Properties

Degree 24
Conductor $ 2^{24} \cdot 3^{36} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 72·13-s + 33·16-s + 558·25-s − 240·37-s + 2.20e3·49-s − 216·52-s + 144·61-s − 421·64-s + 156·73-s + 516·97-s + 1.67e3·100-s + 3.21e3·109-s − 6.11e3·121-s + 127-s + 131-s + 137-s + 139-s − 720·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.24e4·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 3/8·4-s − 1.53·13-s + 0.515·16-s + 4.46·25-s − 1.06·37-s + 6.41·49-s − 0.576·52-s + 0.302·61-s − 0.822·64-s + 0.250·73-s + 0.540·97-s + 1.67·100-s + 2.82·109-s − 4.59·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.399·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 5.65·169-s + 0.000439·173-s + 0.000417·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{24} \cdot 3^{36}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((24,\ 2^{24} \cdot 3^{36} ,\ ( \ : [3/2]^{12} ),\ 1 )\)
\(L(2)\)  \(\approx\)  \(9.64284\)
\(L(\frac12)\)  \(\approx\)  \(9.64284\)
\(L(\frac{5}{2})\)   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 24. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad2 \( 1 - 3 T^{2} - 3 p^{3} T^{4} + 37 p^{4} T^{6} - 3 p^{9} T^{8} - 3 p^{12} T^{10} + p^{18} T^{12} \)
3 \( 1 \)
good5 \( ( 1 - 279 T^{2} + 14742 T^{4} + 9281 p^{3} T^{6} + 14742 p^{6} T^{8} - 279 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
7 \( ( 1 - 1101 T^{2} + 652854 T^{4} - 259162985 T^{6} + 652854 p^{6} T^{8} - 1101 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
11 \( ( 1 + 3057 T^{2} + 6556998 T^{4} + 9226981429 T^{6} + 6556998 p^{6} T^{8} + 3057 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
13 \( ( 1 + 18 T + 3915 T^{2} + 8332 T^{3} + 3915 p^{3} T^{4} + 18 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
17 \( ( 1 - 11946 T^{2} + 72737391 T^{4} - 332667352460 T^{6} + 72737391 p^{6} T^{8} - 11946 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
19 \( ( 1 - 438 p T^{2} + 146058135 T^{4} - 772095813500 T^{6} + 146058135 p^{6} T^{8} - 438 p^{13} T^{10} + p^{18} T^{12} )^{2} \)
23 \( ( 1 + 52314 T^{2} + 1249837599 T^{4} + 18511268917228 T^{6} + 1249837599 p^{6} T^{8} + 52314 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
29 \( ( 1 - 117954 T^{2} + 6353615223 T^{4} - 198355607069564 T^{6} + 6353615223 p^{6} T^{8} - 117954 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
31 \( ( 1 - 45741 T^{2} + 1957054854 T^{4} - 51110094297449 T^{6} + 1957054854 p^{6} T^{8} - 45741 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
37 \( ( 1 + 60 T + 83559 T^{2} - 2089640 T^{3} + 83559 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
41 \( ( 1 - 189606 T^{2} + 21529296543 T^{4} - 1820253257865236 T^{6} + 21529296543 p^{6} T^{8} - 189606 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
43 \( ( 1 + 27042 T^{2} + 15896713575 T^{4} + 267144985943548 T^{6} + 15896713575 p^{6} T^{8} + 27042 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
47 \( ( 1 + 377142 T^{2} + 74876518767 T^{4} + 9593174285133748 T^{6} + 74876518767 p^{6} T^{8} + 377142 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
53 \( ( 1 - 398679 T^{2} + 82394238966 T^{4} - 238475483639575 p T^{6} + 82394238966 p^{6} T^{8} - 398679 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
59 \( ( 1 + 275982 T^{2} + 30860919687 T^{4} - 3157264383153500 T^{6} + 30860919687 p^{6} T^{8} + 275982 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
61 \( ( 1 - 36 T + 351135 T^{2} - 84569384 T^{3} + 351135 p^{3} T^{4} - 36 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
67 \( ( 1 - 1341390 T^{2} + 846750752247 T^{4} - 319903491280410596 T^{6} + 846750752247 p^{6} T^{8} - 1341390 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
71 \( ( 1 + 1203654 T^{2} + 624403431231 T^{4} + 229814346307125076 T^{6} + 624403431231 p^{6} T^{8} + 1203654 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
73 \( ( 1 - 39 T + 955110 T^{2} - 15631571 T^{3} + 955110 p^{3} T^{4} - 39 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
79 \( ( 1 - 1716006 T^{2} + 1350264931311 T^{4} - 731914440695115860 T^{6} + 1350264931311 p^{6} T^{8} - 1716006 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
83 \( ( 1 + 1600737 T^{2} + 1256006483670 T^{4} + 724491266389094437 T^{6} + 1256006483670 p^{6} T^{8} + 1600737 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
89 \( ( 1 + 123846 T^{2} + 872233472127 T^{4} - 77031279330160556 T^{6} + 872233472127 p^{6} T^{8} + 123846 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
97 \( ( 1 - 129 T + 1814286 T^{2} - 533624789 T^{3} + 1814286 p^{3} T^{4} - 129 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.51604350393361402217808887671, −4.08720074572229285262052802827, −4.07148051659690706220469174416, −3.91012189583593365728573490993, −3.90722803729044424184236502567, −3.84583490796172822610150843420, −3.57949107565351646971696693997, −3.38190806725150980036246672811, −3.28983459568688344859573850367, −3.11382846557793374289897324461, −2.82343097697133271625410694667, −2.80286244317110914645129734090, −2.62597603549689278839354057073, −2.51266461732731790883521016926, −2.40815448797661654392573465023, −2.33707485222832699916042660376, −2.12991481930781626514433842052, −1.75961764882611012292001948525, −1.45882833618538772919744254467, −1.40338959761437196051778358859, −1.11507430563042471571746425774, −0.926458814277290342799473864912, −0.78317034622281289097070280781, −0.40363863507521334178840110270, −0.30432762571761501793149754270, 0.30432762571761501793149754270, 0.40363863507521334178840110270, 0.78317034622281289097070280781, 0.926458814277290342799473864912, 1.11507430563042471571746425774, 1.40338959761437196051778358859, 1.45882833618538772919744254467, 1.75961764882611012292001948525, 2.12991481930781626514433842052, 2.33707485222832699916042660376, 2.40815448797661654392573465023, 2.51266461732731790883521016926, 2.62597603549689278839354057073, 2.80286244317110914645129734090, 2.82343097697133271625410694667, 3.11382846557793374289897324461, 3.28983459568688344859573850367, 3.38190806725150980036246672811, 3.57949107565351646971696693997, 3.84583490796172822610150843420, 3.90722803729044424184236502567, 3.91012189583593365728573490993, 4.07148051659690706220469174416, 4.08720074572229285262052802827, 4.51604350393361402217808887671

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

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