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

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·4-s + 36·13-s + 24·16-s + 684·25-s + 516·37-s + 1.69e3·49-s − 216·52-s − 972·61-s + 344·64-s + 660·73-s + 2.53e3·97-s − 4.10e3·100-s − 1.17e3·109-s − 8.14e3·121-s + 127-s + 131-s + 137-s + 139-s − 3.09e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7.03e3·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 3/4·4-s + 0.768·13-s + 3/8·16-s + 5.47·25-s + 2.29·37-s + 4.95·49-s − 0.576·52-s − 2.04·61-s + 0.671·64-s + 1.05·73-s + 2.65·97-s − 4.10·100-s − 1.03·109-s − 6.12·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 1.71·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 3.20·169-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  =  \((24,\ 2^{24} \cdot 3^{36} ,\ ( \ : [3/2]^{12} ),\ 1 )\)
\(L(2)\)  \(\approx\)  \(12.1460\)
\(L(\frac12)\)  \(\approx\)  \(12.1460\)
\(L(\frac{5}{2})\)   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 \( 1 + 3 p T^{2} + 3 p^{2} T^{4} - 13 p^{5} T^{6} + 3 p^{8} T^{8} + 3 p^{13} T^{10} + p^{18} T^{12} \)
3 \( 1 \)
good5 \( ( 1 - 342 T^{2} + 2943 p^{2} T^{4} - 11181908 T^{6} + 2943 p^{8} T^{8} - 342 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
7 \( ( 1 - 849 T^{2} + 387966 T^{4} - 141880421 T^{6} + 387966 p^{6} T^{8} - 849 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
11 \( ( 1 + 4074 T^{2} + 757941 p T^{4} + 12032309932 T^{6} + 757941 p^{7} T^{8} + 4074 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
13 \( ( 1 - 9 T + 1962 T^{2} + 66427 T^{3} + 1962 p^{3} T^{4} - 9 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
17 \( ( 1 - 13134 T^{2} + 78982671 T^{4} - 357919940036 T^{6} + 78982671 p^{6} T^{8} - 13134 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
19 \( ( 1 - 18825 T^{2} + 202841862 T^{4} - 1656627349277 T^{6} + 202841862 p^{6} T^{8} - 18825 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
23 \( ( 1 + 6306 T^{2} + 110681535 T^{4} + 2149854095644 T^{6} + 110681535 p^{6} T^{8} + 6306 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
29 \( ( 1 - 1494 p T^{2} + 1075658103 T^{4} - 29470174509188 T^{6} + 1075658103 p^{6} T^{8} - 1494 p^{13} T^{10} + p^{18} T^{12} )^{2} \)
31 \( ( 1 - 153750 T^{2} + 10409070351 T^{4} - 400107422263412 T^{6} + 10409070351 p^{6} T^{8} - 153750 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
37 \( ( 1 - 129 T + 39378 T^{2} + 7527427 T^{3} + 39378 p^{3} T^{4} - 129 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
41 \( ( 1 - 232950 T^{2} + 25214641887 T^{4} - 1908694392304628 T^{6} + 25214641887 p^{6} T^{8} - 232950 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
43 \( ( 1 - 262254 T^{2} + 37283382183 T^{4} - 3613899564944228 T^{6} + 37283382183 p^{6} T^{8} - 262254 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
47 \( ( 1 + 179538 T^{2} + 22658226447 T^{4} + 3067316968586236 T^{6} + 22658226447 p^{6} T^{8} + 179538 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
53 \( ( 1 - 768750 T^{2} + 259538463591 T^{4} - 49798991602056548 T^{6} + 259538463591 p^{6} T^{8} - 768750 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
59 \( ( 1 + 523338 T^{2} + 180986988135 T^{4} + 40857650874211948 T^{6} + 180986988135 p^{6} T^{8} + 523338 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
61 \( ( 1 + 243 T + 648738 T^{2} + 106648063 T^{3} + 648738 p^{3} T^{4} + 243 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
67 \( ( 1 - 1637409 T^{2} + 1163514522486 T^{4} - 457988207385885221 T^{6} + 1163514522486 p^{6} T^{8} - 1637409 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
71 \( ( 1 + 377994 T^{2} + 13348121343 T^{4} - 1180461843095924 T^{6} + 13348121343 p^{6} T^{8} + 377994 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
73 \( ( 1 - 165 T + 834942 T^{2} - 185551841 T^{3} + 834942 p^{3} T^{4} - 165 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
79 \( ( 1 - 2108721 T^{2} + 2206739657838 T^{4} - 1369032710710548773 T^{6} + 2206739657838 p^{6} T^{8} - 2108721 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
83 \( ( 1 + 290850 T^{2} + 143373074583 T^{4} + 347024073827678524 T^{6} + 143373074583 p^{6} T^{8} + 290850 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
89 \( ( 1 - 1956702 T^{2} + 2321980636575 T^{4} - 1921108256399844452 T^{6} + 2321980636575 p^{6} T^{8} - 1956702 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
97 \( ( 1 - 633 T + 2697438 T^{2} - 1126776701 T^{3} + 2697438 p^{3} T^{4} - 633 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.28710094511918766906851660678, −4.23111620663900938470190089629, −4.14525769418205933127639072043, −4.07821942464880267484139673298, −3.88495086884369451655871156996, −3.84216745294029808907480061040, −3.52230809503140834792851672167, −3.34629416603008498227498878650, −3.29528533310982126948121642415, −3.22388443119844686436524485819, −2.88365945927751206353901038063, −2.86202390242230195024421659298, −2.67538019982293563439824287863, −2.53346962644923761559213104593, −2.34739037736927164646439130798, −2.34727550402845558925190920252, −2.13797049366818692189561971823, −1.61712412330441366289475659958, −1.42594036397854903215337505308, −1.29542689310078805923529202204, −1.17635159623305234534222665133, −0.814293681843886388339478191890, −0.70683793051078551320981593027, −0.64194721744895297822515307471, −0.28910281676697173826437296512, 0.28910281676697173826437296512, 0.64194721744895297822515307471, 0.70683793051078551320981593027, 0.814293681843886388339478191890, 1.17635159623305234534222665133, 1.29542689310078805923529202204, 1.42594036397854903215337505308, 1.61712412330441366289475659958, 2.13797049366818692189561971823, 2.34727550402845558925190920252, 2.34739037736927164646439130798, 2.53346962644923761559213104593, 2.67538019982293563439824287863, 2.86202390242230195024421659298, 2.88365945927751206353901038063, 3.22388443119844686436524485819, 3.29528533310982126948121642415, 3.34629416603008498227498878650, 3.52230809503140834792851672167, 3.84216745294029808907480061040, 3.88495086884369451655871156996, 4.07821942464880267484139673298, 4.14525769418205933127639072043, 4.23111620663900938470190089629, 4.28710094511918766906851660678

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

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