Properties

Degree 24
Conductor $ 2^{24} \cdot 17^{12} \cdot 59^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 12

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 3·5-s + 2·7-s − 7·9-s − 5·11-s + 9·13-s − 12·15-s − 12·17-s − 15·19-s − 8·21-s − 22·23-s − 23·25-s + 48·27-s − 17·29-s + 5·31-s + 20·33-s + 6·35-s − 2·37-s − 36·39-s − 18·41-s − 16·43-s − 21·45-s − 60·47-s − 35·49-s + 48·51-s − 24·53-s − 15·55-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.34·5-s + 0.755·7-s − 7/3·9-s − 1.50·11-s + 2.49·13-s − 3.09·15-s − 2.91·17-s − 3.44·19-s − 1.74·21-s − 4.58·23-s − 4.59·25-s + 9.23·27-s − 3.15·29-s + 0.898·31-s + 3.48·33-s + 1.01·35-s − 0.328·37-s − 5.76·39-s − 2.81·41-s − 2.43·43-s − 3.13·45-s − 8.75·47-s − 5·49-s + 6.72·51-s − 3.29·53-s − 2.02·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 17^{12} \cdot 59^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 17^{12} \cdot 59^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{24} \cdot 17^{12} \cdot 59^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4012} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  12
Selberg data  =  $(24,\ 2^{24} \cdot 17^{12} \cdot 59^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \(F_p\) is a polynomial of degree 24. If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 23.
$p$$F_p$
bad2 \( 1 \)
17 \( ( 1 + T )^{12} \)
59 \( ( 1 - T )^{12} \)
good3 \( 1 + 4 T + 23 T^{2} + 8 p^{2} T^{3} + 649 T^{5} + 28 p^{2} T^{4} + 1738 T^{6} + 3847 T^{7} + 8630 T^{8} + 5662 p T^{9} + 33622 T^{10} + 60443 T^{11} + 109222 T^{12} + 60443 p T^{13} + 33622 p^{2} T^{14} + 5662 p^{4} T^{15} + 8630 p^{4} T^{16} + 3847 p^{5} T^{17} + 1738 p^{6} T^{18} + 649 p^{7} T^{19} + 28 p^{10} T^{20} + 8 p^{11} T^{21} + 23 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
5 \( 1 - 3 T + 32 T^{2} - 73 T^{3} + 462 T^{4} - 824 T^{5} + 4263 T^{6} - 6398 T^{7} + 31078 T^{8} - 43703 T^{9} + 198957 T^{10} - 268946 T^{11} + 1092688 T^{12} - 268946 p T^{13} + 198957 p^{2} T^{14} - 43703 p^{3} T^{15} + 31078 p^{4} T^{16} - 6398 p^{5} T^{17} + 4263 p^{6} T^{18} - 824 p^{7} T^{19} + 462 p^{8} T^{20} - 73 p^{9} T^{21} + 32 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
7 \( 1 - 2 T + 39 T^{2} - 89 T^{3} + 869 T^{4} - 2026 T^{5} + 13586 T^{6} - 31006 T^{7} + 163043 T^{8} - 352767 T^{9} + 1557036 T^{10} - 3119131 T^{11} + 12073898 T^{12} - 3119131 p T^{13} + 1557036 p^{2} T^{14} - 352767 p^{3} T^{15} + 163043 p^{4} T^{16} - 31006 p^{5} T^{17} + 13586 p^{6} T^{18} - 2026 p^{7} T^{19} + 869 p^{8} T^{20} - 89 p^{9} T^{21} + 39 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 + 5 T + 74 T^{2} + 367 T^{3} + 2889 T^{4} + 13297 T^{5} + 6943 p T^{6} + 319559 T^{7} + 1507262 T^{8} + 516323 p T^{9} + 2113752 p T^{10} + 649564 p^{2} T^{11} + 2361512 p^{2} T^{12} + 649564 p^{3} T^{13} + 2113752 p^{3} T^{14} + 516323 p^{4} T^{15} + 1507262 p^{4} T^{16} + 319559 p^{5} T^{17} + 6943 p^{7} T^{18} + 13297 p^{7} T^{19} + 2889 p^{8} T^{20} + 367 p^{9} T^{21} + 74 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 - 9 T + 103 T^{2} - 583 T^{3} + 4242 T^{4} - 19416 T^{5} + 117521 T^{6} - 478145 T^{7} + 2549845 T^{8} - 9334119 T^{9} + 44278587 T^{10} - 146176544 T^{11} + 629059858 T^{12} - 146176544 p T^{13} + 44278587 p^{2} T^{14} - 9334119 p^{3} T^{15} + 2549845 p^{4} T^{16} - 478145 p^{5} T^{17} + 117521 p^{6} T^{18} - 19416 p^{7} T^{19} + 4242 p^{8} T^{20} - 583 p^{9} T^{21} + 103 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 15 T + 227 T^{2} + 2177 T^{3} + 20285 T^{4} + 148441 T^{5} + 1059503 T^{6} + 6406841 T^{7} + 38109495 T^{8} + 199026663 T^{9} + 1030099984 T^{10} + 4762169337 T^{11} + 21905988513 T^{12} + 4762169337 p T^{13} + 1030099984 p^{2} T^{14} + 199026663 p^{3} T^{15} + 38109495 p^{4} T^{16} + 6406841 p^{5} T^{17} + 1059503 p^{6} T^{18} + 148441 p^{7} T^{19} + 20285 p^{8} T^{20} + 2177 p^{9} T^{21} + 227 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 + 22 T + 367 T^{2} + 4259 T^{3} + 41017 T^{4} + 318344 T^{5} + 2094237 T^{6} + 11260048 T^{7} + 49236660 T^{8} + 153000611 T^{9} + 207740031 T^{10} - 1281777529 T^{11} - 10103437610 T^{12} - 1281777529 p T^{13} + 207740031 p^{2} T^{14} + 153000611 p^{3} T^{15} + 49236660 p^{4} T^{16} + 11260048 p^{5} T^{17} + 2094237 p^{6} T^{18} + 318344 p^{7} T^{19} + 41017 p^{8} T^{20} + 4259 p^{9} T^{21} + 367 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 + 17 T + 267 T^{2} + 2896 T^{3} + 29860 T^{4} + 259073 T^{5} + 2154919 T^{6} + 15978321 T^{7} + 113636417 T^{8} + 25517484 p T^{9} + 159812613 p T^{10} + 26911758984 T^{11} + 150329209016 T^{12} + 26911758984 p T^{13} + 159812613 p^{3} T^{14} + 25517484 p^{4} T^{15} + 113636417 p^{4} T^{16} + 15978321 p^{5} T^{17} + 2154919 p^{6} T^{18} + 259073 p^{7} T^{19} + 29860 p^{8} T^{20} + 2896 p^{9} T^{21} + 267 p^{10} T^{22} + 17 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 - 5 T + 174 T^{2} - 612 T^{3} + 14514 T^{4} - 35717 T^{5} + 843245 T^{6} - 1558202 T^{7} + 39777604 T^{8} - 59702625 T^{9} + 1569775628 T^{10} - 2000648013 T^{11} + 52495876228 T^{12} - 2000648013 p T^{13} + 1569775628 p^{2} T^{14} - 59702625 p^{3} T^{15} + 39777604 p^{4} T^{16} - 1558202 p^{5} T^{17} + 843245 p^{6} T^{18} - 35717 p^{7} T^{19} + 14514 p^{8} T^{20} - 612 p^{9} T^{21} + 174 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 + 2 T + 276 T^{2} + 652 T^{3} + 38837 T^{4} + 96256 T^{5} + 3631670 T^{6} + 8930412 T^{7} + 249282483 T^{8} + 584164766 T^{9} + 13169887841 T^{10} + 28427662304 T^{11} + 547481700896 T^{12} + 28427662304 p T^{13} + 13169887841 p^{2} T^{14} + 584164766 p^{3} T^{15} + 249282483 p^{4} T^{16} + 8930412 p^{5} T^{17} + 3631670 p^{6} T^{18} + 96256 p^{7} T^{19} + 38837 p^{8} T^{20} + 652 p^{9} T^{21} + 276 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 + 18 T + 513 T^{2} + 6752 T^{3} + 112032 T^{4} + 1180612 T^{5} + 14535318 T^{6} + 128286403 T^{7} + 1278488494 T^{8} + 9670533526 T^{9} + 81232337198 T^{10} + 12976454388 p T^{11} + 3842570017322 T^{12} + 12976454388 p^{2} T^{13} + 81232337198 p^{2} T^{14} + 9670533526 p^{3} T^{15} + 1278488494 p^{4} T^{16} + 128286403 p^{5} T^{17} + 14535318 p^{6} T^{18} + 1180612 p^{7} T^{19} + 112032 p^{8} T^{20} + 6752 p^{9} T^{21} + 513 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 + 16 T + 452 T^{2} + 5166 T^{3} + 84594 T^{4} + 768632 T^{5} + 9494587 T^{6} + 73309881 T^{7} + 755743740 T^{8} + 5146333609 T^{9} + 46182597778 T^{10} + 280738811836 T^{11} + 2228528119624 T^{12} + 280738811836 p T^{13} + 46182597778 p^{2} T^{14} + 5146333609 p^{3} T^{15} + 755743740 p^{4} T^{16} + 73309881 p^{5} T^{17} + 9494587 p^{6} T^{18} + 768632 p^{7} T^{19} + 84594 p^{8} T^{20} + 5166 p^{9} T^{21} + 452 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 + 60 T + 2010 T^{2} + 47866 T^{3} + 899112 T^{4} + 14051168 T^{5} + 188897608 T^{6} + 2231635363 T^{7} + 23513775589 T^{8} + 223192326289 T^{9} + 1921928283489 T^{10} + 15081365904050 T^{11} + 108125238818798 T^{12} + 15081365904050 p T^{13} + 1921928283489 p^{2} T^{14} + 223192326289 p^{3} T^{15} + 23513775589 p^{4} T^{16} + 2231635363 p^{5} T^{17} + 188897608 p^{6} T^{18} + 14051168 p^{7} T^{19} + 899112 p^{8} T^{20} + 47866 p^{9} T^{21} + 2010 p^{10} T^{22} + 60 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 + 24 T + 585 T^{2} + 10035 T^{3} + 156536 T^{4} + 2090218 T^{5} + 25483383 T^{6} + 280685847 T^{7} + 2849940379 T^{8} + 26691947343 T^{9} + 232136355273 T^{10} + 1878007036003 T^{11} + 14158182047281 T^{12} + 1878007036003 p T^{13} + 232136355273 p^{2} T^{14} + 26691947343 p^{3} T^{15} + 2849940379 p^{4} T^{16} + 280685847 p^{5} T^{17} + 25483383 p^{6} T^{18} + 2090218 p^{7} T^{19} + 156536 p^{8} T^{20} + 10035 p^{9} T^{21} + 585 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 + 17 T + 523 T^{2} + 7084 T^{3} + 125632 T^{4} + 1423665 T^{5} + 19094435 T^{6} + 187175044 T^{7} + 2096912119 T^{8} + 18193464886 T^{9} + 177833498575 T^{10} + 1384227028687 T^{11} + 12060055117058 T^{12} + 1384227028687 p T^{13} + 177833498575 p^{2} T^{14} + 18193464886 p^{3} T^{15} + 2096912119 p^{4} T^{16} + 187175044 p^{5} T^{17} + 19094435 p^{6} T^{18} + 1423665 p^{7} T^{19} + 125632 p^{8} T^{20} + 7084 p^{9} T^{21} + 523 p^{10} T^{22} + 17 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 22 T + 741 T^{2} + 12614 T^{3} + 247452 T^{4} + 3454128 T^{5} + 50461200 T^{6} + 596546933 T^{7} + 7086482547 T^{8} + 72239490641 T^{9} + 726632463270 T^{10} + 6441622270486 T^{11} + 55947235078290 T^{12} + 6441622270486 p T^{13} + 726632463270 p^{2} T^{14} + 72239490641 p^{3} T^{15} + 7086482547 p^{4} T^{16} + 596546933 p^{5} T^{17} + 50461200 p^{6} T^{18} + 3454128 p^{7} T^{19} + 247452 p^{8} T^{20} + 12614 p^{9} T^{21} + 741 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 + 10 T + 574 T^{2} + 6474 T^{3} + 164602 T^{4} + 1912514 T^{5} + 31427999 T^{6} + 346712137 T^{7} + 4403217165 T^{8} + 43628604814 T^{9} + 467550791406 T^{10} + 4063454064165 T^{11} + 37945230477130 T^{12} + 4063454064165 p T^{13} + 467550791406 p^{2} T^{14} + 43628604814 p^{3} T^{15} + 4403217165 p^{4} T^{16} + 346712137 p^{5} T^{17} + 31427999 p^{6} T^{18} + 1912514 p^{7} T^{19} + 164602 p^{8} T^{20} + 6474 p^{9} T^{21} + 574 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 - 4 T + 437 T^{2} - 576 T^{3} + 88919 T^{4} + 116331 T^{5} + 11939372 T^{6} + 45208590 T^{7} + 1236816657 T^{8} + 7281945960 T^{9} + 106422782444 T^{10} + 762970891832 T^{11} + 8103868987534 T^{12} + 762970891832 p T^{13} + 106422782444 p^{2} T^{14} + 7281945960 p^{3} T^{15} + 1236816657 p^{4} T^{16} + 45208590 p^{5} T^{17} + 11939372 p^{6} T^{18} + 116331 p^{7} T^{19} + 88919 p^{8} T^{20} - 576 p^{9} T^{21} + 437 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 48 T + 1625 T^{2} - 40426 T^{3} + 839442 T^{4} - 14827026 T^{5} + 231784702 T^{6} - 3237232009 T^{7} + 41158789322 T^{8} - 478465878064 T^{9} + 5137583259438 T^{10} - 51004021734028 T^{11} + 470673531133144 T^{12} - 51004021734028 p T^{13} + 5137583259438 p^{2} T^{14} - 478465878064 p^{3} T^{15} + 41158789322 p^{4} T^{16} - 3237232009 p^{5} T^{17} + 231784702 p^{6} T^{18} - 14827026 p^{7} T^{19} + 839442 p^{8} T^{20} - 40426 p^{9} T^{21} + 1625 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 + 20 T + 745 T^{2} + 12329 T^{3} + 264212 T^{4} + 3753535 T^{5} + 59783574 T^{6} + 742649018 T^{7} + 9656633440 T^{8} + 105856008277 T^{9} + 1175051350486 T^{10} + 11401113299177 T^{11} + 110512481376744 T^{12} + 11401113299177 p T^{13} + 1175051350486 p^{2} T^{14} + 105856008277 p^{3} T^{15} + 9656633440 p^{4} T^{16} + 742649018 p^{5} T^{17} + 59783574 p^{6} T^{18} + 3753535 p^{7} T^{19} + 264212 p^{8} T^{20} + 12329 p^{9} T^{21} + 745 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 + 29 T + 765 T^{2} + 13253 T^{3} + 201887 T^{4} + 2435499 T^{5} + 24592778 T^{6} + 189383588 T^{7} + 775452159 T^{8} - 5841612153 T^{9} - 191711192048 T^{10} - 2742591163352 T^{11} - 29526628703276 T^{12} - 2742591163352 p T^{13} - 191711192048 p^{2} T^{14} - 5841612153 p^{3} T^{15} + 775452159 p^{4} T^{16} + 189383588 p^{5} T^{17} + 24592778 p^{6} T^{18} + 2435499 p^{7} T^{19} + 201887 p^{8} T^{20} + 13253 p^{9} T^{21} + 765 p^{10} T^{22} + 29 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 + 26 T + 994 T^{2} + 18412 T^{3} + 419067 T^{4} + 6226726 T^{5} + 107948306 T^{6} + 1365044803 T^{7} + 19716930881 T^{8} + 218636377270 T^{9} + 2739861393601 T^{10} + 26981021125004 T^{11} + 298758142387224 T^{12} + 26981021125004 p T^{13} + 2739861393601 p^{2} T^{14} + 218636377270 p^{3} T^{15} + 19716930881 p^{4} T^{16} + 1365044803 p^{5} T^{17} + 107948306 p^{6} T^{18} + 6226726 p^{7} T^{19} + 419067 p^{8} T^{20} + 18412 p^{9} T^{21} + 994 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.12264288376511381627467511916, −2.78469006648003287618148161453, −2.74291983693506004095876174417, −2.65424623835679030794815915834, −2.53745503599766375077134988684, −2.53526196608386433408047714065, −2.50266244068807000163940849076, −2.46799208411170099191147324624, −2.27403976753859290886056420512, −2.25723290941815816973851536964, −2.23775178143282389072848044736, −2.09277038921347771377113420179, −1.93279542512458449066619225962, −1.88214908499253841231632912509, −1.86596972900636316965011824082, −1.81665253346831263061175210437, −1.70608737982944304768203963650, −1.48091515573725862236542189392, −1.47039421325910029539278827958, −1.37948633664658586709166330200, −1.37343934252468851156360781326, −1.37245905431549897903115807811, −1.33433004256844929769903933298, −1.20268058637710312574565973240, −0.844065227341930856662468257343, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.844065227341930856662468257343, 1.20268058637710312574565973240, 1.33433004256844929769903933298, 1.37245905431549897903115807811, 1.37343934252468851156360781326, 1.37948633664658586709166330200, 1.47039421325910029539278827958, 1.48091515573725862236542189392, 1.70608737982944304768203963650, 1.81665253346831263061175210437, 1.86596972900636316965011824082, 1.88214908499253841231632912509, 1.93279542512458449066619225962, 2.09277038921347771377113420179, 2.23775178143282389072848044736, 2.25723290941815816973851536964, 2.27403976753859290886056420512, 2.46799208411170099191147324624, 2.50266244068807000163940849076, 2.53526196608386433408047714065, 2.53745503599766375077134988684, 2.65424623835679030794815915834, 2.74291983693506004095876174417, 2.78469006648003287618148161453, 3.12264288376511381627467511916

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

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