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Properties

Label 24-2880e12-1.1-c1e12-0-1
Degree $24$
Conductor $3.256\times 10^{41}$
Sign $1$
Analytic cond. $2.18793\times 10^{16}$
Root an. cond. $4.79550$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s + 4·13-s − 8·25-s − 32·31-s + 20·37-s − 16·43-s + 32·49-s + 32·67-s + 44·73-s − 32·91-s + 28·97-s + 24·103-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 64·175-s + 179-s + 181-s + ⋯
L(s)  = 1  − 3.02·7-s + 1.10·13-s − 8/5·25-s − 5.74·31-s + 3.28·37-s − 2.43·43-s + 32/7·49-s + 3.90·67-s + 5.14·73-s − 3.35·91-s + 2.84·97-s + 2.36·103-s + 4.72·121-s + 0.0887·127-s + 0.0873·131-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 + 8/13·169-s + 0.0760·173-s + 4.83·175-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{72} \cdot 3^{24} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(2.18793\times 10^{16}\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{72} \cdot 3^{24} \cdot 5^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4308129935\)
\(L(\frac12)\) \(\approx\) \(0.4308129935\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + 8 T^{2} + 43 T^{4} + 272 T^{6} + 43 p^{2} T^{8} + 8 p^{4} T^{10} + p^{6} T^{12} \)
good7 \( ( 1 + 4 T + 8 T^{2} + 12 T^{3} - 13 T^{4} - 40 T^{5} + 16 T^{6} - 40 p T^{7} - 13 p^{2} T^{8} + 12 p^{3} T^{9} + 8 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
11 \( ( 1 - 26 T^{2} + 247 T^{4} - 1676 T^{6} + 247 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
13 \( ( 1 - 2 T + 2 T^{2} - 10 T^{3} + 87 T^{4} - 300 T^{5} + 476 T^{6} - 300 p T^{7} + 87 p^{2} T^{8} - 10 p^{3} T^{9} + 2 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
17 \( 1 + 198 T^{4} + 19919 T^{8} - 19177580 T^{12} + 19919 p^{4} T^{16} + 198 p^{8} T^{20} + p^{12} T^{24} \)
19 \( ( 1 - 50 T^{2} + 1063 T^{4} - 16988 T^{6} + 1063 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
23 \( 1 - 218 T^{4} - 18449 T^{8} + 47552596 T^{12} - 18449 p^{4} T^{16} - 218 p^{8} T^{20} + p^{12} T^{24} \)
29 \( ( 1 + 120 T^{2} + 6395 T^{4} + 217840 T^{6} + 6395 p^{2} T^{8} + 120 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 + 8 T + 77 T^{2} + 336 T^{3} + 77 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
37 \( ( 1 - 10 T + 50 T^{2} + 6 p T^{3} - 793 T^{4} - 14492 T^{5} + 209212 T^{6} - 14492 p T^{7} - 793 p^{2} T^{8} + 6 p^{4} T^{9} + 50 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( ( 1 - 176 T^{2} + 14387 T^{4} - 725856 T^{6} + 14387 p^{2} T^{8} - 176 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 + 8 T + 32 T^{2} - 296 T^{3} - 1365 T^{4} + 15888 T^{5} + 214592 T^{6} + 15888 p T^{7} - 1365 p^{2} T^{8} - 296 p^{3} T^{9} + 32 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( 1 - 12730 T^{4} + 68618447 T^{8} - 200477780844 T^{12} + 68618447 p^{4} T^{16} - 12730 p^{8} T^{20} + p^{12} T^{24} \)
53 \( 1 - 4650 T^{4} + 29489375 T^{8} - 74776815308 T^{12} + 29489375 p^{4} T^{16} - 4650 p^{8} T^{20} + p^{12} T^{24} \)
59 \( ( 1 - 70 T^{2} + 9175 T^{4} - 492340 T^{6} + 9175 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 71 T^{2} - 416 T^{3} + 71 p T^{4} + p^{3} T^{6} )^{4} \)
67 \( ( 1 - 16 T + 128 T^{2} - 1200 T^{3} + 5915 T^{4} - 1184 T^{5} - 18176 T^{6} - 1184 p T^{7} + 5915 p^{2} T^{8} - 1200 p^{3} T^{9} + 128 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 170 T^{2} + 19295 T^{4} - 1610700 T^{6} + 19295 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 22 T + 242 T^{2} - 2790 T^{3} + 34671 T^{4} - 322292 T^{5} + 2592092 T^{6} - 322292 p T^{7} + 34671 p^{2} T^{8} - 2790 p^{3} T^{9} + 242 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 58 T^{2} + 7471 T^{4} - 900844 T^{6} + 7471 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( 1 - 5450 T^{4} - 34445825 T^{8} + 679018810612 T^{12} - 34445825 p^{4} T^{16} - 5450 p^{8} T^{20} + p^{12} T^{24} \)
89 \( ( 1 + 208 T^{2} + 34003 T^{4} + 3269536 T^{6} + 34003 p^{2} T^{8} + 208 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( ( 1 - 14 T + 98 T^{2} - 1326 T^{3} - 6273 T^{4} + 214652 T^{5} - 1511236 T^{6} + 214652 p T^{7} - 6273 p^{2} T^{8} - 1326 p^{3} T^{9} + 98 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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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

−2.65753967236408114909206162340, −2.64810936833266384445415146235, −2.58060455316754494111709944550, −2.40367894770227961598272253597, −2.37911175976318953348095351973, −2.23813545977118386335821464596, −2.11418098606563546942470525430, −1.98562241982969328579993240174, −1.94753564935328089294227865358, −1.94177017996433306792390728558, −1.84851303185708790000077461775, −1.67682981630158909270522919179, −1.61489301129091999097055696163, −1.50903898606169923662101155155, −1.45392632527134597631906112261, −1.24538404620366401573694352498, −1.20401920139198979487071754777, −0.864014007397062373734065823046, −0.796451132832484406520535147245, −0.77866167873510901697282899600, −0.72952261331429580848836731549, −0.44583210892901465036861115012, −0.36393880354218971288828420471, −0.33635345645803958554736127058, −0.04633895723086844495443861016, 0.04633895723086844495443861016, 0.33635345645803958554736127058, 0.36393880354218971288828420471, 0.44583210892901465036861115012, 0.72952261331429580848836731549, 0.77866167873510901697282899600, 0.796451132832484406520535147245, 0.864014007397062373734065823046, 1.20401920139198979487071754777, 1.24538404620366401573694352498, 1.45392632527134597631906112261, 1.50903898606169923662101155155, 1.61489301129091999097055696163, 1.67682981630158909270522919179, 1.84851303185708790000077461775, 1.94177017996433306792390728558, 1.94753564935328089294227865358, 1.98562241982969328579993240174, 2.11418098606563546942470525430, 2.23813545977118386335821464596, 2.37911175976318953348095351973, 2.40367894770227961598272253597, 2.58060455316754494111709944550, 2.64810936833266384445415146235, 2.65753967236408114909206162340

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

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