Dirichlet series
| L(s) = 1 | + 3·3-s + 56·7-s + 3·9-s − 300·19-s + 168·21-s + 354·25-s + 930·31-s + 764·37-s + 1.01e3·43-s + 1.40e3·49-s − 900·57-s + 2.35e3·61-s + 168·63-s − 792·67-s − 2.90e3·73-s + 1.06e3·75-s − 1.67e3·79-s + 207·81-s + 2.79e3·93-s − 888·103-s − 3.18e3·109-s + 2.29e3·111-s − 4.74e3·121-s + 127-s + 3.03e3·129-s + 131-s − 1.68e4·133-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3.02·7-s + 1/9·9-s − 3.62·19-s + 1.74·21-s + 2.83·25-s + 5.38·31-s + 3.39·37-s + 3.58·43-s + 4.08·49-s − 2.09·57-s + 4.94·61-s + 0.335·63-s − 1.44·67-s − 4.65·73-s + 1.63·75-s − 2.38·79-s + 0.283·81-s + 3.11·93-s − 0.849·103-s − 2.79·109-s + 1.95·111-s − 3.56·121-s + 0.000698·127-s + 2.07·129-s + 0.000666·131-s − 10.9·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 3^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.68522\times 10^{15}\) |
| Root analytic conductor: | \(4.45248\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(46.33578612\) |
| \(L(\frac12)\) | \(\approx\) | \(46.33578612\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T + 2 p T^{2} - p^{2} T^{3} - 22 p^{2} T^{4} + 95 p^{3} T^{5} - 1334 p^{3} T^{6} + 95 p^{6} T^{7} - 22 p^{8} T^{8} - p^{11} T^{9} + 2 p^{13} T^{10} - p^{16} T^{11} + p^{18} T^{12} \) | |
| 7 | \( ( 1 - 4 p T + 68 p T^{2} - 220 p^{2} T^{3} + 68 p^{4} T^{4} - 4 p^{7} T^{5} + p^{9} T^{6} )^{2} \) | |
| good | 5 | \( 1 - 354 T^{2} + 53346 T^{4} - 3913744 T^{6} + 238386078 T^{8} - 13245019542 p T^{10} + 12349753291374 T^{12} - 13245019542 p^{7} T^{14} + 238386078 p^{12} T^{16} - 3913744 p^{18} T^{18} + 53346 p^{24} T^{20} - 354 p^{30} T^{22} + p^{36} T^{24} \) |
| 11 | \( 1 + 4742 T^{2} + 10147682 T^{4} + 18336975648 T^{6} + 35136366245806 T^{8} + 55422032379705850 T^{10} + 74318354148055569358 T^{12} + 55422032379705850 p^{6} T^{14} + 35136366245806 p^{12} T^{16} + 18336975648 p^{18} T^{18} + 10147682 p^{24} T^{20} + 4742 p^{30} T^{22} + p^{36} T^{24} \) | |
| 13 | \( ( 1 - 8847 T^{2} + 36037359 T^{4} - 94069474274 T^{6} + 36037359 p^{6} T^{8} - 8847 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 17 | \( 1 - 17217 T^{2} + 160729722 T^{4} - 865949822635 T^{6} + 2480649769543932 T^{8} + 2994987661331912811 T^{10} - \)\(48\!\cdots\!48\)\( T^{12} + 2994987661331912811 p^{6} T^{14} + 2480649769543932 p^{12} T^{16} - 865949822635 p^{18} T^{18} + 160729722 p^{24} T^{20} - 17217 p^{30} T^{22} + p^{36} T^{24} \) | |
| 19 | \( ( 1 + 150 T + 30330 T^{2} + 3424500 T^{3} + 468855630 T^{4} + 40538051670 T^{5} + 4025485422214 T^{6} + 40538051670 p^{3} T^{7} + 468855630 p^{6} T^{8} + 3424500 p^{9} T^{9} + 30330 p^{12} T^{10} + 150 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 23 | \( 1 + 58691 T^{2} + 1855192226 T^{4} + 43669490183457 T^{6} + 835010896549582396 T^{8} + \)\(13\!\cdots\!91\)\( T^{10} + \)\(17\!\cdots\!88\)\( T^{12} + \)\(13\!\cdots\!91\)\( p^{6} T^{14} + 835010896549582396 p^{12} T^{16} + 43669490183457 p^{18} T^{18} + 1855192226 p^{24} T^{20} + 58691 p^{30} T^{22} + p^{36} T^{24} \) | |
| 29 | \( ( 1 - 26333 T^{2} + 912806819 T^{4} - 27528483699758 T^{6} + 912806819 p^{6} T^{8} - 26333 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 15 p T + 176469 T^{2} - 1565910 p T^{3} + 12033403527 T^{4} - 2446560037821 T^{5} + 457872227799046 T^{6} - 2446560037821 p^{3} T^{7} + 12033403527 p^{6} T^{8} - 1565910 p^{10} T^{9} + 176469 p^{12} T^{10} - 15 p^{16} T^{11} + p^{18} T^{12} )^{2} \) | |
| 37 | \( ( 1 - 382 T - 886 p T^{2} + 5434056 T^{3} + 8807187550 T^{4} - 649676877530 T^{5} - 374210188787594 T^{6} - 649676877530 p^{3} T^{7} + 8807187550 p^{6} T^{8} + 5434056 p^{9} T^{9} - 886 p^{13} T^{10} - 382 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 41 | \( ( 1 + 240738 T^{2} + 28558186959 T^{4} + 1368012595132 p^{2} T^{6} + 28558186959 p^{6} T^{8} + 240738 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 43 | \( ( 1 - 253 T + 215237 T^{2} - 33567598 T^{3} + 215237 p^{3} T^{4} - 253 p^{6} T^{5} + p^{9} T^{6} )^{4} \) | |
| 47 | \( 1 - 437385 T^{2} + 98505882894 T^{4} - 16642251751296271 T^{6} + \)\(23\!\cdots\!92\)\( T^{8} - \)\(28\!\cdots\!97\)\( T^{10} + \)\(30\!\cdots\!08\)\( T^{12} - \)\(28\!\cdots\!97\)\( p^{6} T^{14} + \)\(23\!\cdots\!92\)\( p^{12} T^{16} - 16642251751296271 p^{18} T^{18} + 98505882894 p^{24} T^{20} - 437385 p^{30} T^{22} + p^{36} T^{24} \) | |
| 53 | \( 1 + 362162 T^{2} + 31342470242 T^{4} + 2991238457561520 T^{6} + \)\(17\!\cdots\!22\)\( T^{8} + \)\(23\!\cdots\!54\)\( T^{10} + \)\(11\!\cdots\!62\)\( T^{12} + \)\(23\!\cdots\!54\)\( p^{6} T^{14} + \)\(17\!\cdots\!22\)\( p^{12} T^{16} + 2991238457561520 p^{18} T^{18} + 31342470242 p^{24} T^{20} + 362162 p^{30} T^{22} + p^{36} T^{24} \) | |
| 59 | \( 1 - 661854 T^{2} + 202275596298 T^{4} - 40619516236130848 T^{6} + \)\(74\!\cdots\!38\)\( T^{8} - \)\(16\!\cdots\!86\)\( T^{10} + \)\(35\!\cdots\!06\)\( T^{12} - \)\(16\!\cdots\!86\)\( p^{6} T^{14} + \)\(74\!\cdots\!38\)\( p^{12} T^{16} - 40619516236130848 p^{18} T^{18} + 202275596298 p^{24} T^{20} - 661854 p^{30} T^{22} + p^{36} T^{24} \) | |
| 61 | \( ( 1 - 1179 T + 941982 T^{2} - 564310665 T^{3} + 276007099296 T^{4} - 128710086478551 T^{5} + 56963412393227764 T^{6} - 128710086478551 p^{3} T^{7} + 276007099296 p^{6} T^{8} - 564310665 p^{9} T^{9} + 941982 p^{12} T^{10} - 1179 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( ( 1 + 396 T - 208818 T^{2} - 528240452 T^{3} - 109541196954 T^{4} + 71767532435832 T^{5} + 113262131816538126 T^{6} + 71767532435832 p^{3} T^{7} - 109541196954 p^{6} T^{8} - 528240452 p^{9} T^{9} - 208818 p^{12} T^{10} + 396 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 1922318 T^{2} + 1602890240687 T^{4} - 746575415906526884 T^{6} + 1602890240687 p^{6} T^{8} - 1922318 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 1452 T + 1911888 T^{2} + 1755642240 T^{3} + 1485047372688 T^{4} + 1012976615051412 T^{5} + 691104732915920110 T^{6} + 1012976615051412 p^{3} T^{7} + 1485047372688 p^{6} T^{8} + 1755642240 p^{9} T^{9} + 1911888 p^{12} T^{10} + 1452 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 837 T - 518715 T^{2} - 592038158 T^{3} + 210634653159 T^{4} + 180227265033825 T^{5} - 13471835399034906 T^{6} + 180227265033825 p^{3} T^{7} + 210634653159 p^{6} T^{8} - 592038158 p^{9} T^{9} - 518715 p^{12} T^{10} + 837 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 83 | \( ( 1 + 2862735 T^{2} + 3680795850915 T^{4} + 2710866692924348218 T^{6} + 3680795850915 p^{6} T^{8} + 2862735 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 89 | \( 1 - 1635561 T^{2} + 1787180462010 T^{4} - 817237014874197619 T^{6} - \)\(17\!\cdots\!36\)\( T^{8} + \)\(75\!\cdots\!95\)\( T^{10} - \)\(70\!\cdots\!84\)\( T^{12} + \)\(75\!\cdots\!95\)\( p^{6} T^{14} - \)\(17\!\cdots\!36\)\( p^{12} T^{16} - 817237014874197619 p^{18} T^{18} + 1787180462010 p^{24} T^{20} - 1635561 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( ( 1 - 3316347 T^{2} + 4922505333747 T^{4} - 4971002050523297522 T^{6} + 4922505333747 p^{6} T^{8} - 3316347 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
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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
−3.20958424895438755211897050278, −3.13771542317066004459633119502, −3.04781559496624860243982888139, −2.92059807705488800198905509585, −2.72260839566187879189768621720, −2.66247382358156692672086102588, −2.65836803083807922302193598516, −2.59422633061877871388284351459, −2.57680342924427340015158676599, −2.40589288554845625328020620534, −2.31527482245549568783838564309, −2.30489252008179561673121799832, −1.92054676043969794898203573390, −1.59201948696641822343299596588, −1.56433660804343454577520562739, −1.50311117982220118156072339866, −1.48642386162484006500971061297, −1.35216618616488050546472995676, −1.10741071718529465301268641954, −0.916672806243928519755480437737, −0.836447948393048858269632719189, −0.67290571318109831008641384082, −0.64578155283160375270690354003, −0.26096952344869793127910113342, −0.23946650388903427875713569665, 0.23946650388903427875713569665, 0.26096952344869793127910113342, 0.64578155283160375270690354003, 0.67290571318109831008641384082, 0.836447948393048858269632719189, 0.916672806243928519755480437737, 1.10741071718529465301268641954, 1.35216618616488050546472995676, 1.48642386162484006500971061297, 1.50311117982220118156072339866, 1.56433660804343454577520562739, 1.59201948696641822343299596588, 1.92054676043969794898203573390, 2.30489252008179561673121799832, 2.31527482245549568783838564309, 2.40589288554845625328020620534, 2.57680342924427340015158676599, 2.59422633061877871388284351459, 2.65836803083807922302193598516, 2.66247382358156692672086102588, 2.72260839566187879189768621720, 2.92059807705488800198905509585, 3.04781559496624860243982888139, 3.13771542317066004459633119502, 3.20958424895438755211897050278
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.