Properties

Degree 24
Conductor $ 3^{24} \cdot 7^{12} \cdot 11^{24} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 12

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 12·7-s − 20·13-s − 16-s − 12·19-s − 14·25-s + 24·28-s − 16·31-s + 4·37-s − 16·43-s + 78·49-s + 40·52-s − 44·61-s + 8·64-s − 52·73-s + 24·76-s + 8·79-s + 240·91-s − 32·97-s + 28·100-s − 28·103-s − 40·109-s + 12·112-s + 32·124-s + 127-s + 131-s + 144·133-s + ⋯
L(s)  = 1  − 4-s − 4.53·7-s − 5.54·13-s − 1/4·16-s − 2.75·19-s − 2.79·25-s + 4.53·28-s − 2.87·31-s + 0.657·37-s − 2.43·43-s + 78/7·49-s + 5.54·52-s − 5.63·61-s + 64-s − 6.08·73-s + 2.75·76-s + 0.900·79-s + 25.1·91-s − 3.24·97-s + 14/5·100-s − 2.75·103-s − 3.83·109-s + 1.13·112-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 12.4·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(3^{24} \cdot 7^{12} \cdot 11^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  12
Selberg data  =  $(24,\ 3^{24} \cdot 7^{12} \cdot 11^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 24. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( ( 1 + T )^{12} \)
11 \( 1 \)
good2 \( 1 + p T^{2} + 5 T^{4} + p^{2} T^{6} + p^{3} T^{8} - 3 p^{3} T^{10} - 3 p^{2} T^{12} - 3 p^{5} T^{14} + p^{7} T^{16} + p^{8} T^{18} + 5 p^{8} T^{20} + p^{11} T^{22} + p^{12} T^{24} \)
5 \( 1 + 14 T^{2} + 101 T^{4} + 98 p T^{6} + 2846 T^{8} + 20574 T^{10} + 123321 T^{12} + 20574 p^{2} T^{14} + 2846 p^{4} T^{16} + 98 p^{7} T^{18} + 101 p^{8} T^{20} + 14 p^{10} T^{22} + p^{12} T^{24} \)
13 \( ( 1 + 10 T + 83 T^{2} + 500 T^{3} + 2621 T^{4} + 870 p T^{5} + 44436 T^{6} + 870 p^{2} T^{7} + 2621 p^{2} T^{8} + 500 p^{3} T^{9} + 83 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
17 \( 1 + 90 T^{2} + 4261 T^{4} + 146590 T^{6} + 4020830 T^{8} + 89661290 T^{10} + 1660695689 T^{12} + 89661290 p^{2} T^{14} + 4020830 p^{4} T^{16} + 146590 p^{6} T^{18} + 4261 p^{8} T^{20} + 90 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 + 6 T + 106 T^{2} + 530 T^{3} + 4853 T^{4} + 19396 T^{5} + 121592 T^{6} + 19396 p T^{7} + 4853 p^{2} T^{8} + 530 p^{3} T^{9} + 106 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 168 T^{2} + 13726 T^{4} + 723160 T^{6} + 27775475 T^{8} + 839711408 T^{10} + 21022991420 T^{12} + 839711408 p^{2} T^{14} + 27775475 p^{4} T^{16} + 723160 p^{6} T^{18} + 13726 p^{8} T^{20} + 168 p^{10} T^{22} + p^{12} T^{24} \)
29 \( 1 + 158 T^{2} + 12523 T^{4} + 676610 T^{6} + 28938587 T^{8} + 1051503368 T^{10} + 32911943606 T^{12} + 1051503368 p^{2} T^{14} + 28938587 p^{4} T^{16} + 676610 p^{6} T^{18} + 12523 p^{8} T^{20} + 158 p^{10} T^{22} + p^{12} T^{24} \)
31 \( ( 1 + 8 T + 110 T^{2} + 688 T^{3} + 4823 T^{4} + 26760 T^{5} + 147540 T^{6} + 26760 p T^{7} + 4823 p^{2} T^{8} + 688 p^{3} T^{9} + 110 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 2 T + 91 T^{2} - 314 T^{3} + 5891 T^{4} - 17852 T^{5} + 259334 T^{6} - 17852 p T^{7} + 5891 p^{2} T^{8} - 314 p^{3} T^{9} + 91 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 + 178 T^{2} + 13639 T^{4} + 560026 T^{6} + 12799187 T^{8} + 170859508 T^{10} + 2999880794 T^{12} + 170859508 p^{2} T^{14} + 12799187 p^{4} T^{16} + 560026 p^{6} T^{18} + 13639 p^{8} T^{20} + 178 p^{10} T^{22} + p^{12} T^{24} \)
43 \( ( 1 + 8 T + 152 T^{2} + 1072 T^{3} + 11384 T^{4} + 75288 T^{5} + 582042 T^{6} + 75288 p T^{7} + 11384 p^{2} T^{8} + 1072 p^{3} T^{9} + 152 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( 1 + 332 T^{2} + 57032 T^{4} + 6587500 T^{6} + 563615924 T^{8} + 37403435100 T^{10} + 1967941380078 T^{12} + 37403435100 p^{2} T^{14} + 563615924 p^{4} T^{16} + 6587500 p^{6} T^{18} + 57032 p^{8} T^{20} + 332 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 382 T^{2} + 66487 T^{4} + 7011646 T^{6} + 511558763 T^{8} + 29260051156 T^{10} + 1541317998842 T^{12} + 29260051156 p^{2} T^{14} + 511558763 p^{4} T^{16} + 7011646 p^{6} T^{18} + 66487 p^{8} T^{20} + 382 p^{10} T^{22} + p^{12} T^{24} \)
59 \( 1 + 388 T^{2} + 79864 T^{4} + 11127724 T^{6} + 1159312964 T^{8} + 94699812532 T^{10} + 6212827842398 T^{12} + 94699812532 p^{2} T^{14} + 1159312964 p^{4} T^{16} + 11127724 p^{6} T^{18} + 79864 p^{8} T^{20} + 388 p^{10} T^{22} + p^{12} T^{24} \)
61 \( ( 1 + 22 T + 418 T^{2} + 5566 T^{3} + 62885 T^{4} + 613252 T^{5} + 5052104 T^{6} + 613252 p T^{7} + 62885 p^{2} T^{8} + 5566 p^{3} T^{9} + 418 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( ( 1 + 196 T^{2} + 64 T^{3} + 21428 T^{4} + 1352 T^{5} + 1671734 T^{6} + 1352 p T^{7} + 21428 p^{2} T^{8} + 64 p^{3} T^{9} + 196 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
71 \( 1 + 72 T^{2} + 12670 T^{4} + 721048 T^{6} + 105137939 T^{8} + 4570036592 T^{10} + 536793730748 T^{12} + 4570036592 p^{2} T^{14} + 105137939 p^{4} T^{16} + 721048 p^{6} T^{18} + 12670 p^{8} T^{20} + 72 p^{10} T^{22} + p^{12} T^{24} \)
73 \( ( 1 + 26 T + 622 T^{2} + 9206 T^{3} + 127301 T^{4} + 1316864 T^{5} + 12781136 T^{6} + 1316864 p T^{7} + 127301 p^{2} T^{8} + 9206 p^{3} T^{9} + 622 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 4 T + 106 T^{2} + 596 T^{3} + 1187 T^{4} + 48536 T^{5} + 293876 T^{6} + 48536 p T^{7} + 1187 p^{2} T^{8} + 596 p^{3} T^{9} + 106 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( 1 + 260 T^{2} + 54704 T^{4} + 7982020 T^{6} + 1013286620 T^{8} + 104330092020 T^{10} + 9478839943518 T^{12} + 104330092020 p^{2} T^{14} + 1013286620 p^{4} T^{16} + 7982020 p^{6} T^{18} + 54704 p^{8} T^{20} + 260 p^{10} T^{22} + p^{12} T^{24} \)
89 \( 1 + 422 T^{2} + 99085 T^{4} + 17225714 T^{6} + 2365521806 T^{8} + 269042562326 T^{10} + 25988928108833 T^{12} + 269042562326 p^{2} T^{14} + 2365521806 p^{4} T^{16} + 17225714 p^{6} T^{18} + 99085 p^{8} T^{20} + 422 p^{10} T^{22} + p^{12} T^{24} \)
97 \( ( 1 + 16 T + 459 T^{2} + 3698 T^{3} + 62861 T^{4} + 231678 T^{5} + 5290852 T^{6} + 231678 p T^{7} + 62861 p^{2} T^{8} + 3698 p^{3} T^{9} + 459 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.75666672746211419244918912599, −2.75534786612670994661075632843, −2.60944891874790410328853462814, −2.52103433828345248627029927701, −2.47326748524991650489704820275, −2.46009245322973443884706117277, −2.35414879769725317781390371831, −2.17712480792533747252314886275, −2.11359987935115270506727766361, −2.03351600018119106076756898013, −2.02005973807493615419063412101, −2.01110840601971037895919551009, −1.97223858083670739913636496463, −1.94844355894843665033444127406, −1.83585691466481492731517289442, −1.51285815626070692238415219090, −1.47793753599588380973586044960, −1.30129153509314971184286979385, −1.29355637187449226561532577675, −1.17124046046932608003922132603, −1.13711805602241134585005758803, −1.11679647210910848847303413702, −0.974535851661545281747643310737, −0.892746536841264087878115119980, −0.791772636095629880227749112336, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.791772636095629880227749112336, 0.892746536841264087878115119980, 0.974535851661545281747643310737, 1.11679647210910848847303413702, 1.13711805602241134585005758803, 1.17124046046932608003922132603, 1.29355637187449226561532577675, 1.30129153509314971184286979385, 1.47793753599588380973586044960, 1.51285815626070692238415219090, 1.83585691466481492731517289442, 1.94844355894843665033444127406, 1.97223858083670739913636496463, 2.01110840601971037895919551009, 2.02005973807493615419063412101, 2.03351600018119106076756898013, 2.11359987935115270506727766361, 2.17712480792533747252314886275, 2.35414879769725317781390371831, 2.46009245322973443884706117277, 2.47326748524991650489704820275, 2.52103433828345248627029927701, 2.60944891874790410328853462814, 2.75534786612670994661075632843, 2.75666672746211419244918912599

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

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