Properties

Label 24-78e12-1.1-c1e12-0-0
Degree $24$
Conductor $5.071\times 10^{22}$
Sign $1$
Analytic cond. $0.00340769$
Root an. cond. $0.789197$
Motivic weight $1$
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  − 12·7-s − 3·16-s − 12·19-s − 12·27-s + 12·31-s + 12·37-s + 72·49-s − 12·67-s − 12·73-s + 72·79-s − 60·97-s + 60·109-s + 36·112-s + 127-s + 131-s + 144·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 4.53·7-s − 3/4·16-s − 2.75·19-s − 2.30·27-s + 2.15·31-s + 1.97·37-s + 72/7·49-s − 1.46·67-s − 1.40·73-s + 8.10·79-s − 6.09·97-s + 5.74·109-s + 3.40·112-s + 0.0887·127-s + 0.0873·131-s + 12.4·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{12} \cdot 3^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(0.00340769\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{12} \cdot 3^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1822824465\)
\(L(\frac12)\) \(\approx\) \(0.1822824465\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{3} \)
3 \( ( 1 + 2 p T^{3} + p^{3} T^{6} )^{2} \)
13 \( ( 1 + 21 T^{2} + 24 T^{3} + 21 p T^{4} + p^{3} T^{6} )^{2} \)
good5 \( 1 - 48 T^{4} + 48 p T^{8} + 17426 T^{12} + 48 p^{5} T^{16} - 48 p^{8} T^{20} + p^{12} T^{24} \)
7 \( ( 1 + 6 T + 18 T^{2} + 50 T^{3} + 72 T^{4} - 6 T^{5} - 82 T^{6} - 6 p T^{7} + 72 p^{2} T^{8} + 50 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
11 \( 1 - 210 T^{4} + 20175 T^{8} - 2226076 T^{12} + 20175 p^{4} T^{16} - 210 p^{8} T^{20} + p^{12} T^{24} \)
17 \( ( 1 + 48 T^{2} + 1392 T^{4} + 28258 T^{6} + 1392 p^{2} T^{8} + 48 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 + 6 T + 18 T^{2} + 98 T^{3} + 315 T^{4} + 444 T^{5} + 1796 T^{6} + 444 p T^{7} + 315 p^{2} T^{8} + 98 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( ( 1 + 102 T^{2} + 4947 T^{4} + 143692 T^{6} + 4947 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - 138 T^{2} + 8763 T^{4} - 324628 T^{6} + 8763 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - 6 T + 18 T^{2} + 70 T^{3} - 765 T^{4} - 1164 T^{5} + 23204 T^{6} - 1164 p T^{7} - 765 p^{2} T^{8} + 70 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 6 T + 18 T^{2} - 194 T^{3} + 1440 T^{4} - 2946 T^{5} + 10574 T^{6} - 2946 p T^{7} + 1440 p^{2} T^{8} - 194 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( ( 1 - 24 T + 288 T^{2} - 1880 T^{3} + 3219 T^{4} + 62544 T^{5} - 660928 T^{6} + 62544 p T^{7} + 3219 p^{2} T^{8} - 1880 p^{3} T^{9} + 288 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )( 1 + 24 T + 288 T^{2} + 1880 T^{3} + 3219 T^{4} - 62544 T^{5} - 660928 T^{6} - 62544 p T^{7} + 3219 p^{2} T^{8} + 1880 p^{3} T^{9} + 288 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} ) \)
43 \( ( 1 - 96 T^{2} + 6432 T^{4} - 356834 T^{6} + 6432 p^{2} T^{8} - 96 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( 1 - 1488 T^{4} - 3298512 T^{8} + 19035456770 T^{12} - 3298512 p^{4} T^{16} - 1488 p^{8} T^{20} + p^{12} T^{24} \)
53 \( ( 1 + 6 T^{2} - 2037 T^{4} - 114388 T^{6} - 2037 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( 1 + 6486 T^{4} + 34394271 T^{8} + 144883123508 T^{12} + 34394271 p^{4} T^{16} + 6486 p^{8} T^{20} + p^{12} T^{24} \)
61 \( ( 1 + 111 T^{2} + 192 T^{3} + 111 p T^{4} + p^{3} T^{6} )^{4} \)
67 \( ( 1 + 6 T + 18 T^{2} + 194 T^{3} + 6327 T^{4} + 46596 T^{5} + 184508 T^{6} + 46596 p T^{7} + 6327 p^{2} T^{8} + 194 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( 1 - 1776 T^{4} - 14518128 T^{8} + 198423496802 T^{12} - 14518128 p^{4} T^{16} - 1776 p^{8} T^{20} + p^{12} T^{24} \)
73 \( ( 1 + 6 T + 18 T^{2} + 422 T^{3} + 12627 T^{4} + 47100 T^{5} + 144356 T^{6} + 47100 p T^{7} + 12627 p^{2} T^{8} + 422 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 18 T + 291 T^{2} - 2832 T^{3} + 291 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( 1 - 11730 T^{4} - 16328721 T^{8} + 684976092644 T^{12} - 16328721 p^{4} T^{16} - 11730 p^{8} T^{20} + p^{12} T^{24} \)
89 \( 1 - 10650 T^{4} + 39050799 T^{8} - 164591505196 T^{12} + 39050799 p^{4} T^{16} - 10650 p^{8} T^{20} + p^{12} T^{24} \)
97 \( ( 1 + 30 T + 450 T^{2} + 5278 T^{3} + 64575 T^{4} + 781188 T^{5} + 8305532 T^{6} + 781188 p T^{7} + 64575 p^{2} T^{8} + 5278 p^{3} T^{9} + 450 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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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

−5.59306071394779687634853653645, −5.38877139318244346406308799767, −5.19456813851508398182693589493, −5.09243984126932801547268944305, −4.85920092912969207655615249622, −4.67785942395056560720689591604, −4.56914739323237194620808803590, −4.40623838732767721349959991332, −4.39666394941560438579021476071, −4.28252569387665086331582308519, −3.85016665786878510166097608901, −3.74591266769627696479587624501, −3.70342917805571943037463869945, −3.66170171373167405329985139525, −3.64454303557230889729777468219, −3.35316169535355541291048359399, −2.98354879085718255505348241214, −2.77631224388767136300601664668, −2.73922884705104817990920566359, −2.63780849197529667581591007760, −2.58061358722273534316177629584, −2.16294521231970995815487519377, −1.85229661636046153072729210994, −1.78387708746019777554034924115, −0.65791028884537277568242105239, 0.65791028884537277568242105239, 1.78387708746019777554034924115, 1.85229661636046153072729210994, 2.16294521231970995815487519377, 2.58061358722273534316177629584, 2.63780849197529667581591007760, 2.73922884705104817990920566359, 2.77631224388767136300601664668, 2.98354879085718255505348241214, 3.35316169535355541291048359399, 3.64454303557230889729777468219, 3.66170171373167405329985139525, 3.70342917805571943037463869945, 3.74591266769627696479587624501, 3.85016665786878510166097608901, 4.28252569387665086331582308519, 4.39666394941560438579021476071, 4.40623838732767721349959991332, 4.56914739323237194620808803590, 4.67785942395056560720689591604, 4.85920092912969207655615249622, 5.09243984126932801547268944305, 5.19456813851508398182693589493, 5.38877139318244346406308799767, 5.59306071394779687634853653645

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

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