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Properties

Label 24-12e36-1.1-c3e12-0-1
Degree $24$
Conductor $7.088\times 10^{38}$
Sign $1$
Analytic cond. $1.26158\times 10^{24}$
Root an. cond. $10.0972$
Motivic weight $3$
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  − 36·13-s + 684·25-s − 516·37-s + 1.69e3·49-s + 972·61-s + 660·73-s + 2.53e3·97-s + 1.17e3·109-s − 8.14e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7.03e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 0.768·13-s + 5.47·25-s − 2.29·37-s + 4.95·49-s + 2.04·61-s + 1.05·73-s + 2.65·97-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 + 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 + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{72} \cdot 3^{36}\)
Sign: $1$
Analytic conductor: \(1.26158\times 10^{24}\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{72} \cdot 3^{36} ,\ ( \ : [3/2]^{12} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(46.24097535\)
\(L(\frac12)\) \(\approx\) \(46.24097535\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.48974629328439464553798176509, −2.41551052048171527004548509925, −2.31426223490808048615016910313, −2.31416328275322601864253895592, −2.17399040977485203679645840013, −2.04976790867658333803546510200, −2.04636545459466938906964684763, −1.75778430987209610408677528963, −1.66158833478226911071082036854, −1.65464899310556480254622398065, −1.61893284153425038131109370139, −1.38415736317922311881990103554, −1.31442067099428751059591912369, −1.23517694999594514150464862171, −1.02324376577340074732337762254, −1.02077146254926050778930588093, −0.987975044392780657836430929875, −0.908370860573952625871299080967, −0.73422078957385407897000041060, −0.62513428622253896347378804170, −0.52512812925074269123484935662, −0.39104468582357244251784859228, −0.29751181847538410347542223918, −0.28915285894052177901049157177, −0.15091497936038621598552706081, 0.15091497936038621598552706081, 0.28915285894052177901049157177, 0.29751181847538410347542223918, 0.39104468582357244251784859228, 0.52512812925074269123484935662, 0.62513428622253896347378804170, 0.73422078957385407897000041060, 0.908370860573952625871299080967, 0.987975044392780657836430929875, 1.02077146254926050778930588093, 1.02324376577340074732337762254, 1.23517694999594514150464862171, 1.31442067099428751059591912369, 1.38415736317922311881990103554, 1.61893284153425038131109370139, 1.65464899310556480254622398065, 1.66158833478226911071082036854, 1.75778430987209610408677528963, 2.04636545459466938906964684763, 2.04976790867658333803546510200, 2.17399040977485203679645840013, 2.31416328275322601864253895592, 2.31426223490808048615016910313, 2.41551052048171527004548509925, 2.48974629328439464553798176509

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

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