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Properties

Label 24-567e12-1.1-c1e12-0-0
Degree $24$
Conductor $1.104\times 10^{33}$
Sign $1$
Analytic cond. $7.41860\times 10^{7}$
Root an. cond. $2.12779$
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  + 4·4-s + 4·7-s + 24·13-s + 2·16-s − 6·19-s + 3·25-s + 16·28-s + 8·37-s − 10·43-s − 16·49-s + 96·52-s − 6·64-s − 72·67-s − 42·73-s − 24·76-s + 72·79-s + 96·91-s + 60·97-s + 12·100-s + 30·103-s + 20·109-s + 8·112-s − 29·121-s + 127-s + 131-s − 24·133-s + 137-s + ⋯
L(s)  = 1  + 2·4-s + 1.51·7-s + 6.65·13-s + 1/2·16-s − 1.37·19-s + 3/5·25-s + 3.02·28-s + 1.31·37-s − 1.52·43-s − 2.28·49-s + 13.3·52-s − 3/4·64-s − 8.79·67-s − 4.91·73-s − 2.75·76-s + 8.10·79-s + 10.0·91-s + 6.09·97-s + 6/5·100-s + 2.95·103-s + 1.91·109-s + 0.755·112-s − 2.63·121-s + 0.0887·127-s + 0.0873·131-s − 2.08·133-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{48} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(7.41860\times 10^{7}\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{48} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4613696266\)
\(L(\frac12)\) \(\approx\) \(0.4613696266\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( ( 1 - 2 T + 2 p T^{2} - 23 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
good2 \( ( 1 - p T^{2} + 5 T^{4} - 11 T^{6} + 5 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} )^{2} \)
5 \( 1 - 3 T^{2} - 24 T^{4} + 11 p^{2} T^{6} - 297 T^{8} - 3096 T^{10} + 33369 T^{12} - 3096 p^{2} T^{14} - 297 p^{4} T^{16} + 11 p^{8} T^{18} - 24 p^{8} T^{20} - 3 p^{10} T^{22} + p^{12} T^{24} \)
11 \( 1 + 29 T^{2} + 40 p T^{4} + 3147 T^{6} - 10709 T^{8} - 688988 T^{10} - 10240799 T^{12} - 688988 p^{2} T^{14} - 10709 p^{4} T^{16} + 3147 p^{6} T^{18} + 40 p^{9} T^{20} + 29 p^{10} T^{22} + p^{12} T^{24} \)
13 \( ( 1 - 12 T + 90 T^{2} - 504 T^{3} + 2286 T^{4} - 9186 T^{5} + 34189 T^{6} - 9186 p T^{7} + 2286 p^{2} T^{8} - 504 p^{3} T^{9} + 90 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
17 \( 1 - 72 T^{2} + 2664 T^{4} - 74146 T^{6} + 1759536 T^{8} - 36331632 T^{10} + 658037283 T^{12} - 36331632 p^{2} T^{14} + 1759536 p^{4} T^{16} - 74146 p^{6} T^{18} + 2664 p^{8} T^{20} - 72 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 + 3 T + 36 T^{2} + 99 T^{3} + 549 T^{4} + 1788 T^{5} + 8413 T^{6} + 1788 p T^{7} + 549 p^{2} T^{8} + 99 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 44 T^{2} + 2132 T^{4} + 55302 T^{6} + 1548472 T^{8} + 27701944 T^{10} + 755610907 T^{12} + 27701944 p^{2} T^{14} + 1548472 p^{4} T^{16} + 55302 p^{6} T^{18} + 2132 p^{8} T^{20} + 44 p^{10} T^{22} + p^{12} T^{24} \)
29 \( 1 + 137 T^{2} + 10124 T^{4} + 545019 T^{6} + 23670379 T^{8} + 868368652 T^{10} + 27252519457 T^{12} + 868368652 p^{2} T^{14} + 23670379 p^{4} T^{16} + 545019 p^{6} T^{18} + 10124 p^{8} T^{20} + 137 p^{10} T^{22} + p^{12} T^{24} \)
31 \( ( 1 - 48 T^{2} + 3036 T^{4} - 90731 T^{6} + 3036 p^{2} T^{8} - 48 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 4 T - 76 T^{2} + 90 T^{3} + 112 p T^{4} + 688 T^{5} - 181325 T^{6} + 688 p T^{7} + 112 p^{3} T^{8} + 90 p^{3} T^{9} - 76 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 - 132 T^{2} + 8952 T^{4} - 341266 T^{6} + 6248844 T^{8} + 95204628 T^{10} - 9442249725 T^{12} + 95204628 p^{2} T^{14} + 6248844 p^{4} T^{16} - 341266 p^{6} T^{18} + 8952 p^{8} T^{20} - 132 p^{10} T^{22} + p^{12} T^{24} \)
43 \( ( 1 + 5 T - 88 T^{2} - 297 T^{3} + 5755 T^{4} + 9358 T^{5} - 250517 T^{6} + 9358 p T^{7} + 5755 p^{2} T^{8} - 297 p^{3} T^{9} - 88 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( ( 1 + 147 T^{2} + 11625 T^{4} + 631267 T^{6} + 11625 p^{2} T^{8} + 147 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( 1 + 200 T^{2} + 21164 T^{4} + 1425678 T^{6} + 67732372 T^{8} + 2466420796 T^{10} + 102771938515 T^{12} + 2466420796 p^{2} T^{14} + 67732372 p^{4} T^{16} + 1425678 p^{6} T^{18} + 21164 p^{8} T^{20} + 200 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 + 291 T^{2} + 38085 T^{4} + 2876659 T^{6} + 38085 p^{2} T^{8} + 291 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 237 T^{2} + 29262 T^{4} - 2210249 T^{6} + 29262 p^{2} T^{8} - 237 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
67 \( ( 1 + 18 T + 216 T^{2} + 1735 T^{3} + 216 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
71 \( ( 1 - 341 T^{2} + 53765 T^{4} - 4892609 T^{6} + 53765 p^{2} T^{8} - 341 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 + 21 T + 342 T^{2} + 4095 T^{3} + 40149 T^{4} + 343806 T^{5} + 3034951 T^{6} + 343806 p T^{7} + 40149 p^{2} T^{8} + 4095 p^{3} T^{9} + 342 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 18 T + 252 T^{2} - 2837 T^{3} + 252 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( 1 - 339 T^{2} + 58560 T^{4} - 7461013 T^{6} + 814401855 T^{8} - 81012917520 T^{10} + 7220222528217 T^{12} - 81012917520 p^{2} T^{14} + 814401855 p^{4} T^{16} - 7461013 p^{6} T^{18} + 58560 p^{8} T^{20} - 339 p^{10} T^{22} + p^{12} T^{24} \)
89 \( 1 - 333 T^{2} + 53955 T^{4} - 6596860 T^{6} + 732206169 T^{8} - 72458005383 T^{10} + 6566534648358 T^{12} - 72458005383 p^{2} T^{14} + 732206169 p^{4} T^{16} - 6596860 p^{6} T^{18} + 53955 p^{8} T^{20} - 333 p^{10} T^{22} + p^{12} T^{24} \)
97 \( ( 1 - 30 T + 639 T^{2} - 10170 T^{3} + 139518 T^{4} - 1666302 T^{5} + 17512459 T^{6} - 1666302 p T^{7} + 139518 p^{2} T^{8} - 10170 p^{3} T^{9} + 639 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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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

−3.59109970645125934694026590097, −3.57943658055821541884673707477, −3.50836004016527483745364906612, −3.37400383133774296846382590047, −3.00421069466786977787219453523, −2.99291149019441613862208943482, −2.86704358431968898285181267000, −2.80522787414609620158412769819, −2.60393662400688885071204849856, −2.57088406414319190033385292951, −2.49753528176611525203539484713, −2.49608697570097362326998558976, −2.02437243152652984721226303184, −2.01190230457740070590023664335, −1.93402863905953657590116285482, −1.77704648410311550017307190382, −1.68902822853762758718033597863, −1.53425412391842197804484969004, −1.38090385509708288300992902466, −1.32602118171480339976357806503, −1.32199534168324962904893241504, −1.14245127268700299734725678936, −0.863321666939330703535475038254, −0.69229145978918922377889589484, −0.03788619948403333798642383304, 0.03788619948403333798642383304, 0.69229145978918922377889589484, 0.863321666939330703535475038254, 1.14245127268700299734725678936, 1.32199534168324962904893241504, 1.32602118171480339976357806503, 1.38090385509708288300992902466, 1.53425412391842197804484969004, 1.68902822853762758718033597863, 1.77704648410311550017307190382, 1.93402863905953657590116285482, 2.01190230457740070590023664335, 2.02437243152652984721226303184, 2.49608697570097362326998558976, 2.49753528176611525203539484713, 2.57088406414319190033385292951, 2.60393662400688885071204849856, 2.80522787414609620158412769819, 2.86704358431968898285181267000, 2.99291149019441613862208943482, 3.00421069466786977787219453523, 3.37400383133774296846382590047, 3.50836004016527483745364906612, 3.57943658055821541884673707477, 3.59109970645125934694026590097

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

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