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Properties

Label 24-756e12-1.1-c1e12-0-1
Degree $24$
Conductor $3.485\times 10^{34}$
Sign $1$
Analytic cond. $2.34199\times 10^{9}$
Root an. cond. $2.45696$
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  − 18·19-s + 15·25-s − 24·37-s − 12·43-s + 18·61-s + 54·73-s + 24·79-s + 18·103-s − 12·109-s − 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 4.12·19-s + 3·25-s − 3.94·37-s − 1.82·43-s + 2.30·61-s + 6.32·73-s + 2.70·79-s + 1.77·103-s − 1.14·109-s − 1.90·121-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 + 5.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{36} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{36} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{36} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(2.34199\times 10^{9}\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{36} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.028531140\)
\(L(\frac12)\) \(\approx\) \(2.028531140\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( ( 1 - p T^{3} + p^{3} T^{6} )^{2} \)
good5 \( 1 - 3 p T^{2} + 96 T^{4} - 409 T^{6} + 1971 T^{8} - 11052 T^{10} + 56769 T^{12} - 11052 p^{2} T^{14} + 1971 p^{4} T^{16} - 409 p^{6} T^{18} + 96 p^{8} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24} \)
11 \( 1 + 21 T^{2} + 120 T^{4} - 1085 T^{6} - 17973 T^{8} - 67788 T^{10} - 69951 T^{12} - 67788 p^{2} T^{14} - 17973 p^{4} T^{16} - 1085 p^{6} T^{18} + 120 p^{8} T^{20} + 21 p^{10} T^{22} + p^{12} T^{24} \)
13 \( ( 1 - 36 T^{2} + 792 T^{4} - 11495 T^{6} + 792 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 + 12 T^{2} - 624 T^{4} - 4882 T^{6} + 283284 T^{8} + 1089036 T^{10} - 84522621 T^{12} + 1089036 p^{2} T^{14} + 283284 p^{4} T^{16} - 4882 p^{6} T^{18} - 624 p^{8} T^{20} + 12 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 + 9 T + 72 T^{2} + 405 T^{3} + 117 p T^{4} + 12546 T^{5} + 56077 T^{6} + 12546 p T^{7} + 117 p^{3} T^{8} + 405 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 84 T^{2} + 3684 T^{4} + 103558 T^{6} + 2039616 T^{8} + 26787600 T^{10} + 378704571 T^{12} + 26787600 p^{2} T^{14} + 2039616 p^{4} T^{16} + 103558 p^{6} T^{18} + 3684 p^{8} T^{20} + 84 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 - 21 T^{2} + 2481 T^{4} - 34153 T^{6} + 2481 p^{2} T^{8} - 21 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 + 30 T^{2} - 30 T^{4} + 1890 T^{5} - 30359 T^{6} + 1890 p T^{7} - 30 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 + 12 T + 48 T^{2} + 130 T^{3} - 468 T^{4} - 19764 T^{5} - 175701 T^{6} - 19764 p T^{7} - 468 p^{2} T^{8} + 130 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( ( 1 + 60 T^{2} + 4836 T^{4} + 196567 T^{6} + 4836 p^{2} T^{8} + 60 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 + 3 T + 111 T^{2} + 245 T^{3} + 111 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
47 \( 1 - 267 T^{2} + 40920 T^{4} - 4343125 T^{6} + 352510407 T^{8} - 22655106144 T^{10} + 1180938336801 T^{12} - 22655106144 p^{2} T^{14} + 352510407 p^{4} T^{16} - 4343125 p^{6} T^{18} + 40920 p^{8} T^{20} - 267 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 84 T^{2} + 2136 T^{4} - 35042 T^{6} - 8443404 T^{8} - 556134516 T^{10} - 29094135357 T^{12} - 556134516 p^{2} T^{14} - 8443404 p^{4} T^{16} - 35042 p^{6} T^{18} + 2136 p^{8} T^{20} + 84 p^{10} T^{22} + p^{12} T^{24} \)
59 \( 1 - 51 T^{2} - 1380 T^{4} + 158351 T^{6} - 7224741 T^{8} + 76185540 T^{10} + 30324904161 T^{12} + 76185540 p^{2} T^{14} - 7224741 p^{4} T^{16} + 158351 p^{6} T^{18} - 1380 p^{8} T^{20} - 51 p^{10} T^{22} + p^{12} T^{24} \)
61 \( ( 1 - 9 T + 135 T^{2} - 972 T^{3} + 9153 T^{4} - 113283 T^{5} + 772702 T^{6} - 113283 p T^{7} + 9153 p^{2} T^{8} - 972 p^{3} T^{9} + 135 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 54 T^{2} - 994 T^{3} - 702 T^{4} + 26838 T^{5} + 653163 T^{6} + 26838 p T^{7} - 702 p^{2} T^{8} - 994 p^{3} T^{9} - 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 237 T^{2} + 29877 T^{4} - 2533201 T^{6} + 29877 p^{2} T^{8} - 237 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 27 T + 522 T^{2} - 7533 T^{3} + 94005 T^{4} - 988578 T^{5} + 9101239 T^{6} - 988578 p T^{7} + 94005 p^{2} T^{8} - 7533 p^{3} T^{9} + 522 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 12 T - 78 T^{2} + 374 T^{3} + 13518 T^{4} + 8802 T^{5} - 1498701 T^{6} + 8802 p T^{7} + 13518 p^{2} T^{8} + 374 p^{3} T^{9} - 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( ( 1 + 123 T^{2} + 12585 T^{4} + 1116115 T^{6} + 12585 p^{2} T^{8} + 123 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 151 T^{2} + 14880 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} )^{3} \)
97 \( ( 1 - 378 T^{2} + 63423 T^{4} - 7014476 T^{6} + 63423 p^{2} T^{8} - 378 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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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

−3.26653326307454291170080140409, −3.26028337373767865831689103333, −3.24323527392908842849655263182, −3.13198235629866008590152670894, −3.00231879372333296607393338905, −2.86760795743681764799666894702, −2.73507863402643284123443737634, −2.59877565754128017419934403976, −2.44246441702264429816000747619, −2.33133079289665050853339521607, −2.21158311085030236538801099781, −2.17384532701422359632799595367, −2.13990056285615274306882285352, −1.97200994478841536384863448490, −1.93752046551127407429621743716, −1.88371535646793331304621111520, −1.55550706521270161200005757834, −1.52411431677430727389562670707, −1.23827794568111362717939462634, −1.21405868163707343823062386748, −0.967623289355127524960642564385, −0.75448469511616770308968461905, −0.60587536992446970911642452948, −0.41593176743631246662813349842, −0.15856062597472183592407514565, 0.15856062597472183592407514565, 0.41593176743631246662813349842, 0.60587536992446970911642452948, 0.75448469511616770308968461905, 0.967623289355127524960642564385, 1.21405868163707343823062386748, 1.23827794568111362717939462634, 1.52411431677430727389562670707, 1.55550706521270161200005757834, 1.88371535646793331304621111520, 1.93752046551127407429621743716, 1.97200994478841536384863448490, 2.13990056285615274306882285352, 2.17384532701422359632799595367, 2.21158311085030236538801099781, 2.33133079289665050853339521607, 2.44246441702264429816000747619, 2.59877565754128017419934403976, 2.73507863402643284123443737634, 2.86760795743681764799666894702, 3.00231879372333296607393338905, 3.13198235629866008590152670894, 3.24323527392908842849655263182, 3.26028337373767865831689103333, 3.26653326307454291170080140409

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

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