Dirichlet series
| L(s) = 1 | + 12·2-s − 9·3-s + 84·4-s − 27·5-s − 108·6-s − 21·7-s + 448·8-s − 24·9-s − 324·10-s + 9·11-s − 756·12-s − 24·13-s − 252·14-s + 243·15-s + 2.01e3·16-s − 102·17-s − 288·18-s − 2.26e3·20-s + 189·21-s + 108·22-s − 264·23-s − 4.03e3·24-s + 78·25-s − 288·26-s + 477·27-s − 1.76e3·28-s − 483·29-s + ⋯ |
| L(s) = 1 | + 4.24·2-s − 1.73·3-s + 21/2·4-s − 2.41·5-s − 7.34·6-s − 1.13·7-s + 19.7·8-s − 8/9·9-s − 10.2·10-s + 0.246·11-s − 18.1·12-s − 0.512·13-s − 4.81·14-s + 4.18·15-s + 63/2·16-s − 1.45·17-s − 3.77·18-s − 25.3·20-s + 1.96·21-s + 1.04·22-s − 2.39·23-s − 34.2·24-s + 0.623·25-s − 2.17·26-s + 3.39·27-s − 11.9·28-s − 3.09·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.97612\times 10^{9}\) |
| Root analytic conductor: | \(6.52681\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{6} \cdot 19^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T )^{6} \) |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + p^{2} T + 35 p T^{2} + 76 p^{2} T^{3} + 1792 p T^{4} + 3214 p^{2} T^{5} + 175067 T^{6} + 3214 p^{5} T^{7} + 1792 p^{7} T^{8} + 76 p^{11} T^{9} + 35 p^{13} T^{10} + p^{17} T^{11} + p^{18} T^{12} \) |
| 5 | \( 1 + 27 T + 651 T^{2} + 11448 T^{3} + 186414 T^{4} + 2446794 T^{5} + 29610421 T^{6} + 2446794 p^{3} T^{7} + 186414 p^{6} T^{8} + 11448 p^{9} T^{9} + 651 p^{12} T^{10} + 27 p^{15} T^{11} + p^{18} T^{12} \) | |
| 7 | \( 1 + 3 p T + 1956 T^{2} + 96 p^{3} T^{3} + 1621869 T^{4} + 21591393 T^{5} + 732988051 T^{6} + 21591393 p^{3} T^{7} + 1621869 p^{6} T^{8} + 96 p^{12} T^{9} + 1956 p^{12} T^{10} + 3 p^{16} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 - 9 T + 3600 T^{2} - 3952 T^{3} + 7810101 T^{4} - 16229913 T^{5} + 12987476791 T^{6} - 16229913 p^{3} T^{7} + 7810101 p^{6} T^{8} - 3952 p^{9} T^{9} + 3600 p^{12} T^{10} - 9 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 + 24 T + 10974 T^{2} + 256354 T^{3} + 54197190 T^{4} + 1102591782 T^{5} + 153362382155 T^{6} + 1102591782 p^{3} T^{7} + 54197190 p^{6} T^{8} + 256354 p^{9} T^{9} + 10974 p^{12} T^{10} + 24 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 6 p T + 19956 T^{2} + 1039538 T^{3} + 125044314 T^{4} + 2731849530 T^{5} + 29935785539 p T^{6} + 2731849530 p^{3} T^{7} + 125044314 p^{6} T^{8} + 1039538 p^{9} T^{9} + 19956 p^{12} T^{10} + 6 p^{16} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 + 264 T + 82101 T^{2} + 14153951 T^{3} + 2587063173 T^{4} + 325362848739 T^{5} + 42249400499698 T^{6} + 325362848739 p^{3} T^{7} + 2587063173 p^{6} T^{8} + 14153951 p^{9} T^{9} + 82101 p^{12} T^{10} + 264 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 483 T + 189003 T^{2} + 52589178 T^{3} + 12633479070 T^{4} + 2452193384982 T^{5} + 419748201919015 T^{6} + 2452193384982 p^{3} T^{7} + 12633479070 p^{6} T^{8} + 52589178 p^{9} T^{9} + 189003 p^{12} T^{10} + 483 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 + 72 T + 89544 T^{2} + 1173673 T^{3} + 3287656695 T^{4} - 213796613868 T^{5} + 89819559580457 T^{6} - 213796613868 p^{3} T^{7} + 3287656695 p^{6} T^{8} + 1173673 p^{9} T^{9} + 89544 p^{12} T^{10} + 72 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 558 T + 298089 T^{2} - 113545809 T^{3} + 38178310329 T^{4} - 10429273769733 T^{5} + 69667380457282 p T^{6} - 10429273769733 p^{3} T^{7} + 38178310329 p^{6} T^{8} - 113545809 p^{9} T^{9} + 298089 p^{12} T^{10} - 558 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 396 T + 370908 T^{2} + 99000422 T^{3} + 53905401432 T^{4} + 10745465145816 T^{5} + 4555259556269047 T^{6} + 10745465145816 p^{3} T^{7} + 53905401432 p^{6} T^{8} + 99000422 p^{9} T^{9} + 370908 p^{12} T^{10} + 396 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 48 p T + 2198586 T^{2} + 1555771900 T^{3} + 809018971350 T^{4} + 324244765768542 T^{5} + 102565723186928597 T^{6} + 324244765768542 p^{3} T^{7} + 809018971350 p^{6} T^{8} + 1555771900 p^{9} T^{9} + 2198586 p^{12} T^{10} + 48 p^{16} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 858 T + 503448 T^{2} - 228595414 T^{3} + 91663455090 T^{4} - 31593426124158 T^{5} + 10309813417641931 T^{6} - 31593426124158 p^{3} T^{7} + 91663455090 p^{6} T^{8} - 228595414 p^{9} T^{9} + 503448 p^{12} T^{10} - 858 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 762 T + 810066 T^{2} + 437818606 T^{3} + 269674011420 T^{4} + 111215488667052 T^{5} + 51019068945222655 T^{6} + 111215488667052 p^{3} T^{7} + 269674011420 p^{6} T^{8} + 437818606 p^{9} T^{9} + 810066 p^{12} T^{10} + 762 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 393 T + 367692 T^{2} + 124294676 T^{3} + 155320097277 T^{4} + 42336876036537 T^{5} + 31058807508581947 T^{6} + 42336876036537 p^{3} T^{7} + 155320097277 p^{6} T^{8} + 124294676 p^{9} T^{9} + 367692 p^{12} T^{10} + 393 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 + 627 T + 436602 T^{2} + 75339464 T^{3} + 76752167697 T^{4} + 31261541869557 T^{5} + 30110394126012045 T^{6} + 31261541869557 p^{3} T^{7} + 76752167697 p^{6} T^{8} + 75339464 p^{9} T^{9} + 436602 p^{12} T^{10} + 627 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 + 2028 T + 2966583 T^{2} + 45295561 p T^{3} + 2609357883009 T^{4} + 1809180999118599 T^{5} + 1089459220750765142 T^{6} + 1809180999118599 p^{3} T^{7} + 2609357883009 p^{6} T^{8} + 45295561 p^{10} T^{9} + 2966583 p^{12} T^{10} + 2028 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 1284 T + 2106021 T^{2} + 1730649085 T^{3} + 1713386949117 T^{4} + 1068257204268645 T^{5} + 787705980917326306 T^{6} + 1068257204268645 p^{3} T^{7} + 1713386949117 p^{6} T^{8} + 1730649085 p^{9} T^{9} + 2106021 p^{12} T^{10} + 1284 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 + 2688 T + 5086995 T^{2} + 6527675181 T^{3} + 6802475766771 T^{4} + 5582031391494687 T^{5} + 3866069611141377538 T^{6} + 5582031391494687 p^{3} T^{7} + 6802475766771 p^{6} T^{8} + 6527675181 p^{9} T^{9} + 5086995 p^{12} T^{10} + 2688 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 969 T + 2609403 T^{2} + 2013103236 T^{3} + 3026668430262 T^{4} + 1839614455913556 T^{5} + 1947670317006949831 T^{6} + 1839614455913556 p^{3} T^{7} + 3026668430262 p^{6} T^{8} + 2013103236 p^{9} T^{9} + 2609403 p^{12} T^{10} + 969 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 + 927 T + 2769498 T^{2} + 1864682002 T^{3} + 3304222784649 T^{4} + 1733021571294591 T^{5} + 2348134965921328765 T^{6} + 1733021571294591 p^{3} T^{7} + 3304222784649 p^{6} T^{8} + 1864682002 p^{9} T^{9} + 2769498 p^{12} T^{10} + 927 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 - 1257 T + 1218441 T^{2} + 214482548 T^{3} - 459034704372 T^{4} + 727101448606356 T^{5} - 261315659354696507 T^{6} + 727101448606356 p^{3} T^{7} - 459034704372 p^{6} T^{8} + 214482548 p^{9} T^{9} + 1218441 p^{12} T^{10} - 1257 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 2403 T + 3246210 T^{2} - 4153205374 T^{3} + 5770588573755 T^{4} - 5998775861114505 T^{5} + 5570975135612099667 T^{6} - 5998775861114505 p^{3} T^{7} + 5770588573755 p^{6} T^{8} - 4153205374 p^{9} T^{9} + 3246210 p^{12} T^{10} - 2403 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−5.84963447528547762569834212044, −5.26598121497325815663023237886, −5.14600474574963263525811320098, −5.07240008667497931889122900487, −4.87110468921137124397625173210, −4.83619138028512304616189862318, −4.70945901951548096337951712023, −4.36705200307461310774753661785, −4.25528286325256194572457016151, −4.20513786849048367827801207391, −3.90277196105496407029275866126, −3.85587134886760726634022853637, −3.59160589690326779879167065036, −3.40412357316195051264656838880, −3.21327469592928817110198737150, −3.20193996798644666048145802800, −2.95163815184606323790623495192, −2.85589566579143381210334018527, −2.83621882268981337765840736320, −2.08126728385046362501121724942, −1.96652361151795310086444698829, −1.87603700662028212676673244483, −1.63835454561042631940280827935, −1.60213134720884143054350690899, −1.13591673433756216730762607304, 0, 0, 0, 0, 0, 0, 1.13591673433756216730762607304, 1.60213134720884143054350690899, 1.63835454561042631940280827935, 1.87603700662028212676673244483, 1.96652361151795310086444698829, 2.08126728385046362501121724942, 2.83621882268981337765840736320, 2.85589566579143381210334018527, 2.95163815184606323790623495192, 3.20193996798644666048145802800, 3.21327469592928817110198737150, 3.40412357316195051264656838880, 3.59160589690326779879167065036, 3.85587134886760726634022853637, 3.90277196105496407029275866126, 4.20513786849048367827801207391, 4.25528286325256194572457016151, 4.36705200307461310774753661785, 4.70945901951548096337951712023, 4.83619138028512304616189862318, 4.87110468921137124397625173210, 5.07240008667497931889122900487, 5.14600474574963263525811320098, 5.26598121497325815663023237886, 5.84963447528547762569834212044
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.