Properties

Label 16-2e24-1.1-c10e8-0-0
Degree $16$
Conductor $16777216$
Sign $1$
Analytic cond. $445516.$
Root an. cond. $2.25451$
Motivic weight $10$
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  + 42·2-s + 480·3-s + 988·4-s + 2.01e4·6-s + 1.67e4·8-s − 1.38e5·9-s + 1.43e5·11-s + 4.74e5·12-s − 2.69e5·16-s − 3.70e5·17-s − 5.82e6·18-s + 1.75e6·19-s + 6.01e6·22-s + 8.05e6·24-s + 2.96e7·25-s − 1.03e8·27-s − 3.10e7·32-s + 6.87e7·33-s − 1.55e7·34-s − 1.37e8·36-s + 7.36e7·38-s + 9.26e7·41-s − 1.01e7·43-s + 1.41e8·44-s − 1.29e8·48-s + 1.08e9·49-s + 1.24e9·50-s + ⋯
L(s)  = 1  + 1.31·2-s + 1.97·3-s + 0.964·4-s + 2.59·6-s + 0.511·8-s − 2.34·9-s + 0.889·11-s + 1.90·12-s − 0.256·16-s − 0.261·17-s − 3.08·18-s + 0.708·19-s + 1.16·22-s + 1.01·24-s + 3.03·25-s − 7.19·27-s − 0.925·32-s + 1.75·33-s − 0.342·34-s − 2.26·36-s + 0.929·38-s + 0.799·41-s − 0.0693·43-s + 0.858·44-s − 0.506·48-s + 3.85·49-s + 3.98·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16777216 ^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16777216 ^{s/2} \, \Gamma_{\C}(s+5)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(16777216\)    =    \(2^{24}\)
Sign: $1$
Analytic conductor: \(445516.\)
Root analytic conductor: \(2.25451\)
Motivic weight: \(10\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 16777216,\ (\ :[5]^{8}),\ 1)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.278401776\)
\(L(\frac12)\) \(\approx\) \(2.278401776\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 21 p T + 97 p^{3} T^{2} - 123 p^{6} T^{3} + 525 p^{10} T^{4} - 123 p^{16} T^{5} + 97 p^{23} T^{6} - 21 p^{31} T^{7} + p^{40} T^{8} \)
good3 \( ( 1 - 80 p T + 17308 p^{2} T^{2} - 405904 p^{4} T^{3} + 53937602 p^{5} T^{4} - 405904 p^{14} T^{5} + 17308 p^{22} T^{6} - 80 p^{31} T^{7} + p^{40} T^{8} )^{2} \)
5 \( 1 - 5933224 p T^{2} + 18980764448956 p^{2} T^{4} - 45074813828010508504 p^{3} T^{6} + \)\(93\!\cdots\!46\)\( p^{4} T^{8} - 45074813828010508504 p^{23} T^{10} + 18980764448956 p^{42} T^{12} - 5933224 p^{61} T^{14} + p^{80} T^{16} \)
7 \( 1 - 1089684872 T^{2} + 95613913648621444 p T^{4} - \)\(81\!\cdots\!88\)\( p^{3} T^{6} + \)\(53\!\cdots\!10\)\( p^{5} T^{8} - \)\(81\!\cdots\!88\)\( p^{23} T^{10} + 95613913648621444 p^{41} T^{12} - 1089684872 p^{60} T^{14} + p^{80} T^{16} \)
11 \( ( 1 - 71664 T + 52584995516 T^{2} - 373693362616752 p T^{3} + \)\(19\!\cdots\!78\)\( T^{4} - 373693362616752 p^{11} T^{5} + 52584995516 p^{20} T^{6} - 71664 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
13 \( 1 - 459250928072 T^{2} + \)\(14\!\cdots\!28\)\( T^{4} - \)\(28\!\cdots\!44\)\( T^{6} + \)\(45\!\cdots\!10\)\( T^{8} - \)\(28\!\cdots\!44\)\( p^{20} T^{10} + \)\(14\!\cdots\!28\)\( p^{40} T^{12} - 459250928072 p^{60} T^{14} + p^{80} T^{16} \)
17 \( ( 1 + 185400 T + 3991264862492 T^{2} - 74312252090724216 T^{3} + \)\(10\!\cdots\!46\)\( T^{4} - 74312252090724216 p^{10} T^{5} + 3991264862492 p^{20} T^{6} + 185400 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
19 \( ( 1 - 876656 T + 38072082076 p^{2} T^{2} - 15629661673389367568 T^{3} + \)\(11\!\cdots\!18\)\( T^{4} - 15629661673389367568 p^{10} T^{5} + 38072082076 p^{22} T^{6} - 876656 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
23 \( 1 - 205026617368712 T^{2} + \)\(20\!\cdots\!08\)\( T^{4} - \)\(13\!\cdots\!44\)\( T^{6} + \)\(65\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!44\)\( p^{20} T^{10} + \)\(20\!\cdots\!08\)\( p^{40} T^{12} - 205026617368712 p^{60} T^{14} + p^{80} T^{16} \)
29 \( 1 - 356194589173448 T^{2} + \)\(30\!\cdots\!68\)\( T^{4} - \)\(29\!\cdots\!36\)\( T^{6} + \)\(43\!\cdots\!30\)\( T^{8} - \)\(29\!\cdots\!36\)\( p^{20} T^{10} + \)\(30\!\cdots\!68\)\( p^{40} T^{12} - 356194589173448 p^{60} T^{14} + p^{80} T^{16} \)
31 \( 1 - 2237050600650248 T^{2} + \)\(38\!\cdots\!68\)\( T^{4} - \)\(44\!\cdots\!56\)\( T^{6} + \)\(41\!\cdots\!70\)\( T^{8} - \)\(44\!\cdots\!56\)\( p^{20} T^{10} + \)\(38\!\cdots\!68\)\( p^{40} T^{12} - 2237050600650248 p^{60} T^{14} + p^{80} T^{16} \)
37 \( 1 - 16551679761900872 T^{2} + \)\(11\!\cdots\!28\)\( T^{4} - \)\(55\!\cdots\!04\)\( T^{6} + \)\(25\!\cdots\!90\)\( T^{8} - \)\(55\!\cdots\!04\)\( p^{20} T^{10} + \)\(11\!\cdots\!28\)\( p^{40} T^{12} - 16551679761900872 p^{60} T^{14} + p^{80} T^{16} \)
41 \( ( 1 - 46334664 T + 39266837717699036 T^{2} - \)\(13\!\cdots\!72\)\( T^{3} + \)\(70\!\cdots\!58\)\( T^{4} - \)\(13\!\cdots\!72\)\( p^{10} T^{5} + 39266837717699036 p^{20} T^{6} - 46334664 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
43 \( ( 1 + 5095312 T + 1817044412344372 p T^{2} + \)\(16\!\cdots\!04\)\( p T^{3} + \)\(24\!\cdots\!90\)\( T^{4} + \)\(16\!\cdots\!04\)\( p^{11} T^{5} + 1817044412344372 p^{21} T^{6} + 5095312 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
47 \( 1 - 134538168632987912 T^{2} + \)\(92\!\cdots\!08\)\( T^{4} - \)\(39\!\cdots\!24\)\( T^{6} + \)\(15\!\cdots\!10\)\( T^{8} - \)\(39\!\cdots\!24\)\( p^{20} T^{10} + \)\(92\!\cdots\!08\)\( p^{40} T^{12} - 134538168632987912 p^{60} T^{14} + p^{80} T^{16} \)
53 \( 1 - 946032264727585352 T^{2} + \)\(44\!\cdots\!68\)\( T^{4} - \)\(13\!\cdots\!24\)\( T^{6} + \)\(27\!\cdots\!10\)\( T^{8} - \)\(13\!\cdots\!24\)\( p^{20} T^{10} + \)\(44\!\cdots\!68\)\( p^{40} T^{12} - 946032264727585352 p^{60} T^{14} + p^{80} T^{16} \)
59 \( ( 1 + 482821008 T + 1514368752837832508 T^{2} + \)\(78\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} + \)\(78\!\cdots\!36\)\( p^{10} T^{5} + 1514368752837832508 p^{20} T^{6} + 482821008 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
61 \( 1 - 2607734110174077128 T^{2} + \)\(31\!\cdots\!88\)\( T^{4} - \)\(28\!\cdots\!36\)\( T^{6} + \)\(22\!\cdots\!50\)\( T^{8} - \)\(28\!\cdots\!36\)\( p^{20} T^{10} + \)\(31\!\cdots\!88\)\( p^{40} T^{12} - 2607734110174077128 p^{60} T^{14} + p^{80} T^{16} \)
67 \( ( 1 + 612791440 T + 6896527760373520252 T^{2} + \)\(31\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!06\)\( T^{4} + \)\(31\!\cdots\!84\)\( p^{10} T^{5} + 6896527760373520252 p^{20} T^{6} + 612791440 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
71 \( 1 - 17068544268164346248 T^{2} + \)\(14\!\cdots\!28\)\( T^{4} - \)\(78\!\cdots\!16\)\( T^{6} + \)\(30\!\cdots\!70\)\( T^{8} - \)\(78\!\cdots\!16\)\( p^{20} T^{10} + \)\(14\!\cdots\!28\)\( p^{40} T^{12} - 17068544268164346248 p^{60} T^{14} + p^{80} T^{16} \)
73 \( ( 1 - 1400036360 T + 8051296934129630812 T^{2} - \)\(11\!\cdots\!04\)\( T^{3} + \)\(55\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!04\)\( p^{10} T^{5} + 8051296934129630812 p^{20} T^{6} - 1400036360 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
79 \( 1 - 45980157940147603208 T^{2} + \)\(10\!\cdots\!08\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{6} + \)\(18\!\cdots\!10\)\( T^{8} - \)\(16\!\cdots\!56\)\( p^{20} T^{10} + \)\(10\!\cdots\!08\)\( p^{40} T^{12} - 45980157940147603208 p^{60} T^{14} + p^{80} T^{16} \)
83 \( ( 1 - 926711280 T + 43393122038825346812 T^{2} - \)\(33\!\cdots\!84\)\( T^{3} + \)\(94\!\cdots\!46\)\( T^{4} - \)\(33\!\cdots\!84\)\( p^{10} T^{5} + 43393122038825346812 p^{20} T^{6} - 926711280 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
89 \( ( 1 - 3081298056 T + 77020954462551268316 T^{2} - \)\(35\!\cdots\!08\)\( T^{3} + \)\(28\!\cdots\!58\)\( T^{4} - \)\(35\!\cdots\!08\)\( p^{10} T^{5} + 77020954462551268316 p^{20} T^{6} - 3081298056 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
97 \( ( 1 + 4848931768 T + \)\(12\!\cdots\!96\)\( T^{2} + \)\(10\!\cdots\!08\)\( T^{3} + \)\(76\!\cdots\!90\)\( T^{4} + \)\(10\!\cdots\!08\)\( p^{10} T^{5} + \)\(12\!\cdots\!96\)\( p^{20} T^{6} + 4848931768 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
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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

−8.599711737628625016956262829656, −8.525140205100311971836864233712, −8.303289285099009083116945148472, −8.012064800682004586603855909817, −7.57975177776673887087490715174, −7.20427641006092238996840351375, −7.13880003209401711245068878383, −6.68746913752879866198755715672, −6.21673432889560886238719746609, −5.96685955414978732359088055394, −5.91629274648150166878305522810, −5.41344494067177331861266813527, −5.15701318819513387851534183383, −4.75474555238725476253197327701, −4.52544855981285131820693375679, −3.81354918231039245425319484914, −3.56545403582249143717818311979, −3.33854134247623960022608252835, −2.87954643047333169984286693771, −2.68311062320705470318713845194, −2.39897984384225522561967519549, −2.34563759868110248128275819356, −1.14962116667345822782600200498, −1.03419648119856735476422706847, −0.13185469860327418994369474580, 0.13185469860327418994369474580, 1.03419648119856735476422706847, 1.14962116667345822782600200498, 2.34563759868110248128275819356, 2.39897984384225522561967519549, 2.68311062320705470318713845194, 2.87954643047333169984286693771, 3.33854134247623960022608252835, 3.56545403582249143717818311979, 3.81354918231039245425319484914, 4.52544855981285131820693375679, 4.75474555238725476253197327701, 5.15701318819513387851534183383, 5.41344494067177331861266813527, 5.91629274648150166878305522810, 5.96685955414978732359088055394, 6.21673432889560886238719746609, 6.68746913752879866198755715672, 7.13880003209401711245068878383, 7.20427641006092238996840351375, 7.57975177776673887087490715174, 8.012064800682004586603855909817, 8.303289285099009083116945148472, 8.525140205100311971836864233712, 8.599711737628625016956262829656

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

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