Properties

Degree 32
Conductor $ 2^{32} \cdot 3^{48} $
Sign $1$
Motivic weight 4
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·4-s + 176·13-s + 120·16-s − 3.63e3·25-s + 80·37-s + 1.52e4·49-s + 2.46e3·52-s − 1.64e3·61-s + 1.60e3·64-s + 80·73-s + 1.48e4·97-s − 5.08e4·100-s + 2.61e4·109-s + 1.32e5·121-s + 127-s + 131-s + 137-s + 139-s + 1.12e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.49e5·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 7/8·4-s + 1.04·13-s + 0.468·16-s − 5.81·25-s + 0.0584·37-s + 6.35·49-s + 0.911·52-s − 0.442·61-s + 0.392·64-s + 0.0150·73-s + 1.57·97-s − 5.08·100-s + 2.20·109-s + 9.08·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 0.0511·148-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 8.71·169-s + 3.34e−5·173-s + 3.12e−5·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{32} \cdot 3^{48}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{32} \cdot 3^{48} ,\ ( \ : [2]^{16} ),\ 1 )\)
\(L(\frac{5}{2})\)  \(\approx\)  \(15.5339\)
\(L(\frac12)\)  \(\approx\)  \(15.5339\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 - 7 p T^{2} + 19 p^{2} T^{4} - 31 p^{5} T^{6} - 11 p^{8} T^{8} - 31 p^{13} T^{10} + 19 p^{18} T^{12} - 7 p^{25} T^{14} + p^{32} T^{16} \)
3 \( 1 \)
good5 \( ( 1 + 1816 T^{2} + 1373884 T^{4} + 428961832 T^{6} + 50788147846 T^{8} + 428961832 p^{8} T^{10} + 1373884 p^{16} T^{12} + 1816 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
7 \( ( 1 - 7628 T^{2} + 4078294 p T^{4} - 90794922320 T^{6} + 5138396437651 p^{2} T^{8} - 90794922320 p^{8} T^{10} + 4078294 p^{17} T^{12} - 7628 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
11 \( ( 1 - 66488 T^{2} + 2449218172 T^{4} - 58905964133768 T^{6} + 1015821535494509830 T^{8} - 58905964133768 p^{8} T^{10} + 2449218172 p^{16} T^{12} - 66488 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
13 \( ( 1 - 44 T + 67090 T^{2} - 74336 p T^{3} + 2192145307 T^{4} - 74336 p^{5} T^{5} + 67090 p^{8} T^{6} - 44 p^{12} T^{7} + p^{16} T^{8} )^{4} \)
17 \( ( 1 + 422296 T^{2} + 87803746876 T^{4} + 11972376766816936 T^{6} + \)\(11\!\cdots\!30\)\( T^{8} + 11972376766816936 p^{8} T^{10} + 87803746876 p^{16} T^{12} + 422296 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
19 \( ( 1 - 298556 T^{2} + 58331731402 T^{4} - 10017587005489136 T^{6} + \)\(14\!\cdots\!39\)\( T^{8} - 10017587005489136 p^{8} T^{10} + 58331731402 p^{16} T^{12} - 298556 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
23 \( ( 1 - 1184696 T^{2} + 509745640444 T^{4} - 84604026939000968 T^{6} + \)\(71\!\cdots\!22\)\( T^{8} - 84604026939000968 p^{8} T^{10} + 509745640444 p^{16} T^{12} - 1184696 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
29 \( ( 1 + 1223560 T^{2} + 603752953756 T^{4} - 126384872678073416 T^{6} - \)\(34\!\cdots\!30\)\( T^{8} - 126384872678073416 p^{8} T^{10} + 603752953756 p^{16} T^{12} + 1223560 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
31 \( ( 1 - 4495112 T^{2} + 10589743228444 T^{4} - 16346503421024387384 T^{6} + \)\(17\!\cdots\!66\)\( T^{8} - 16346503421024387384 p^{8} T^{10} + 10589743228444 p^{16} T^{12} - 4495112 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
37 \( ( 1 - 20 T + 5679442 T^{2} - 534101024 T^{3} + 14604995159131 T^{4} - 534101024 p^{4} T^{5} + 5679442 p^{8} T^{6} - 20 p^{12} T^{7} + p^{16} T^{8} )^{4} \)
41 \( ( 1 + 8498440 T^{2} + 39457953574684 T^{4} + \)\(14\!\cdots\!68\)\( T^{6} + \)\(47\!\cdots\!42\)\( T^{8} + \)\(14\!\cdots\!68\)\( p^{8} T^{10} + 39457953574684 p^{16} T^{12} + 8498440 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
43 \( ( 1 - 16979720 T^{2} + 140301823474588 T^{4} - \)\(77\!\cdots\!28\)\( T^{6} + \)\(30\!\cdots\!14\)\( T^{8} - \)\(77\!\cdots\!28\)\( p^{8} T^{10} + 140301823474588 p^{16} T^{12} - 16979720 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
47 \( ( 1 - 11004728 T^{2} + 98757031754236 T^{4} - \)\(67\!\cdots\!20\)\( T^{6} + \)\(36\!\cdots\!06\)\( T^{8} - \)\(67\!\cdots\!20\)\( p^{8} T^{10} + 98757031754236 p^{16} T^{12} - 11004728 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
53 \( ( 1 + 25978312 T^{2} + 440128249758748 T^{4} + \)\(51\!\cdots\!52\)\( T^{6} + \)\(46\!\cdots\!86\)\( T^{8} + \)\(51\!\cdots\!52\)\( p^{8} T^{10} + 440128249758748 p^{16} T^{12} + 25978312 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
59 \( ( 1 - 31826744 T^{2} + 538590143907196 T^{4} - \)\(92\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!74\)\( T^{8} - \)\(92\!\cdots\!92\)\( p^{8} T^{10} + 538590143907196 p^{16} T^{12} - 31826744 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
61 \( ( 1 + 412 T + 9607138 T^{2} + 3267599488 T^{3} + 216368249231659 T^{4} + 3267599488 p^{4} T^{5} + 9607138 p^{8} T^{6} + 412 p^{12} T^{7} + p^{16} T^{8} )^{4} \)
67 \( ( 1 - 38213180 T^{2} + 1702311028453834 T^{4} - \)\(38\!\cdots\!28\)\( T^{6} + \)\(99\!\cdots\!27\)\( T^{8} - \)\(38\!\cdots\!28\)\( p^{8} T^{10} + 1702311028453834 p^{16} T^{12} - 38213180 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
71 \( ( 1 - 102214088 T^{2} + 3903098982427036 T^{4} - \)\(62\!\cdots\!96\)\( T^{6} + \)\(63\!\cdots\!58\)\( T^{8} - \)\(62\!\cdots\!96\)\( p^{8} T^{10} + 3903098982427036 p^{16} T^{12} - 102214088 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
73 \( ( 1 - 20 T + 67167226 T^{2} + 662878096 T^{3} + 2724353301452611 T^{4} + 662878096 p^{4} T^{5} + 67167226 p^{8} T^{6} - 20 p^{12} T^{7} + p^{16} T^{8} )^{4} \)
79 \( ( 1 - 164533964 T^{2} + 15212503116962266 T^{4} - \)\(95\!\cdots\!24\)\( T^{6} + \)\(43\!\cdots\!71\)\( T^{8} - \)\(95\!\cdots\!24\)\( p^{8} T^{10} + 15212503116962266 p^{16} T^{12} - 164533964 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
83 \( ( 1 - 185955656 T^{2} + 18958921421058076 T^{4} - \)\(13\!\cdots\!16\)\( T^{6} + \)\(73\!\cdots\!26\)\( T^{8} - \)\(13\!\cdots\!16\)\( p^{8} T^{10} + 18958921421058076 p^{16} T^{12} - 185955656 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
89 \( ( 1 + 103821208 T^{2} + 12140769019811644 T^{4} + \)\(81\!\cdots\!28\)\( T^{6} + \)\(66\!\cdots\!06\)\( T^{8} + \)\(81\!\cdots\!28\)\( p^{8} T^{10} + 12140769019811644 p^{16} T^{12} + 103821208 p^{24} T^{14} + p^{32} T^{16} )^{2} \)
97 \( ( 1 - 3716 T + 259176202 T^{2} - 799969063952 T^{3} + 30273937496959507 T^{4} - 799969063952 p^{4} T^{5} + 259176202 p^{8} T^{6} - 3716 p^{12} T^{7} + p^{16} T^{8} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.42752029894637123707944932934, −3.41761799747451218949114043043, −2.93915408115188284564109645471, −2.88197797315220148751088093545, −2.67438788513312079015128622003, −2.64578806305633275587880393088, −2.56509818798426279678375315515, −2.53824696834225361095194473345, −2.29155572848525603302774112394, −2.21436778448499811793107552268, −2.13564889093580539723554982946, −1.91884358153253504944754350470, −1.84783307854502259272649352812, −1.76588040584694720021243693412, −1.72864674593953113121492134860, −1.67074873845682746378983813636, −1.22252485037696754735893871362, −1.15237511634133615653486080100, −0.990339888930384510456613000715, −0.942245181934765983894894160047, −0.69920590090473968277336336871, −0.55839093073197238344604256784, −0.43565287824892026175009391459, −0.28950389968241938465870342288, −0.17217281424127234926720816060, 0.17217281424127234926720816060, 0.28950389968241938465870342288, 0.43565287824892026175009391459, 0.55839093073197238344604256784, 0.69920590090473968277336336871, 0.942245181934765983894894160047, 0.990339888930384510456613000715, 1.15237511634133615653486080100, 1.22252485037696754735893871362, 1.67074873845682746378983813636, 1.72864674593953113121492134860, 1.76588040584694720021243693412, 1.84783307854502259272649352812, 1.91884358153253504944754350470, 2.13564889093580539723554982946, 2.21436778448499811793107552268, 2.29155572848525603302774112394, 2.53824696834225361095194473345, 2.56509818798426279678375315515, 2.64578806305633275587880393088, 2.67438788513312079015128622003, 2.88197797315220148751088093545, 2.93915408115188284564109645471, 3.41761799747451218949114043043, 3.42752029894637123707944932934

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

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