Dirichlet series
| L(s) = 1 | − 3-s + 30·5-s + 24·7-s − 43·9-s + 117·11-s − 59·13-s − 30·15-s + 88·17-s + 105·19-s − 24·21-s − 138·23-s + 525·25-s + 107·27-s − 71·29-s + 396·31-s − 117·33-s + 720·35-s + 57·37-s + 59·39-s + 692·41-s + 778·43-s − 1.29e3·45-s + 248·47-s − 262·49-s − 88·51-s − 49·53-s + 3.51e3·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 2.68·5-s + 1.29·7-s − 1.59·9-s + 3.20·11-s − 1.25·13-s − 0.516·15-s + 1.25·17-s + 1.26·19-s − 0.249·21-s − 1.25·23-s + 21/5·25-s + 0.762·27-s − 0.454·29-s + 2.29·31-s − 0.617·33-s + 3.47·35-s + 0.253·37-s + 0.242·39-s + 2.63·41-s + 2.75·43-s − 4.27·45-s + 0.769·47-s − 0.763·49-s − 0.241·51-s − 0.126·53-s + 8.60·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{12} \cdot 5^{6} \cdot 23^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.99708\times 10^{8}\) |
| Root analytic conductor: | \(5.20969\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{12} \cdot 5^{6} \cdot 23^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(61.13436002\) |
| \(L(\frac12)\) | \(\approx\) | \(61.13436002\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p T )^{6} \) | |
| 23 | \( ( 1 + p T )^{6} \) | |
| good | 3 | \( 1 + T + 44 T^{2} - 20 T^{3} + 1286 T^{4} + 1207 T^{5} + 47458 T^{6} + 1207 p^{3} T^{7} + 1286 p^{6} T^{8} - 20 p^{9} T^{9} + 44 p^{12} T^{10} + p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 24 T + 838 T^{2} - 11427 T^{3} + 174221 T^{4} + 156003 p T^{5} - 870728 T^{6} + 156003 p^{4} T^{7} + 174221 p^{6} T^{8} - 11427 p^{9} T^{9} + 838 p^{12} T^{10} - 24 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 - 117 T + 12631 T^{2} - 852315 T^{3} + 4704321 p T^{4} - 2367522090 T^{5} + 97712990522 T^{6} - 2367522090 p^{3} T^{7} + 4704321 p^{7} T^{8} - 852315 p^{9} T^{9} + 12631 p^{12} T^{10} - 117 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 + 59 T + 7984 T^{2} + 33508 p T^{3} + 36651676 T^{4} + 1601129129 T^{5} + 99900493982 T^{6} + 1601129129 p^{3} T^{7} + 36651676 p^{6} T^{8} + 33508 p^{10} T^{9} + 7984 p^{12} T^{10} + 59 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 - 88 T + 12676 T^{2} - 760549 T^{3} + 88073669 T^{4} - 4202100943 T^{5} + 418388150884 T^{6} - 4202100943 p^{3} T^{7} + 88073669 p^{6} T^{8} - 760549 p^{9} T^{9} + 12676 p^{12} T^{10} - 88 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 - 105 T + 30409 T^{2} - 2878095 T^{3} + 428127011 T^{4} - 35203256994 T^{5} + 3652520275462 T^{6} - 35203256994 p^{3} T^{7} + 428127011 p^{6} T^{8} - 2878095 p^{9} T^{9} + 30409 p^{12} T^{10} - 105 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 71 T + 37093 T^{2} - 2051104 T^{3} + 81389963 T^{4} - 136693406251 T^{5} - 7405245290714 T^{6} - 136693406251 p^{3} T^{7} + 81389963 p^{6} T^{8} - 2051104 p^{9} T^{9} + 37093 p^{12} T^{10} + 71 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 396 T + 144469 T^{2} - 36050289 T^{3} + 9255097835 T^{4} - 1796226959484 T^{5} + 346331508516334 T^{6} - 1796226959484 p^{3} T^{7} + 9255097835 p^{6} T^{8} - 36050289 p^{9} T^{9} + 144469 p^{12} T^{10} - 396 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 57 T + 101362 T^{2} - 6320409 T^{3} + 10115600063 T^{4} - 488767342170 T^{5} + 518952493566796 T^{6} - 488767342170 p^{3} T^{7} + 10115600063 p^{6} T^{8} - 6320409 p^{9} T^{9} + 101362 p^{12} T^{10} - 57 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 - 692 T + 379429 T^{2} - 165253637 T^{3} + 61181902301 T^{4} - 19357104545918 T^{5} + 5471477431297984 T^{6} - 19357104545918 p^{3} T^{7} + 61181902301 p^{6} T^{8} - 165253637 p^{9} T^{9} + 379429 p^{12} T^{10} - 692 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 - 778 T + 430046 T^{2} - 153946182 T^{3} + 43461410359 T^{4} - 9788571582740 T^{5} + 2496089999210404 T^{6} - 9788571582740 p^{3} T^{7} + 43461410359 p^{6} T^{8} - 153946182 p^{9} T^{9} + 430046 p^{12} T^{10} - 778 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 248 T + 392815 T^{2} - 79717130 T^{3} + 78563681855 T^{4} - 13813779548570 T^{5} + 10109538958714018 T^{6} - 13813779548570 p^{3} T^{7} + 78563681855 p^{6} T^{8} - 79717130 p^{9} T^{9} + 392815 p^{12} T^{10} - 248 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 49 T + 502456 T^{2} + 65163193 T^{3} + 109840086479 T^{4} + 22373756707522 T^{5} + 17070291910828768 T^{6} + 22373756707522 p^{3} T^{7} + 109840086479 p^{6} T^{8} + 65163193 p^{9} T^{9} + 502456 p^{12} T^{10} + 49 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 - 1539 T + 1518864 T^{2} - 994791105 T^{3} + 541810450911 T^{4} - 249193553510466 T^{5} + 115168722366095440 T^{6} - 249193553510466 p^{3} T^{7} + 541810450911 p^{6} T^{8} - 994791105 p^{9} T^{9} + 1518864 p^{12} T^{10} - 1539 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 461 T + 596111 T^{2} - 230370345 T^{3} + 169469348971 T^{4} - 1343626488046 p T^{5} + 42899712833241178 T^{6} - 1343626488046 p^{4} T^{7} + 169469348971 p^{6} T^{8} - 230370345 p^{9} T^{9} + 596111 p^{12} T^{10} - 461 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 2815 T + 4452128 T^{2} - 4872792597 T^{3} + 4146424963903 T^{4} - 2889382178993186 T^{5} + 1712325032873427664 T^{6} - 2889382178993186 p^{3} T^{7} + 4146424963903 p^{6} T^{8} - 4872792597 p^{9} T^{9} + 4452128 p^{12} T^{10} - 2815 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 - 378 T + 569509 T^{2} - 191216001 T^{3} + 381211729539 T^{4} - 124077737886786 T^{5} + 139338206797372262 T^{6} - 124077737886786 p^{3} T^{7} + 381211729539 p^{6} T^{8} - 191216001 p^{9} T^{9} + 569509 p^{12} T^{10} - 378 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 518 T + 1072605 T^{2} - 484677948 T^{3} + 572888254531 T^{4} - 240402861105662 T^{5} + 241993859335244734 T^{6} - 240402861105662 p^{3} T^{7} + 572888254531 p^{6} T^{8} - 484677948 p^{9} T^{9} + 1072605 p^{12} T^{10} - 518 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 - 1694 T + 3121374 T^{2} - 2969142330 T^{3} + 3095157196063 T^{4} - 2097601772070044 T^{5} + 1751724273228217924 T^{6} - 2097601772070044 p^{3} T^{7} + 3095157196063 p^{6} T^{8} - 2969142330 p^{9} T^{9} + 3121374 p^{12} T^{10} - 1694 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 1757 T + 3466410 T^{2} - 4073838271 T^{3} + 4758941501335 T^{4} - 4190439246919382 T^{5} + 3559405969025116076 T^{6} - 4190439246919382 p^{3} T^{7} + 4758941501335 p^{6} T^{8} - 4073838271 p^{9} T^{9} + 3466410 p^{12} T^{10} - 1757 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 - 688 T + 2846550 T^{2} - 1389801800 T^{3} + 3820042744543 T^{4} - 1397216204071336 T^{5} + 3219824095633462772 T^{6} - 1397216204071336 p^{3} T^{7} + 3820042744543 p^{6} T^{8} - 1389801800 p^{9} T^{9} + 2846550 p^{12} T^{10} - 688 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 1029 T + 3739417 T^{2} - 2879031123 T^{3} + 6492288004859 T^{4} - 4287238742024904 T^{5} + 7299731723384715862 T^{6} - 4287238742024904 p^{3} T^{7} + 6492288004859 p^{6} T^{8} - 2879031123 p^{9} T^{9} + 3739417 p^{12} T^{10} - 1029 p^{15} T^{11} + p^{18} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.65579850102668702495493224426, −5.20666177529444469121715019019, −5.20278754066355965785352901519, −5.01283511821483990245784835768, −4.93198730557064196388163326441, −4.81102267041065992881618136446, −4.48925635678082116503642681437, −4.20222051297189567624580138600, −4.02629080439980003748035666079, −3.70668388269093711571184706776, −3.57046976245552901935797590937, −3.56963669839685615833936193025, −3.41573148285664040400521784462, −2.73277507163423568914825812359, −2.52836041813346843300399084681, −2.41193222099466150549874589101, −2.38659979681346670942641394753, −2.14503230418175204094958731466, −2.12375553848481785556077138695, −1.40666315457956096689881009585, −1.38147957823064210463765306650, −0.960100165894660079041477703596, −0.921058852467073143159510918342, −0.67385080367682175894757075911, −0.64055829140818006982287892481, 0.64055829140818006982287892481, 0.67385080367682175894757075911, 0.921058852467073143159510918342, 0.960100165894660079041477703596, 1.38147957823064210463765306650, 1.40666315457956096689881009585, 2.12375553848481785556077138695, 2.14503230418175204094958731466, 2.38659979681346670942641394753, 2.41193222099466150549874589101, 2.52836041813346843300399084681, 2.73277507163423568914825812359, 3.41573148285664040400521784462, 3.56963669839685615833936193025, 3.57046976245552901935797590937, 3.70668388269093711571184706776, 4.02629080439980003748035666079, 4.20222051297189567624580138600, 4.48925635678082116503642681437, 4.81102267041065992881618136446, 4.93198730557064196388163326441, 5.01283511821483990245784835768, 5.20278754066355965785352901519, 5.20666177529444469121715019019, 5.65579850102668702495493224426
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.