Dirichlet series
| L(s) = 1 | − 3·3-s + 8·4-s − 3·5-s − 3·7-s + 13·9-s + 12·11-s − 24·12-s − 2·13-s + 9·15-s + 32·16-s − 34·17-s + 9·19-s − 24·20-s + 9·21-s − 6·23-s − 13·25-s − 32·27-s − 24·28-s − 29-s + 18·31-s − 36·33-s + 9·35-s + 104·36-s + 6·39-s − 6·41-s + 11·43-s + 96·44-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 4·4-s − 1.34·5-s − 1.13·7-s + 13/3·9-s + 3.61·11-s − 6.92·12-s − 0.554·13-s + 2.32·15-s + 8·16-s − 8.24·17-s + 2.06·19-s − 5.36·20-s + 1.96·21-s − 1.25·23-s − 2.59·25-s − 6.15·27-s − 4.53·28-s − 0.185·29-s + 3.23·31-s − 6.26·33-s + 1.52·35-s + 52/3·36-s + 0.960·39-s − 0.937·41-s + 1.67·43-s + 14.4·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0216681\) |
| Root analytic conductor: | \(0.852431\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.7846989883\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7846989883\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 + 3 T + 6 T^{2} + 15 T^{3} - 39 T^{4} - 258 T^{5} - 475 T^{6} - 258 p T^{7} - 39 p^{2} T^{8} + 15 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 2 T - 18 T^{2} - 17 T^{3} + 341 T^{4} - 63 T^{5} - 6395 T^{6} - 63 p T^{7} + 341 p^{2} T^{8} - 17 p^{3} T^{9} - 18 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( 1 - p^{3} T^{2} + p^{5} T^{4} - 91 T^{6} + 7 p^{5} T^{8} - 33 p^{4} T^{10} + 1137 T^{12} - 33 p^{6} T^{14} + 7 p^{9} T^{16} - 91 p^{6} T^{18} + p^{13} T^{20} - p^{13} T^{22} + p^{12} T^{24} \) |
| 3 | \( 1 + p T - 4 T^{2} - 19 T^{3} + p T^{4} + 16 p T^{5} - 19 T^{6} - 47 p T^{7} + 22 T^{8} + 73 p T^{9} - 95 p T^{10} + 124 T^{11} + 2353 T^{12} + 124 p T^{13} - 95 p^{3} T^{14} + 73 p^{4} T^{15} + 22 p^{4} T^{16} - 47 p^{6} T^{17} - 19 p^{6} T^{18} + 16 p^{8} T^{19} + p^{9} T^{20} - 19 p^{9} T^{21} - 4 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) | |
| 5 | \( 1 + 3 T + 22 T^{2} + 57 T^{3} + 241 T^{4} + 18 p^{2} T^{5} + 1411 T^{6} + 1311 T^{7} + 3032 T^{8} - 7749 T^{9} - 3571 p T^{10} - 104772 T^{11} - 169399 T^{12} - 104772 p T^{13} - 3571 p^{3} T^{14} - 7749 p^{3} T^{15} + 3032 p^{4} T^{16} + 1311 p^{5} T^{17} + 1411 p^{6} T^{18} + 18 p^{9} T^{19} + 241 p^{8} T^{20} + 57 p^{9} T^{21} + 22 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 12 T + 107 T^{2} - 708 T^{3} + 3961 T^{4} - 18831 T^{5} + 79826 T^{6} - 305121 T^{7} + 98539 p T^{8} - 3659247 T^{9} + 11994110 T^{10} - 39258729 T^{11} + 129081947 T^{12} - 39258729 p T^{13} + 11994110 p^{2} T^{14} - 3659247 p^{3} T^{15} + 98539 p^{5} T^{16} - 305121 p^{5} T^{17} + 79826 p^{6} T^{18} - 18831 p^{7} T^{19} + 3961 p^{8} T^{20} - 708 p^{9} T^{21} + 107 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + p T + 198 T^{2} + 1643 T^{3} + 10919 T^{4} + 59082 T^{5} + 266647 T^{6} + 59082 p T^{7} + 10919 p^{2} T^{8} + 1643 p^{3} T^{9} + 198 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 9 T + 115 T^{2} - 792 T^{3} + 6216 T^{4} - 34521 T^{5} + 202721 T^{6} - 930966 T^{7} + 4436762 T^{8} - 17205552 T^{9} + 72759135 T^{10} - 265082310 T^{11} + 1203810589 T^{12} - 265082310 p T^{13} + 72759135 p^{2} T^{14} - 17205552 p^{3} T^{15} + 4436762 p^{4} T^{16} - 930966 p^{5} T^{17} + 202721 p^{6} T^{18} - 34521 p^{7} T^{19} + 6216 p^{8} T^{20} - 792 p^{9} T^{21} + 115 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 + 3 T + 88 T^{2} + 86 T^{3} + 3150 T^{4} - 1355 T^{5} + 76923 T^{6} - 1355 p T^{7} + 3150 p^{2} T^{8} + 86 p^{3} T^{9} + 88 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 + T - 3 p T^{2} + 178 T^{3} + 128 p T^{4} - 15194 T^{5} - 71197 T^{6} + 410532 T^{7} + 447076 T^{8} - 2082223 T^{9} - 28038240 T^{10} - 61539492 T^{11} + 1484469159 T^{12} - 61539492 p T^{13} - 28038240 p^{2} T^{14} - 2082223 p^{3} T^{15} + 447076 p^{4} T^{16} + 410532 p^{5} T^{17} - 71197 p^{6} T^{18} - 15194 p^{7} T^{19} + 128 p^{9} T^{20} + 178 p^{9} T^{21} - 3 p^{11} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 18 T + 232 T^{2} - 72 p T^{3} + 18099 T^{4} - 118611 T^{5} + 671327 T^{6} - 3244986 T^{7} + 13790561 T^{8} - 53577873 T^{9} + 222950535 T^{10} - 1049414532 T^{11} + 5476860865 T^{12} - 1049414532 p T^{13} + 222950535 p^{2} T^{14} - 53577873 p^{3} T^{15} + 13790561 p^{4} T^{16} - 3244986 p^{5} T^{17} + 671327 p^{6} T^{18} - 118611 p^{7} T^{19} + 18099 p^{8} T^{20} - 72 p^{10} T^{21} + 232 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 297 T^{2} + 43128 T^{4} - 4084528 T^{6} + 281694330 T^{8} - 14898657933 T^{10} + 619410154695 T^{12} - 14898657933 p^{2} T^{14} + 281694330 p^{4} T^{16} - 4084528 p^{6} T^{18} + 43128 p^{8} T^{20} - 297 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 + 6 T + 87 T^{2} + 450 T^{3} + 3253 T^{4} + 22749 T^{5} + 119000 T^{6} + 655563 T^{7} + 2153315 T^{8} - 1902885 T^{9} - 21527678 T^{10} - 258610773 T^{11} - 1564835407 T^{12} - 258610773 p T^{13} - 21527678 p^{2} T^{14} - 1902885 p^{3} T^{15} + 2153315 p^{4} T^{16} + 655563 p^{5} T^{17} + 119000 p^{6} T^{18} + 22749 p^{7} T^{19} + 3253 p^{8} T^{20} + 450 p^{9} T^{21} + 87 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 11 T - 88 T^{2} + 799 T^{3} + 8712 T^{4} - 35151 T^{5} - 615381 T^{6} + 1493311 T^{7} + 26239709 T^{8} - 29370632 T^{9} - 1062722582 T^{10} - 167088622 T^{11} + 50469301069 T^{12} - 167088622 p T^{13} - 1062722582 p^{2} T^{14} - 29370632 p^{3} T^{15} + 26239709 p^{4} T^{16} + 1493311 p^{5} T^{17} - 615381 p^{6} T^{18} - 35151 p^{7} T^{19} + 8712 p^{8} T^{20} + 799 p^{9} T^{21} - 88 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 15 T + 374 T^{2} + 4485 T^{3} + 70585 T^{4} + 717852 T^{5} + 8690861 T^{6} + 76990419 T^{7} + 777130526 T^{8} + 6072726129 T^{9} + 52865829659 T^{10} + 366568670022 T^{11} + 2805436179305 T^{12} + 366568670022 p T^{13} + 52865829659 p^{2} T^{14} + 6072726129 p^{3} T^{15} + 777130526 p^{4} T^{16} + 76990419 p^{5} T^{17} + 8690861 p^{6} T^{18} + 717852 p^{7} T^{19} + 70585 p^{8} T^{20} + 4485 p^{9} T^{21} + 374 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 8 T - 216 T^{2} - 1192 T^{3} + 31204 T^{4} + 105740 T^{5} - 3167758 T^{6} - 5889708 T^{7} + 250320748 T^{8} + 208010656 T^{9} - 16435908756 T^{10} - 4149850428 T^{11} + 923207251827 T^{12} - 4149850428 p T^{13} - 16435908756 p^{2} T^{14} + 208010656 p^{3} T^{15} + 250320748 p^{4} T^{16} - 5889708 p^{5} T^{17} - 3167758 p^{6} T^{18} + 105740 p^{7} T^{19} + 31204 p^{8} T^{20} - 1192 p^{9} T^{21} - 216 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 155 T^{2} + 24317 T^{4} - 2572015 T^{6} + 230267981 T^{8} - 17468522907 T^{10} + 1098167895897 T^{12} - 17468522907 p^{2} T^{14} + 230267981 p^{4} T^{16} - 2572015 p^{6} T^{18} + 24317 p^{8} T^{20} - 155 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 - 5 T - 266 T^{2} + 887 T^{3} + 40211 T^{4} - 80704 T^{5} - 4493895 T^{6} + 5470553 T^{7} + 403359860 T^{8} - 265514337 T^{9} - 30729415795 T^{10} + 6069628418 T^{11} + 2020503763113 T^{12} + 6069628418 p T^{13} - 30729415795 p^{2} T^{14} - 265514337 p^{3} T^{15} + 403359860 p^{4} T^{16} + 5470553 p^{5} T^{17} - 4493895 p^{6} T^{18} - 80704 p^{7} T^{19} + 40211 p^{8} T^{20} + 887 p^{9} T^{21} - 266 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 15 T + 295 T^{2} - 3300 T^{3} + 37385 T^{4} - 337086 T^{5} + 2917472 T^{6} - 22840887 T^{7} + 168746996 T^{8} - 1188097515 T^{9} + 6974328225 T^{10} - 52016179137 T^{11} + 290088466863 T^{12} - 52016179137 p T^{13} + 6974328225 p^{2} T^{14} - 1188097515 p^{3} T^{15} + 168746996 p^{4} T^{16} - 22840887 p^{5} T^{17} + 2917472 p^{6} T^{18} - 337086 p^{7} T^{19} + 37385 p^{8} T^{20} - 3300 p^{9} T^{21} + 295 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 30 T + 653 T^{2} - 10590 T^{3} + 145585 T^{4} - 1771113 T^{5} + 19362440 T^{6} - 197718039 T^{7} + 1886667311 T^{8} - 17315970285 T^{9} + 153782228042 T^{10} - 1330750537803 T^{11} + 11347613768747 T^{12} - 1330750537803 p T^{13} + 153782228042 p^{2} T^{14} - 17315970285 p^{3} T^{15} + 1886667311 p^{4} T^{16} - 197718039 p^{5} T^{17} + 19362440 p^{6} T^{18} - 1771113 p^{7} T^{19} + 145585 p^{8} T^{20} - 10590 p^{9} T^{21} + 653 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 42 T + 1120 T^{2} - 22344 T^{3} + 367763 T^{4} - 5157765 T^{5} + 63488279 T^{6} - 699015660 T^{7} + 7009629269 T^{8} - 65233277895 T^{9} + 575574698283 T^{10} - 4944431307318 T^{11} + 42178220425467 T^{12} - 4944431307318 p T^{13} + 575574698283 p^{2} T^{14} - 65233277895 p^{3} T^{15} + 7009629269 p^{4} T^{16} - 699015660 p^{5} T^{17} + 63488279 p^{6} T^{18} - 5157765 p^{7} T^{19} + 367763 p^{8} T^{20} - 22344 p^{9} T^{21} + 1120 p^{10} T^{22} - 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 35 T + 407 T^{2} + 1988 T^{3} + 21243 T^{4} + 209523 T^{5} - 2661180 T^{6} - 47561854 T^{7} - 79767523 T^{8} + 341374970 T^{9} - 12037425323 T^{10} + 160472632948 T^{11} + 4058664204571 T^{12} + 160472632948 p T^{13} - 12037425323 p^{2} T^{14} + 341374970 p^{3} T^{15} - 79767523 p^{4} T^{16} - 47561854 p^{5} T^{17} - 2661180 p^{6} T^{18} + 209523 p^{7} T^{19} + 21243 p^{8} T^{20} + 1988 p^{9} T^{21} + 407 p^{10} T^{22} + 35 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 533 T^{2} + 130220 T^{4} - 19632136 T^{6} + 2123453318 T^{8} - 188332543809 T^{10} + 15642556571895 T^{12} - 188332543809 p^{2} T^{14} + 2123453318 p^{4} T^{16} - 19632136 p^{6} T^{18} + 130220 p^{8} T^{20} - 533 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 638 T^{2} + 182947 T^{4} - 30717524 T^{6} + 3345408557 T^{8} - 264875421410 T^{10} + 20634099784895 T^{12} - 264875421410 p^{2} T^{14} + 3345408557 p^{4} T^{16} - 30717524 p^{6} T^{18} + 182947 p^{8} T^{20} - 638 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 + 3 T + 406 T^{2} + 1209 T^{3} + 93212 T^{4} + 404427 T^{5} + 13886771 T^{6} + 91101243 T^{7} + 1491993611 T^{8} + 15578221158 T^{9} + 128844085584 T^{10} + 1984216560294 T^{11} + 11260776987123 T^{12} + 1984216560294 p T^{13} + 128844085584 p^{2} T^{14} + 15578221158 p^{3} T^{15} + 1491993611 p^{4} T^{16} + 91101243 p^{5} T^{17} + 13886771 p^{6} T^{18} + 404427 p^{7} T^{19} + 93212 p^{8} T^{20} + 1209 p^{9} T^{21} + 406 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−5.27188710414152442926613317590, −5.08561285753788235415578127806, −4.98267779755160012875900137534, −4.95940164439146905501957439871, −4.62307160903893714184466982493, −4.35783861857568312560413271774, −4.34501875962581247912489375981, −4.33935097469234951997144780227, −4.09078694384969952072980629077, −3.98474407140208452410106251378, −3.97848288246132326619559631352, −3.93187325209653416162588145160, −3.86502760398039012710136254043, −3.52586268377497243789815218450, −3.20470804234942167004027587377, −3.10132776733547695672129432767, −2.98211847019348852409088570769, −2.49236745136808444183187113798, −2.30329242869913836253238432207, −2.28239241536009428385497676937, −2.10531560573797820735164018391, −2.01020331761217424593089926431, −1.93215821454881907906339766027, −1.36878862363922681752185491903, −1.26612256890437437309622037185, 1.26612256890437437309622037185, 1.36878862363922681752185491903, 1.93215821454881907906339766027, 2.01020331761217424593089926431, 2.10531560573797820735164018391, 2.28239241536009428385497676937, 2.30329242869913836253238432207, 2.49236745136808444183187113798, 2.98211847019348852409088570769, 3.10132776733547695672129432767, 3.20470804234942167004027587377, 3.52586268377497243789815218450, 3.86502760398039012710136254043, 3.93187325209653416162588145160, 3.97848288246132326619559631352, 3.98474407140208452410106251378, 4.09078694384969952072980629077, 4.33935097469234951997144780227, 4.34501875962581247912489375981, 4.35783861857568312560413271774, 4.62307160903893714184466982493, 4.95940164439146905501957439871, 4.98267779755160012875900137534, 5.08561285753788235415578127806, 5.27188710414152442926613317590
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.