Dirichlet series
| L(s) = 1 | − 16·13-s + 64·19-s + 88·25-s − 160·31-s − 56·37-s − 64·43-s + 42·49-s − 104·61-s − 64·67-s − 64·73-s − 32·79-s − 96·97-s + 608·103-s − 192·109-s + 788·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 924·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.23·13-s + 3.36·19-s + 3.51·25-s − 5.16·31-s − 1.51·37-s − 1.48·43-s + 6/7·49-s − 1.70·61-s − 0.955·67-s − 0.876·73-s − 0.405·79-s − 0.989·97-s + 5.90·103-s − 1.76·109-s + 6.51·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 5.46·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{72} \cdot 3^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.09219\times 10^{24}\) |
| Root analytic conductor: | \(10.4816\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{72} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(2.726364837\) |
| \(L(\frac12)\) | \(\approx\) | \(2.726364837\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 - p T^{2} )^{6} \) | |
| good | 5 | \( 1 - 88 T^{2} + 4446 T^{4} - 172312 T^{6} + 1143123 p T^{8} - 163186704 T^{10} + 4221625796 T^{12} - 163186704 p^{4} T^{14} + 1143123 p^{9} T^{16} - 172312 p^{12} T^{18} + 4446 p^{16} T^{20} - 88 p^{20} T^{22} + p^{24} T^{24} \) |
| 11 | \( 1 - 788 T^{2} + 295946 T^{4} - 73143140 T^{6} + 13895300031 T^{8} - 2185947504008 T^{10} + 288996636615340 T^{12} - 2185947504008 p^{4} T^{14} + 13895300031 p^{8} T^{16} - 73143140 p^{12} T^{18} + 295946 p^{16} T^{20} - 788 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( ( 1 + 8 T + 558 T^{2} + 5000 T^{3} + 160239 T^{4} + 1297840 T^{5} + 31879908 T^{6} + 1297840 p^{2} T^{7} + 160239 p^{4} T^{8} + 5000 p^{6} T^{9} + 558 p^{8} T^{10} + 8 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 17 | \( 1 - 1464 T^{2} + 53854 p T^{4} - 326348920 T^{6} + 5091900175 p T^{8} - 25140504641744 T^{10} + 7794388955640452 T^{12} - 25140504641744 p^{4} T^{14} + 5091900175 p^{9} T^{16} - 326348920 p^{12} T^{18} + 53854 p^{17} T^{20} - 1464 p^{20} T^{22} + p^{24} T^{24} \) | |
| 19 | \( ( 1 - 32 T + 2086 T^{2} - 47776 T^{3} + 92453 p T^{4} - 30861376 T^{5} + 819613076 T^{6} - 30861376 p^{2} T^{7} + 92453 p^{5} T^{8} - 47776 p^{6} T^{9} + 2086 p^{8} T^{10} - 32 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 23 | \( 1 - 3412 T^{2} + 5510506 T^{4} - 5616049124 T^{6} + 4124050945759 T^{8} - 2441569189763464 T^{10} + 1313577222733938284 T^{12} - 2441569189763464 p^{4} T^{14} + 4124050945759 p^{8} T^{16} - 5616049124 p^{12} T^{18} + 5510506 p^{16} T^{20} - 3412 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( 1 - 4576 T^{2} + 10941414 T^{4} - 18431088480 T^{6} + 24224689239919 T^{8} - 901199848585664 p T^{10} + 23813832473001809620 T^{12} - 901199848585664 p^{5} T^{14} + 24224689239919 p^{8} T^{16} - 18431088480 p^{12} T^{18} + 10941414 p^{16} T^{20} - 4576 p^{20} T^{22} + p^{24} T^{24} \) | |
| 31 | \( ( 1 + 80 T + 4854 T^{2} + 204688 T^{3} + 7908175 T^{4} + 253266208 T^{5} + 8282803636 T^{6} + 253266208 p^{2} T^{7} + 7908175 p^{4} T^{8} + 204688 p^{6} T^{9} + 4854 p^{8} T^{10} + 80 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 37 | \( ( 1 + 28 T + 5978 T^{2} + 158572 T^{3} + 16822575 T^{4} + 392980952 T^{5} + 28773500620 T^{6} + 392980952 p^{2} T^{7} + 16822575 p^{4} T^{8} + 158572 p^{6} T^{9} + 5978 p^{8} T^{10} + 28 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( 1 - 7352 T^{2} + 32864510 T^{4} - 105328464248 T^{6} + 273805951205439 T^{8} - 590413137426539984 T^{10} + \)\(10\!\cdots\!84\)\( T^{12} - 590413137426539984 p^{4} T^{14} + 273805951205439 p^{8} T^{16} - 105328464248 p^{12} T^{18} + 32864510 p^{16} T^{20} - 7352 p^{20} T^{22} + p^{24} T^{24} \) | |
| 43 | \( ( 1 + 32 T + 7126 T^{2} + 257440 T^{3} + 24503903 T^{4} + 910093120 T^{5} + 54452273972 T^{6} + 910093120 p^{2} T^{7} + 24503903 p^{4} T^{8} + 257440 p^{6} T^{9} + 7126 p^{8} T^{10} + 32 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 47 | \( 1 - 10156 T^{2} + 38899074 T^{4} - 59018079996 T^{6} + 10144766384239 T^{8} - 44523315051525208 T^{10} + \)\(35\!\cdots\!76\)\( T^{12} - 44523315051525208 p^{4} T^{14} + 10144766384239 p^{8} T^{16} - 59018079996 p^{12} T^{18} + 38899074 p^{16} T^{20} - 10156 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( 1 - 18560 T^{2} + 172143654 T^{4} - 1049067123328 T^{6} + 4738424436817071 T^{8} - 17106375657857312000 T^{10} + \)\(51\!\cdots\!88\)\( T^{12} - 17106375657857312000 p^{4} T^{14} + 4738424436817071 p^{8} T^{16} - 1049067123328 p^{12} T^{18} + 172143654 p^{16} T^{20} - 18560 p^{20} T^{22} + p^{24} T^{24} \) | |
| 59 | \( 1 - 14284 T^{2} + 117930210 T^{4} - 654122951644 T^{6} + 2810159914812687 T^{8} - 10069521608166307224 T^{10} + \)\(34\!\cdots\!52\)\( T^{12} - 10069521608166307224 p^{4} T^{14} + 2810159914812687 p^{8} T^{16} - 654122951644 p^{12} T^{18} + 117930210 p^{16} T^{20} - 14284 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 + 52 T + 14546 T^{2} + 759172 T^{3} + 109319583 T^{4} + 5061800936 T^{5} + 502112234044 T^{6} + 5061800936 p^{2} T^{7} + 109319583 p^{4} T^{8} + 759172 p^{6} T^{9} + 14546 p^{8} T^{10} + 52 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( ( 1 + 32 T + 15614 T^{2} + 337440 T^{3} + 123909103 T^{4} + 2460293824 T^{5} + 674310911428 T^{6} + 2460293824 p^{2} T^{7} + 123909103 p^{4} T^{8} + 337440 p^{6} T^{9} + 15614 p^{8} T^{10} + 32 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 71 | \( 1 - 28436 T^{2} + 438962666 T^{4} - 4707270790820 T^{6} + 38898275137186911 T^{8} - \)\(25\!\cdots\!72\)\( T^{10} + \)\(14\!\cdots\!04\)\( T^{12} - \)\(25\!\cdots\!72\)\( p^{4} T^{14} + 38898275137186911 p^{8} T^{16} - 4707270790820 p^{12} T^{18} + 438962666 p^{16} T^{20} - 28436 p^{20} T^{22} + p^{24} T^{24} \) | |
| 73 | \( ( 1 + 32 T + 14174 T^{2} + 972512 T^{3} + 92720223 T^{4} + 9711613696 T^{5} + 495039728068 T^{6} + 9711613696 p^{2} T^{7} + 92720223 p^{4} T^{8} + 972512 p^{6} T^{9} + 14174 p^{8} T^{10} + 32 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 16 T + 10414 T^{2} - 1360 p T^{3} + 39411839 T^{4} - 5975985760 T^{5} + 161763756068 T^{6} - 5975985760 p^{2} T^{7} + 39411839 p^{4} T^{8} - 1360 p^{7} T^{9} + 10414 p^{8} T^{10} + 16 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 83 | \( 1 - 45228 T^{2} + 1051862498 T^{4} - 16711593024188 T^{6} + 200629425132245199 T^{8} - \)\(19\!\cdots\!24\)\( T^{10} + \)\(14\!\cdots\!20\)\( T^{12} - \)\(19\!\cdots\!24\)\( p^{4} T^{14} + 200629425132245199 p^{8} T^{16} - 16711593024188 p^{12} T^{18} + 1051862498 p^{16} T^{20} - 45228 p^{20} T^{22} + p^{24} T^{24} \) | |
| 89 | \( 1 - 64696 T^{2} + 2018996734 T^{4} - 40646358567544 T^{6} + 593437857442633407 T^{8} - \)\(66\!\cdots\!08\)\( T^{10} + \)\(59\!\cdots\!36\)\( T^{12} - \)\(66\!\cdots\!08\)\( p^{4} T^{14} + 593437857442633407 p^{8} T^{16} - 40646358567544 p^{12} T^{18} + 2018996734 p^{16} T^{20} - 64696 p^{20} T^{22} + p^{24} T^{24} \) | |
| 97 | \( ( 1 + 48 T + 30238 T^{2} + 1616880 T^{3} + 464601471 T^{4} + 24248528864 T^{5} + 5040207919044 T^{6} + 24248528864 p^{2} T^{7} + 464601471 p^{4} T^{8} + 1616880 p^{6} T^{9} + 30238 p^{8} T^{10} + 48 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
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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
−2.48883920387043821521334298136, −2.17445539769778502427280974121, −2.12645242604985291514465861950, −1.98036306530490486720982154253, −1.88977775590849504492163033965, −1.88958218071960813009789455760, −1.79525009611579286607102610988, −1.79018327278306906048935444230, −1.62670797877387145325411480706, −1.62244216078169191635409964875, −1.61167583483632912294024276989, −1.41233775932342088911349448128, −1.22584883680640322808939060536, −1.19036749768292613876854799507, −1.16324710083267696075217577846, −0.978978764275975843488107164826, −0.897309340814179025285925814481, −0.821059685828650322005875945311, −0.67333335865805538276427794693, −0.61187404405621629515749140144, −0.46761224481575814216747637029, −0.42833359991169466254374776545, −0.26781032095522049955209016194, −0.21405545560697388361896135544, −0.05135009309011323468070128093, 0.05135009309011323468070128093, 0.21405545560697388361896135544, 0.26781032095522049955209016194, 0.42833359991169466254374776545, 0.46761224481575814216747637029, 0.61187404405621629515749140144, 0.67333335865805538276427794693, 0.821059685828650322005875945311, 0.897309340814179025285925814481, 0.978978764275975843488107164826, 1.16324710083267696075217577846, 1.19036749768292613876854799507, 1.22584883680640322808939060536, 1.41233775932342088911349448128, 1.61167583483632912294024276989, 1.62244216078169191635409964875, 1.62670797877387145325411480706, 1.79018327278306906048935444230, 1.79525009611579286607102610988, 1.88958218071960813009789455760, 1.88977775590849504492163033965, 1.98036306530490486720982154253, 2.12645242604985291514465861950, 2.17445539769778502427280974121, 2.48883920387043821521334298136
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.