Dirichlet series
| L(s) = 1 | − 16·3-s + 80·9-s − 64·11-s − 96·19-s − 40·25-s + 112·27-s + 1.02e3·33-s + 96·41-s + 176·43-s + 304·49-s + 1.53e3·57-s + 32·59-s + 560·67-s + 320·73-s + 640·75-s − 2.80e3·81-s − 48·83-s + 96·89-s + 448·97-s − 5.12e3·99-s + 112·107-s + 288·113-s + 1.16e3·121-s − 1.53e3·123-s + 127-s − 2.81e3·129-s + 131-s + ⋯ |
| L(s) = 1 | − 5.33·3-s + 80/9·9-s − 5.81·11-s − 5.05·19-s − 8/5·25-s + 4.14·27-s + 31.0·33-s + 2.34·41-s + 4.09·43-s + 6.20·49-s + 26.9·57-s + 0.542·59-s + 8.35·67-s + 4.38·73-s + 8.53·75-s − 34.6·81-s − 0.578·83-s + 1.07·89-s + 4.61·97-s − 51.7·99-s + 1.04·107-s + 2.54·113-s + 9.65·121-s − 12.4·123-s + 0.00787·127-s − 21.8·129-s + 0.00763·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{128} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.79424\times 10^{24}\) |
| Root analytic conductor: | \(5.90571\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{128} \cdot 5^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.03791403475\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03791403475\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + p T^{2} )^{8} \) | |
| good | 3 | \( ( 1 + 8 T + 56 T^{2} + 88 p T^{3} + 1180 T^{4} + 4472 T^{5} + 16328 T^{6} + 17960 p T^{7} + 2086 p^{4} T^{8} + 17960 p^{3} T^{9} + 16328 p^{4} T^{10} + 4472 p^{6} T^{11} + 1180 p^{8} T^{12} + 88 p^{11} T^{13} + 56 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
| 7 | \( 1 - 304 T^{2} + 49016 T^{4} - 5553680 T^{6} + 498874652 T^{8} - 37676627120 T^{10} + 2467707210312 T^{12} - 142726652364816 T^{14} + 7380181971427654 T^{16} - 142726652364816 p^{4} T^{18} + 2467707210312 p^{8} T^{20} - 37676627120 p^{12} T^{22} + 498874652 p^{16} T^{24} - 5553680 p^{20} T^{26} + 49016 p^{24} T^{28} - 304 p^{28} T^{30} + p^{32} T^{32} \) | |
| 11 | \( ( 1 + 32 T + 952 T^{2} + 19424 T^{3} + 360412 T^{4} + 5491872 T^{5} + 78205320 T^{6} + 87946272 p T^{7} + 11313867654 T^{8} + 87946272 p^{3} T^{9} + 78205320 p^{4} T^{10} + 5491872 p^{6} T^{11} + 360412 p^{8} T^{12} + 19424 p^{10} T^{13} + 952 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 13 | \( 1 - 944 T^{2} + 508536 T^{4} - 200560656 T^{6} + 4867022604 p T^{8} - 1289808191856 p T^{10} + 3838575569766984 T^{12} - 773135016282532752 T^{14} + \)\(13\!\cdots\!50\)\( T^{16} - 773135016282532752 p^{4} T^{18} + 3838575569766984 p^{8} T^{20} - 1289808191856 p^{13} T^{22} + 4867022604 p^{17} T^{24} - 200560656 p^{20} T^{26} + 508536 p^{24} T^{28} - 944 p^{28} T^{30} + p^{32} T^{32} \) | |
| 17 | \( ( 1 + 920 T^{2} - 1024 T^{3} + 482908 T^{4} - 2569216 T^{5} + 180440104 T^{6} - 1384073216 T^{7} + 53607862470 T^{8} - 1384073216 p^{2} T^{9} + 180440104 p^{4} T^{10} - 2569216 p^{6} T^{11} + 482908 p^{8} T^{12} - 1024 p^{10} T^{13} + 920 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 19 | \( ( 1 + 48 T + 2264 T^{2} + 64784 T^{3} + 1878172 T^{4} + 42673712 T^{5} + 54810616 p T^{6} + 21032073616 T^{7} + 443288946054 T^{8} + 21032073616 p^{2} T^{9} + 54810616 p^{5} T^{10} + 42673712 p^{6} T^{11} + 1878172 p^{8} T^{12} + 64784 p^{10} T^{13} + 2264 p^{12} T^{14} + 48 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 - 3888 T^{2} + 7955064 T^{4} - 490605168 p T^{6} + 12361205009692 T^{8} - 11064434609235632 T^{10} + 363641861555362808 p T^{12} - \)\(54\!\cdots\!36\)\( T^{14} + \)\(30\!\cdots\!54\)\( T^{16} - \)\(54\!\cdots\!36\)\( p^{4} T^{18} + 363641861555362808 p^{9} T^{20} - 11064434609235632 p^{12} T^{22} + 12361205009692 p^{16} T^{24} - 490605168 p^{21} T^{26} + 7955064 p^{24} T^{28} - 3888 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( 1 - 6512 T^{2} + 20950520 T^{4} - 43677911888 T^{6} + 65805636311196 T^{8} - 76313988616836720 T^{10} + 72144379825780853448 T^{12} - \)\(60\!\cdots\!60\)\( T^{14} + \)\(49\!\cdots\!46\)\( T^{16} - \)\(60\!\cdots\!60\)\( p^{4} T^{18} + 72144379825780853448 p^{8} T^{20} - 76313988616836720 p^{12} T^{22} + 65805636311196 p^{16} T^{24} - 43677911888 p^{20} T^{26} + 20950520 p^{24} T^{28} - 6512 p^{28} T^{30} + p^{32} T^{32} \) | |
| 31 | \( 1 - 11024 T^{2} + 58849912 T^{4} - 202519110448 T^{6} + 504762028525340 T^{8} - 970414651855115280 T^{10} + 48232488550621370808 p T^{12} - \)\(18\!\cdots\!08\)\( T^{14} + \)\(20\!\cdots\!14\)\( p^{2} T^{16} - \)\(18\!\cdots\!08\)\( p^{4} T^{18} + 48232488550621370808 p^{9} T^{20} - 970414651855115280 p^{12} T^{22} + 504762028525340 p^{16} T^{24} - 202519110448 p^{20} T^{26} + 58849912 p^{24} T^{28} - 11024 p^{28} T^{30} + p^{32} T^{32} \) | |
| 37 | \( 1 - 8048 T^{2} + 33554168 T^{4} - 96692518736 T^{6} + 214283037705116 T^{8} - 388089210730827888 T^{10} + \)\(60\!\cdots\!68\)\( T^{12} - \)\(86\!\cdots\!56\)\( T^{14} + \)\(11\!\cdots\!54\)\( T^{16} - \)\(86\!\cdots\!56\)\( p^{4} T^{18} + \)\(60\!\cdots\!68\)\( p^{8} T^{20} - 388089210730827888 p^{12} T^{22} + 214283037705116 p^{16} T^{24} - 96692518736 p^{20} T^{26} + 33554168 p^{24} T^{28} - 8048 p^{28} T^{30} + p^{32} T^{32} \) | |
| 41 | \( ( 1 - 48 T + 6776 T^{2} - 248848 T^{3} + 23784604 T^{4} - 743771056 T^{5} + 57834774088 T^{6} - 1553618752016 T^{7} + 107458518845382 T^{8} - 1553618752016 p^{2} T^{9} + 57834774088 p^{4} T^{10} - 743771056 p^{6} T^{11} + 23784604 p^{8} T^{12} - 248848 p^{10} T^{13} + 6776 p^{12} T^{14} - 48 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 88 T + 10904 T^{2} - 584760 T^{3} + 41315356 T^{4} - 1470855176 T^{5} + 81897391464 T^{6} - 2142946814824 T^{7} + 135038752556166 T^{8} - 2142946814824 p^{2} T^{9} + 81897391464 p^{4} T^{10} - 1470855176 p^{6} T^{11} + 41315356 p^{8} T^{12} - 584760 p^{10} T^{13} + 10904 p^{12} T^{14} - 88 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 47 | \( 1 - 20336 T^{2} + 207670136 T^{4} - 30107912752 p T^{6} + 7215051578850332 T^{8} - 29246104650132675312 T^{10} + \)\(97\!\cdots\!64\)\( T^{12} - \)\(27\!\cdots\!48\)\( T^{14} + \)\(65\!\cdots\!90\)\( T^{16} - \)\(27\!\cdots\!48\)\( p^{4} T^{18} + \)\(97\!\cdots\!64\)\( p^{8} T^{20} - 29246104650132675312 p^{12} T^{22} + 7215051578850332 p^{16} T^{24} - 30107912752 p^{21} T^{26} + 207670136 p^{24} T^{28} - 20336 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 - 29872 T^{2} + 435357048 T^{4} - 78003178192 p T^{6} + 28784975106602652 T^{8} - \)\(15\!\cdots\!80\)\( T^{10} + \)\(68\!\cdots\!72\)\( T^{12} - \)\(25\!\cdots\!32\)\( T^{14} + \)\(77\!\cdots\!90\)\( T^{16} - \)\(25\!\cdots\!32\)\( p^{4} T^{18} + \)\(68\!\cdots\!72\)\( p^{8} T^{20} - \)\(15\!\cdots\!80\)\( p^{12} T^{22} + 28784975106602652 p^{16} T^{24} - 78003178192 p^{21} T^{26} + 435357048 p^{24} T^{28} - 29872 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( ( 1 - 16 T + 14424 T^{2} - 123312 T^{3} + 106948380 T^{4} - 6965296 p T^{5} + 555854451432 T^{6} - 1028420088880 T^{7} + 2204072588055046 T^{8} - 1028420088880 p^{2} T^{9} + 555854451432 p^{4} T^{10} - 6965296 p^{7} T^{11} + 106948380 p^{8} T^{12} - 123312 p^{10} T^{13} + 14424 p^{12} T^{14} - 16 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 61 | \( 1 - 38768 T^{2} + 739627512 T^{4} - 9245594103120 T^{6} + 85028219842655388 T^{8} - \)\(61\!\cdots\!04\)\( T^{10} + \)\(35\!\cdots\!04\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{14} + \)\(70\!\cdots\!90\)\( T^{16} - \)\(17\!\cdots\!00\)\( p^{4} T^{18} + \)\(35\!\cdots\!04\)\( p^{8} T^{20} - \)\(61\!\cdots\!04\)\( p^{12} T^{22} + 85028219842655388 p^{16} T^{24} - 9245594103120 p^{20} T^{26} + 739627512 p^{24} T^{28} - 38768 p^{28} T^{30} + p^{32} T^{32} \) | |
| 67 | \( ( 1 - 280 T + 48472 T^{2} - 5380280 T^{3} + 420742172 T^{4} - 18820044680 T^{5} - 74589407832 T^{6} + 102869654427480 T^{7} - 9472269788316922 T^{8} + 102869654427480 p^{2} T^{9} - 74589407832 p^{4} T^{10} - 18820044680 p^{6} T^{11} + 420742172 p^{8} T^{12} - 5380280 p^{10} T^{13} + 48472 p^{12} T^{14} - 280 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 71 | \( 1 - 51600 T^{2} + 1323871352 T^{4} - 22419807308976 T^{6} + 280539512089956892 T^{8} - \)\(27\!\cdots\!32\)\( T^{10} + \)\(21\!\cdots\!92\)\( T^{12} - \)\(14\!\cdots\!24\)\( T^{14} + \)\(79\!\cdots\!06\)\( T^{16} - \)\(14\!\cdots\!24\)\( p^{4} T^{18} + \)\(21\!\cdots\!92\)\( p^{8} T^{20} - \)\(27\!\cdots\!32\)\( p^{12} T^{22} + 280539512089956892 p^{16} T^{24} - 22419807308976 p^{20} T^{26} + 1323871352 p^{24} T^{28} - 51600 p^{28} T^{30} + p^{32} T^{32} \) | |
| 73 | \( ( 1 - 160 T + 41560 T^{2} - 5063392 T^{3} + 743540572 T^{4} - 72298781088 T^{5} + 7663253056616 T^{6} - 606120688885472 T^{7} + 50409130991220806 T^{8} - 606120688885472 p^{2} T^{9} + 7663253056616 p^{4} T^{10} - 72298781088 p^{6} T^{11} + 743540572 p^{8} T^{12} - 5063392 p^{10} T^{13} + 41560 p^{12} T^{14} - 160 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( 1 - 46608 T^{2} + 1016109688 T^{4} - 13642586212400 T^{6} + 127120421230055708 T^{8} - \)\(90\!\cdots\!00\)\( T^{10} + \)\(56\!\cdots\!40\)\( T^{12} - \)\(34\!\cdots\!12\)\( T^{14} + \)\(21\!\cdots\!62\)\( T^{16} - \)\(34\!\cdots\!12\)\( p^{4} T^{18} + \)\(56\!\cdots\!40\)\( p^{8} T^{20} - \)\(90\!\cdots\!00\)\( p^{12} T^{22} + 127120421230055708 p^{16} T^{24} - 13642586212400 p^{20} T^{26} + 1016109688 p^{24} T^{28} - 46608 p^{28} T^{30} + p^{32} T^{32} \) | |
| 83 | \( ( 1 + 24 T + 41912 T^{2} + 1147320 T^{3} + 834583708 T^{4} + 22945159560 T^{5} + 10267229349960 T^{6} + 258219735515304 T^{7} + 85140581561116422 T^{8} + 258219735515304 p^{2} T^{9} + 10267229349960 p^{4} T^{10} + 22945159560 p^{6} T^{11} + 834583708 p^{8} T^{12} + 1147320 p^{10} T^{13} + 41912 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 48 T + 27064 T^{2} - 1479824 T^{3} + 417520412 T^{4} - 27657160752 T^{5} + 4725903785224 T^{6} - 319950661730576 T^{7} + 41247738959908294 T^{8} - 319950661730576 p^{2} T^{9} + 4725903785224 p^{4} T^{10} - 27657160752 p^{6} T^{11} + 417520412 p^{8} T^{12} - 1479824 p^{10} T^{13} + 27064 p^{12} T^{14} - 48 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 224 T + 61848 T^{2} - 10373792 T^{3} + 1822566748 T^{4} - 240792740576 T^{5} + 31877413855272 T^{6} - 3439868084683168 T^{7} + 366932138198951878 T^{8} - 3439868084683168 p^{2} T^{9} + 31877413855272 p^{4} T^{10} - 240792740576 p^{6} T^{11} + 1822566748 p^{8} T^{12} - 10373792 p^{10} T^{13} + 61848 p^{12} T^{14} - 224 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.30224963534603550997259969141, −2.24326303244730838796842548534, −2.17644453134306557119205913334, −2.16596692933658737729031530468, −2.08477544007690855001390420268, −1.93546472520253578017903910696, −1.93235610418465558290647358144, −1.65591233226583532965507286917, −1.62187087615530382586610145806, −1.36097655326721575874319833230, −1.30402824860800848984797975870, −1.10034667967409306229854866997, −1.08022923061149769967974439302, −0.895907144903720883719504620922, −0.890941478534640441132047559949, −0.824844109479814385614278614028, −0.77475024436579006963728265929, −0.68664134342442000687894542810, −0.58007470367898423316585479227, −0.37227805227857136205624090924, −0.34478333403359271829313284085, −0.31911469381919260754027774054, −0.27765804459736386546883698490, −0.13126582163566716463673535080, −0.097557574456500542869977483153, 0.097557574456500542869977483153, 0.13126582163566716463673535080, 0.27765804459736386546883698490, 0.31911469381919260754027774054, 0.34478333403359271829313284085, 0.37227805227857136205624090924, 0.58007470367898423316585479227, 0.68664134342442000687894542810, 0.77475024436579006963728265929, 0.824844109479814385614278614028, 0.890941478534640441132047559949, 0.895907144903720883719504620922, 1.08022923061149769967974439302, 1.10034667967409306229854866997, 1.30402824860800848984797975870, 1.36097655326721575874319833230, 1.62187087615530382586610145806, 1.65591233226583532965507286917, 1.93235610418465558290647358144, 1.93546472520253578017903910696, 2.08477544007690855001390420268, 2.16596692933658737729031530468, 2.17644453134306557119205913334, 2.24326303244730838796842548534, 2.30224963534603550997259969141
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.