Properties

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

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 2·4-s − 6·5-s + 15·8-s − 18·10-s − 46·13-s + 45·16-s − 12·17-s − 12·20-s + 103·25-s − 138·26-s − 42·29-s + 6·32-s − 36·34-s + 56·37-s − 90·40-s − 84·41-s − 167·49-s + 309·50-s − 92·52-s + 72·53-s − 126·58-s − 34·61-s + 9·64-s + 276·65-s − 24·68-s + 116·73-s + ⋯
L(s)  = 1  + 3/2·2-s + 1/2·4-s − 6/5·5-s + 15/8·8-s − 9/5·10-s − 3.53·13-s + 2.81·16-s − 0.705·17-s − 3/5·20-s + 4.11·25-s − 5.30·26-s − 1.44·29-s + 3/16·32-s − 1.05·34-s + 1.51·37-s − 9/4·40-s − 2.04·41-s − 3.40·49-s + 6.17·50-s − 1.76·52-s + 1.35·53-s − 2.17·58-s − 0.557·61-s + 9/64·64-s + 4.24·65-s − 0.352·68-s + 1.58·73-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{32} \cdot 3^{48}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{32} \cdot 3^{48} ,\ ( \ : [1]^{16} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(5.05603\)
\(L(\frac12)\)  \(\approx\)  \(5.05603\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) 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 \( 1 - 3 T + 7 T^{2} - 15 p T^{3} + 19 p^{2} T^{4} - 9 p^{4} T^{5} + 53 p^{3} T^{6} - 57 p^{4} T^{7} + 97 p^{4} T^{8} - 57 p^{6} T^{9} + 53 p^{7} T^{10} - 9 p^{10} T^{11} + 19 p^{10} T^{12} - 15 p^{11} T^{13} + 7 p^{12} T^{14} - 3 p^{14} T^{15} + p^{16} T^{16} \)
3 \( 1 \)
good5 \( ( 1 + 3 T - 38 T^{2} + 201 T^{3} + 299 p T^{4} - 6984 T^{5} + 24112 T^{6} + 36276 p T^{7} - 866084 T^{8} + 36276 p^{3} T^{9} + 24112 p^{4} T^{10} - 6984 p^{6} T^{11} + 299 p^{9} T^{12} + 201 p^{10} T^{13} - 38 p^{12} T^{14} + 3 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
7 \( 1 + 167 T^{2} + 1884 p T^{4} + 76561 p T^{6} + 4595825 T^{8} - 715412784 T^{10} - 48284089874 T^{12} - 2073288609886 T^{14} - 88693520972328 T^{16} - 2073288609886 p^{4} T^{18} - 48284089874 p^{8} T^{20} - 715412784 p^{12} T^{22} + 4595825 p^{16} T^{24} + 76561 p^{21} T^{26} + 1884 p^{25} T^{28} + 167 p^{28} T^{30} + p^{32} T^{32} \)
11 \( 1 + 524 T^{2} + 140094 T^{4} + 24486904 T^{6} + 2972860745 T^{8} + 215973352008 T^{10} - 2384433396482 T^{12} - 3310557806176204 T^{14} - 541778612591523276 T^{16} - 3310557806176204 p^{4} T^{18} - 2384433396482 p^{8} T^{20} + 215973352008 p^{12} T^{22} + 2972860745 p^{16} T^{24} + 24486904 p^{20} T^{26} + 140094 p^{24} T^{28} + 524 p^{28} T^{30} + p^{32} T^{32} \)
13 \( ( 1 + 23 T - 276 T^{2} - 5009 T^{3} + 162641 T^{4} + 1572720 T^{5} - 34351730 T^{6} - 33339262 T^{7} + 8566506504 T^{8} - 33339262 p^{2} T^{9} - 34351730 p^{4} T^{10} + 1572720 p^{6} T^{11} + 162641 p^{8} T^{12} - 5009 p^{10} T^{13} - 276 p^{12} T^{14} + 23 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
17 \( ( 1 + 3 T + 2 p^{2} T^{2} + 6549 T^{3} + 169242 T^{4} + 6549 p^{2} T^{5} + 2 p^{6} T^{6} + 3 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
19 \( ( 1 - 1673 T^{2} + 1559890 T^{4} - 937716023 T^{6} + 401371470970 T^{8} - 937716023 p^{4} T^{10} + 1559890 p^{8} T^{12} - 1673 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
23 \( 1 + 2687 T^{2} + 161724 p T^{4} + 3461634655 T^{6} + 2442367009985 T^{8} + 1429403163693456 T^{10} + 759315200173466974 T^{12} + \)\(39\!\cdots\!18\)\( T^{14} + \)\(20\!\cdots\!24\)\( T^{16} + \)\(39\!\cdots\!18\)\( p^{4} T^{18} + 759315200173466974 p^{8} T^{20} + 1429403163693456 p^{12} T^{22} + 2442367009985 p^{16} T^{24} + 3461634655 p^{20} T^{26} + 161724 p^{25} T^{28} + 2687 p^{28} T^{30} + p^{32} T^{32} \)
29 \( ( 1 + 21 T - 2432 T^{2} - 19167 T^{3} + 4062037 T^{4} + 228384 p T^{5} - 4703569190 T^{6} - 3469324686 T^{7} + 4129311885376 T^{8} - 3469324686 p^{2} T^{9} - 4703569190 p^{4} T^{10} + 228384 p^{7} T^{11} + 4062037 p^{8} T^{12} - 19167 p^{10} T^{13} - 2432 p^{12} T^{14} + 21 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
31 \( 1 + 4715 T^{2} + 10729584 T^{4} + 17585852527 T^{6} + 25170831884477 T^{8} + 32193274685973408 T^{10} + 37136398859226646834 T^{12} + \)\(40\!\cdots\!98\)\( T^{14} + \)\(41\!\cdots\!28\)\( T^{16} + \)\(40\!\cdots\!98\)\( p^{4} T^{18} + 37136398859226646834 p^{8} T^{20} + 32193274685973408 p^{12} T^{22} + 25170831884477 p^{16} T^{24} + 17585852527 p^{20} T^{26} + 10729584 p^{24} T^{28} + 4715 p^{28} T^{30} + p^{32} T^{32} \)
37 \( ( 1 - 14 T + 3040 T^{2} - 66050 T^{3} + 123814 p T^{4} - 66050 p^{2} T^{5} + 3040 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
41 \( ( 1 + 42 T - 4424 T^{2} - 125040 T^{3} + 14943835 T^{4} + 232367040 T^{5} - 36025798940 T^{6} - 132403432026 T^{7} + 71285616374608 T^{8} - 132403432026 p^{2} T^{9} - 36025798940 p^{4} T^{10} + 232367040 p^{6} T^{11} + 14943835 p^{8} T^{12} - 125040 p^{10} T^{13} - 4424 p^{12} T^{14} + 42 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
43 \( 1 + 10292 T^{2} + 52819302 T^{4} + 202099368136 T^{6} + 665872758097265 T^{8} + 1877680374529631208 T^{10} + \)\(45\!\cdots\!90\)\( T^{12} + \)\(10\!\cdots\!64\)\( T^{14} + \)\(19\!\cdots\!28\)\( T^{16} + \)\(10\!\cdots\!64\)\( p^{4} T^{18} + \)\(45\!\cdots\!90\)\( p^{8} T^{20} + 1877680374529631208 p^{12} T^{22} + 665872758097265 p^{16} T^{24} + 202099368136 p^{20} T^{26} + 52819302 p^{24} T^{28} + 10292 p^{28} T^{30} + p^{32} T^{32} \)
47 \( 1 + 12983 T^{2} + 90223140 T^{4} + 428939892679 T^{6} + 33021025933951 p T^{8} + 4556649670813575888 T^{10} + \)\(11\!\cdots\!54\)\( T^{12} + \)\(26\!\cdots\!06\)\( T^{14} + \)\(58\!\cdots\!04\)\( T^{16} + \)\(26\!\cdots\!06\)\( p^{4} T^{18} + \)\(11\!\cdots\!54\)\( p^{8} T^{20} + 4556649670813575888 p^{12} T^{22} + 33021025933951 p^{17} T^{24} + 428939892679 p^{20} T^{26} + 90223140 p^{24} T^{28} + 12983 p^{28} T^{30} + p^{32} T^{32} \)
53 \( ( 1 - 18 T + 10016 T^{2} - 126558 T^{3} + 40472766 T^{4} - 126558 p^{2} T^{5} + 10016 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
59 \( 1 + 18092 T^{2} + 2981418 p T^{4} + 1133035156984 T^{6} + 5240167912552121 T^{8} + 17175194950853605128 T^{10} + \)\(35\!\cdots\!18\)\( T^{12} + \)\(21\!\cdots\!16\)\( T^{14} - \)\(83\!\cdots\!68\)\( T^{16} + \)\(21\!\cdots\!16\)\( p^{4} T^{18} + \)\(35\!\cdots\!18\)\( p^{8} T^{20} + 17175194950853605128 p^{12} T^{22} + 5240167912552121 p^{16} T^{24} + 1133035156984 p^{20} T^{26} + 2981418 p^{25} T^{28} + 18092 p^{28} T^{30} + p^{32} T^{32} \)
61 \( ( 1 + 17 T - 3582 T^{2} - 125549 T^{3} - 18305593 T^{4} - 350178696 T^{5} - 12273736208 T^{6} + 2060229775052 T^{7} + 599123836260396 T^{8} + 2060229775052 p^{2} T^{9} - 12273736208 p^{4} T^{10} - 350178696 p^{6} T^{11} - 18305593 p^{8} T^{12} - 125549 p^{10} T^{13} - 3582 p^{12} T^{14} + 17 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
67 \( 1 + 25148 T^{2} + 330341646 T^{4} + 3017899283800 T^{6} + 21559886998912025 T^{8} + \)\(12\!\cdots\!60\)\( T^{10} + \)\(66\!\cdots\!50\)\( T^{12} + \)\(31\!\cdots\!76\)\( T^{14} + \)\(14\!\cdots\!68\)\( T^{16} + \)\(31\!\cdots\!76\)\( p^{4} T^{18} + \)\(66\!\cdots\!50\)\( p^{8} T^{20} + \)\(12\!\cdots\!60\)\( p^{12} T^{22} + 21559886998912025 p^{16} T^{24} + 3017899283800 p^{20} T^{26} + 330341646 p^{24} T^{28} + 25148 p^{28} T^{30} + p^{32} T^{32} \)
71 \( ( 1 - 12968 T^{2} + 137492380 T^{4} - 912038324888 T^{6} + 5454839368725190 T^{8} - 912038324888 p^{4} T^{10} + 137492380 p^{8} T^{12} - 12968 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
73 \( ( 1 - 29 T + 7774 T^{2} - 620747 T^{3} + 46170850 T^{4} - 620747 p^{2} T^{5} + 7774 p^{4} T^{6} - 29 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
79 \( 1 + 293 p T^{2} + 328058736 T^{4} + 3136165224559 T^{6} + 21315067551811709 T^{8} + 91269506056145418912 T^{10} + \)\(59\!\cdots\!06\)\( T^{12} - \)\(27\!\cdots\!10\)\( T^{14} - \)\(25\!\cdots\!44\)\( T^{16} - \)\(27\!\cdots\!10\)\( p^{4} T^{18} + \)\(59\!\cdots\!06\)\( p^{8} T^{20} + 91269506056145418912 p^{12} T^{22} + 21315067551811709 p^{16} T^{24} + 3136165224559 p^{20} T^{26} + 328058736 p^{24} T^{28} + 293 p^{29} T^{30} + p^{32} T^{32} \)
83 \( 1 + 17795 T^{2} + 28452672 T^{4} - 239952438809 T^{6} + 12519030554664557 T^{8} + 90364909803409302048 T^{10} - \)\(32\!\cdots\!46\)\( T^{12} + \)\(11\!\cdots\!22\)\( T^{14} + \)\(52\!\cdots\!80\)\( T^{16} + \)\(11\!\cdots\!22\)\( p^{4} T^{18} - \)\(32\!\cdots\!46\)\( p^{8} T^{20} + 90364909803409302048 p^{12} T^{22} + 12519030554664557 p^{16} T^{24} - 239952438809 p^{20} T^{26} + 28452672 p^{24} T^{28} + 17795 p^{28} T^{30} + p^{32} T^{32} \)
89 \( ( 1 - 96 T + 27716 T^{2} - 1940016 T^{3} + 308634966 T^{4} - 1940016 p^{2} T^{5} + 27716 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
97 \( ( 1 + 74 T - 29868 T^{2} - 1540328 T^{3} + 607324823 T^{4} + 20751693936 T^{5} - 8122914917000 T^{6} - 74589814314322 T^{7} + 88320904907559480 T^{8} - 74589814314322 p^{2} T^{9} - 8122914917000 p^{4} T^{10} + 20751693936 p^{6} T^{11} + 607324823 p^{8} T^{12} - 1540328 p^{10} T^{13} - 29868 p^{12} T^{14} + 74 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.84709874097132258207358421828, −3.69949853456844443332037280636, −3.68954253217821728205388211764, −3.68131866756282995835750877444, −3.56555384082555777691372785635, −3.37673861890857071640482653232, −3.31536556979562851607850691726, −3.22520660201977415540696030214, −3.04559944863135219147735352834, −2.92745591367415112590191307043, −2.77561568027857382393465918018, −2.54290796454051181040245522256, −2.52533217853251119660899958403, −2.41955482269049163010347208014, −2.41419398628813141766989360955, −2.11785979711276322974283643624, −2.02593455528903372581063402357, −1.90088102603402957475671973919, −1.58243423292837904645637517620, −1.37514981626963821634744133611, −1.32359757136277887153429086983, −1.23891170162293392259847744849, −0.65264821412231623908511162464, −0.35416633113750459101995979611, −0.34330138188566151270547277990, 0.34330138188566151270547277990, 0.35416633113750459101995979611, 0.65264821412231623908511162464, 1.23891170162293392259847744849, 1.32359757136277887153429086983, 1.37514981626963821634744133611, 1.58243423292837904645637517620, 1.90088102603402957475671973919, 2.02593455528903372581063402357, 2.11785979711276322974283643624, 2.41419398628813141766989360955, 2.41955482269049163010347208014, 2.52533217853251119660899958403, 2.54290796454051181040245522256, 2.77561568027857382393465918018, 2.92745591367415112590191307043, 3.04559944863135219147735352834, 3.22520660201977415540696030214, 3.31536556979562851607850691726, 3.37673861890857071640482653232, 3.56555384082555777691372785635, 3.68131866756282995835750877444, 3.68954253217821728205388211764, 3.69949853456844443332037280636, 3.84709874097132258207358421828

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

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