e15·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.11·5-s − 0.0908·9-s + 1.06·13-s + 0.0319·17-s − 2.76·25-s − 1.01·29-s − 1.47·37-s + 3.39·41-s + 0.101·45-s + 2.27·49-s − 2.53·53-s − 2.10·61-s − 1.18·65-s − 1.08·73-s + 1.14·81-s − 0.0355·85-s + 2.40·89-s + 1.18·97-s − 4.29·101-s − 4.06·109-s + 1.47·113-s − 0.0967·117-s + 3.51·121-s + 4.36·125-s + 1.12·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+7)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(65536\)    =    \(2^{16}\)
Sign: $1$
Analytic conductor: \(156591.\)
Root analytic conductor: \(4.46011\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 65536,\ (\ :7, 7, 7, 7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(1.914686033\)
\(L(\frac12)\) \(\approx\) \(1.914686033\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
good3$D_4\times C_2$ \( 1 + 144820 p T^{2} - 35797601642 p^{6} T^{4} + 144820 p^{29} T^{6} + p^{56} T^{8} \)
5$D_{4}$ \( ( 1 + 348 p^{3} T + 2254488382 p T^{2} + 348 p^{17} T^{3} + p^{28} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 219949408156 p T^{2} + \)\(25\!\cdots\!42\)\( p^{2} T^{4} - 219949408156 p^{29} T^{6} + p^{56} T^{8} \)
11$D_4\times C_2$ \( 1 - 1334996682152740 T^{2} + \)\(60\!\cdots\!82\)\( p^{2} T^{4} - 1334996682152740 p^{28} T^{6} + p^{56} T^{8} \)
13$D_{4}$ \( ( 1 - 2571716 p T + 2891929558782 p^{3} T^{2} - 2571716 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 6558276 T + 267198985045368902 T^{2} - 6558276 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 2057925827207927140 T^{2} + \)\(23\!\cdots\!22\)\( T^{4} - 2057925827207927140 p^{28} T^{6} + p^{56} T^{8} \)
23$D_4\times C_2$ \( 1 - 11694673362806264260 T^{2} + \)\(27\!\cdots\!22\)\( T^{4} - 11694673362806264260 p^{28} T^{6} + p^{56} T^{8} \)
29$D_{4}$ \( ( 1 + 8750892876 T + \)\(38\!\cdots\!46\)\( T^{2} + 8750892876 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!60\)\( T^{2} + \)\(11\!\cdots\!82\)\( T^{4} - \)\(14\!\cdots\!60\)\( p^{28} T^{6} + p^{56} T^{8} \)
37$D_{4}$ \( ( 1 + 69992867180 T + \)\(13\!\cdots\!38\)\( T^{2} + 69992867180 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 331044205284 T + \)\(92\!\cdots\!26\)\( T^{2} - 331044205284 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(11\!\cdots\!20\)\( T^{2} + \)\(93\!\cdots\!42\)\( T^{4} - \)\(11\!\cdots\!20\)\( p^{28} T^{6} + p^{56} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(36\!\cdots\!52\)\( T^{2} + \)\(15\!\cdots\!58\)\( T^{4} - \)\(36\!\cdots\!52\)\( p^{28} T^{6} + p^{56} T^{8} \)
53$D_{4}$ \( ( 1 + 1487586812076 T + \)\(17\!\cdots\!22\)\( T^{2} + 1487586812076 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(18\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!02\)\( T^{4} - \)\(18\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
61$D_{4}$ \( ( 1 + 3312485425868 T + \)\(20\!\cdots\!98\)\( T^{2} + 3312485425868 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(86\!\cdots\!80\)\( T^{2} + \)\(42\!\cdots\!42\)\( T^{4} - \)\(86\!\cdots\!80\)\( p^{28} T^{6} + p^{56} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(22\!\cdots\!48\)\( T^{2} + \)\(25\!\cdots\!98\)\( T^{4} - \)\(22\!\cdots\!48\)\( p^{28} T^{6} + p^{56} T^{8} \)
73$D_{4}$ \( ( 1 + 6016566294556 T + \)\(75\!\cdots\!62\)\( T^{2} + 6016566294556 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - \)\(88\!\cdots\!24\)\( T^{2} + \)\(43\!\cdots\!66\)\( T^{4} - \)\(88\!\cdots\!24\)\( p^{28} T^{6} + p^{56} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!72\)\( T^{2} + \)\(14\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!72\)\( p^{28} T^{6} + p^{56} T^{8} \)
89$D_{4}$ \( ( 1 - 53153507248548 T + \)\(31\!\cdots\!18\)\( T^{2} - 53153507248548 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 47896582968068 T + \)\(11\!\cdots\!94\)\( T^{2} - 47896582968068 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.99161325938826647377532661251, −10.92370624039253234251768414750, −10.33982256220592295368242034744, −9.512641265430048249617750059054, −9.489952441810520366919434483927, −9.216778914712456825368533961197, −8.448335467308203625550200473171, −8.267987962335364485487411124730, −7.72810643792124718052605879701, −7.44283707668746807684508058381, −7.34342879203946709853948257381, −6.39110768148752104426081865930, −6.13893938342966794431321113816, −5.65439992470305527846048369084, −5.46084371360132242137460410328, −4.38714630813622324253000123576, −4.33865198919160258519561371351, −3.84612891401976161861012803843, −3.45355918093293927699225143285, −2.99555730261388886154641435701, −2.24486047925941343751596615248, −1.80056172841325941168831364504, −1.37141060323344517318940352315, −0.54038587704151075640160233648, −0.35680627879401868361208211450, 0.35680627879401868361208211450, 0.54038587704151075640160233648, 1.37141060323344517318940352315, 1.80056172841325941168831364504, 2.24486047925941343751596615248, 2.99555730261388886154641435701, 3.45355918093293927699225143285, 3.84612891401976161861012803843, 4.33865198919160258519561371351, 4.38714630813622324253000123576, 5.46084371360132242137460410328, 5.65439992470305527846048369084, 6.13893938342966794431321113816, 6.39110768148752104426081865930, 7.34342879203946709853948257381, 7.44283707668746807684508058381, 7.72810643792124718052605879701, 8.267987962335364485487411124730, 8.448335467308203625550200473171, 9.216778914712456825368533961197, 9.489952441810520366919434483927, 9.512641265430048249617750059054, 10.33982256220592295368242034744, 10.92370624039253234251768414750, 10.99161325938826647377532661251

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