Dirichlet series
| L(s) = 1 | + 8·4-s + 3·5-s − 3·7-s − 12·11-s − 2·13-s + 32·16-s + 34·17-s + 9·19-s + 24·20-s + 6·23-s − 13·25-s − 24·28-s + 29-s + 18·31-s − 9·35-s + 6·41-s + 11·43-s − 96·44-s + 15·47-s + 3·49-s − 16·52-s + 8·53-s − 36·55-s + 5·61-s + 91·64-s − 6·65-s + 15·67-s + ⋯ |
| L(s) = 1 | + 4·4-s + 1.34·5-s − 1.13·7-s − 3.61·11-s − 0.554·13-s + 8·16-s + 8.24·17-s + 2.06·19-s + 5.36·20-s + 1.25·23-s − 2.59·25-s − 4.53·28-s + 0.185·29-s + 3.23·31-s − 1.52·35-s + 0.937·41-s + 1.67·43-s − 14.4·44-s + 2.18·47-s + 3/7·49-s − 2.21·52-s + 1.09·53-s − 4.85·55-s + 0.640·61-s + 91/8·64-s − 0.744·65-s + 1.83·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 7^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.11972\times 10^{9}\) |
| Root analytic conductor: | \(2.55729\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.116447788\) |
| \(L(\frac12)\) | \(\approx\) | \(8.116447788\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 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 | \( 1 - p^{3} T^{2} + p^{5} T^{4} - 91 T^{6} + 7 p^{5} T^{8} - 33 p^{4} T^{10} + 1137 T^{12} - 33 p^{6} T^{14} + 7 p^{9} T^{16} - 91 p^{6} T^{18} + p^{13} T^{20} - p^{13} T^{22} + p^{12} T^{24} \) |
| 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
−3.17475792984947280895478761636, −3.06515776834280678016909051824, −2.94494743388972853970230904706, −2.88077051547403674997552547796, −2.87645381401872364939381441834, −2.80057601968878910493028596866, −2.79904311032790012841988304807, −2.54112554313928427341224084084, −2.48588288085043609780181726987, −2.46840036802076855308653574056, −2.43102875079830899643973831151, −2.25482742446617083838257937533, −2.20542738563265175864166555907, −2.06848503092773324202582424983, −1.72366327285081231684991845575, −1.67730944594870939693081538607, −1.60519038849866369859789861956, −1.30725486933704495124244846736, −1.27838400328866267579680090126, −1.18173748128029979898072282486, −0.991752743627407461824956959151, −0.960484497347189269153804137071, −0.959918025541785863799192756997, −0.66073079644487281432416975410, −0.092538717615619190739759191759, 0.092538717615619190739759191759, 0.66073079644487281432416975410, 0.959918025541785863799192756997, 0.960484497347189269153804137071, 0.991752743627407461824956959151, 1.18173748128029979898072282486, 1.27838400328866267579680090126, 1.30725486933704495124244846736, 1.60519038849866369859789861956, 1.67730944594870939693081538607, 1.72366327285081231684991845575, 2.06848503092773324202582424983, 2.20542738563265175864166555907, 2.25482742446617083838257937533, 2.43102875079830899643973831151, 2.46840036802076855308653574056, 2.48588288085043609780181726987, 2.54112554313928427341224084084, 2.79904311032790012841988304807, 2.80057601968878910493028596866, 2.87645381401872364939381441834, 2.88077051547403674997552547796, 2.94494743388972853970230904706, 3.06515776834280678016909051824, 3.17475792984947280895478761636
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.