Dirichlet series
| L(s) = 1 | − 2-s − 8·3-s − 2·4-s + 8·6-s + 11·8-s + 44·9-s + 32·11-s + 16·12-s + 11·16-s − 80·17-s − 44·18-s + 56·19-s − 32·22-s − 88·24-s − 92·25-s − 208·27-s − 34·32-s − 256·33-s + 80·34-s − 88·36-s − 56·38-s − 256·41-s − 64·44-s − 88·48-s + 92·50-s + 640·51-s + 208·54-s + ⋯ |
| L(s) = 1 | − 1/2·2-s − 8/3·3-s − 1/2·4-s + 4/3·6-s + 11/8·8-s + 44/9·9-s + 2.90·11-s + 4/3·12-s + 0.687·16-s − 4.70·17-s − 2.44·18-s + 2.94·19-s − 1.45·22-s − 3.66·24-s − 3.67·25-s − 7.70·27-s − 1.06·32-s − 7.75·33-s + 2.35·34-s − 2.44·36-s − 1.47·38-s − 6.24·41-s − 1.45·44-s − 1.83·48-s + 1.83·50-s + 12.5·51-s + 3.85·54-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.87037\times 10^{16}\) |
| Root analytic conductor: | \(3.26821\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 7^{32} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.203462480\) |
| \(L(\frac12)\) | \(\approx\) | \(1.203462480\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + T + 3 T^{2} - 3 p T^{3} - 11 p T^{4} - 11 p^{2} T^{5} - 5 p^{2} T^{6} + 7 p^{4} T^{7} + 23 p^{4} T^{8} + 7 p^{6} T^{9} - 5 p^{6} T^{10} - 11 p^{8} T^{11} - 11 p^{9} T^{12} - 3 p^{11} T^{13} + 3 p^{12} T^{14} + p^{14} T^{15} + p^{16} T^{16} \) |
| 7 | \( 1 \) | |
| good | 3 | \( ( 1 + 4 T + 2 T^{2} - 14 T^{4} - 140 T^{5} + 224 p T^{6} + 3140 T^{7} + 3439 T^{8} + 3140 p^{2} T^{9} + 224 p^{5} T^{10} - 140 p^{6} T^{11} - 14 p^{8} T^{12} + 2 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
| 5 | \( 1 + 92 T^{2} + 24 p^{3} T^{4} + 78776 T^{6} + 824826 p T^{8} + 144056708 T^{10} + 591854528 p T^{12} + 91907152084 T^{14} + 3067686313171 T^{16} + 91907152084 p^{4} T^{18} + 591854528 p^{9} T^{20} + 144056708 p^{12} T^{22} + 824826 p^{17} T^{24} + 78776 p^{20} T^{26} + 24 p^{27} T^{28} + 92 p^{28} T^{30} + p^{32} T^{32} \) | |
| 11 | \( ( 1 - 16 T - 72 T^{2} + 4640 T^{3} - 20750 T^{4} - 644496 T^{5} + 8328608 T^{6} + 43860880 T^{7} - 1416483549 T^{8} + 43860880 p^{2} T^{9} + 8328608 p^{4} T^{10} - 644496 p^{6} T^{11} - 20750 p^{8} T^{12} + 4640 p^{10} T^{13} - 72 p^{12} T^{14} - 16 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 444 T^{2} + 142936 T^{4} - 35361044 T^{6} + 6575436334 T^{8} - 35361044 p^{4} T^{10} + 142936 p^{8} T^{12} - 444 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 40 T + 292 T^{2} - 10704 T^{3} - 137846 T^{4} + 3266264 T^{5} + 77813520 T^{6} + 47796952 T^{7} - 13592851501 T^{8} + 47796952 p^{2} T^{9} + 77813520 p^{4} T^{10} + 3266264 p^{6} T^{11} - 137846 p^{8} T^{12} - 10704 p^{10} T^{13} + 292 p^{12} T^{14} + 40 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 28 T - 926 T^{2} + 14784 T^{3} + 1071538 T^{4} - 10273676 T^{5} - 557016672 T^{6} + 569025380 T^{7} + 272464898159 T^{8} + 569025380 p^{2} T^{9} - 557016672 p^{4} T^{10} - 10273676 p^{6} T^{11} + 1071538 p^{8} T^{12} + 14784 p^{10} T^{13} - 926 p^{12} T^{14} - 28 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 + 1744 T^{2} + 1769380 T^{4} + 1394487904 T^{6} + 800412311114 T^{8} + 293154906699184 T^{10} + 41608418133156240 T^{12} - 40513926748661757392 T^{14} - \)\(37\!\cdots\!61\)\( T^{16} - 40513926748661757392 p^{4} T^{18} + 41608418133156240 p^{8} T^{20} + 293154906699184 p^{12} T^{22} + 800412311114 p^{16} T^{24} + 1394487904 p^{20} T^{26} + 1769380 p^{24} T^{28} + 1744 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( ( 1 - 3384 T^{2} + 6555580 T^{4} - 8754768776 T^{6} + 8490907402822 T^{8} - 8754768776 p^{4} T^{10} + 6555580 p^{8} T^{12} - 3384 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 31 | \( 1 + 3944 T^{2} + 7116516 T^{4} + 8387060656 T^{6} + 8423414263178 T^{8} + 8514445544397720 T^{10} + 8434364772846028816 T^{12} + \)\(79\!\cdots\!12\)\( T^{14} + \)\(74\!\cdots\!23\)\( T^{16} + \)\(79\!\cdots\!12\)\( p^{4} T^{18} + 8434364772846028816 p^{8} T^{20} + 8514445544397720 p^{12} T^{22} + 8423414263178 p^{16} T^{24} + 8387060656 p^{20} T^{26} + 7116516 p^{24} T^{28} + 3944 p^{28} T^{30} + p^{32} T^{32} \) | |
| 37 | \( 1 + 3512 T^{2} + 3145476 T^{4} - 4974960880 T^{6} - 8515936106358 T^{8} + 12549151402549448 T^{10} + 35685999046008816016 T^{12} + \)\(80\!\cdots\!88\)\( T^{14} - \)\(37\!\cdots\!25\)\( T^{16} + \)\(80\!\cdots\!88\)\( p^{4} T^{18} + 35685999046008816016 p^{8} T^{20} + 12549151402549448 p^{12} T^{22} - 8515936106358 p^{16} T^{24} - 4974960880 p^{20} T^{26} + 3145476 p^{24} T^{28} + 3512 p^{28} T^{30} + p^{32} T^{32} \) | |
| 41 | \( ( 1 + 64 T + 4956 T^{2} + 221760 T^{3} + 11848326 T^{4} + 221760 p^{2} T^{5} + 4956 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 43 | \( ( 1 + 4680 T^{2} - 58016 T^{3} + 10251086 T^{4} - 58016 p^{2} T^{5} + 4680 p^{4} T^{6} + p^{8} T^{8} )^{4} \) | |
| 47 | \( 1 + 8392 T^{2} + 30858916 T^{4} + 77629558512 T^{6} + 181309996781322 T^{8} + 338519134932094776 T^{10} + \)\(32\!\cdots\!52\)\( T^{12} + \)\(46\!\cdots\!92\)\( T^{14} - \)\(14\!\cdots\!37\)\( T^{16} + \)\(46\!\cdots\!92\)\( p^{4} T^{18} + \)\(32\!\cdots\!52\)\( p^{8} T^{20} + 338519134932094776 p^{12} T^{22} + 181309996781322 p^{16} T^{24} + 77629558512 p^{20} T^{26} + 30858916 p^{24} T^{28} + 8392 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 + 18920 T^{2} + 195169572 T^{4} + 1399932528560 T^{6} + 7743546865787466 T^{8} + 34991808675085792280 T^{10} + \)\(13\!\cdots\!12\)\( T^{12} + \)\(44\!\cdots\!80\)\( T^{14} + \)\(13\!\cdots\!79\)\( T^{16} + \)\(44\!\cdots\!80\)\( p^{4} T^{18} + \)\(13\!\cdots\!12\)\( p^{8} T^{20} + 34991808675085792280 p^{12} T^{22} + 7743546865787466 p^{16} T^{24} + 1399932528560 p^{20} T^{26} + 195169572 p^{24} T^{28} + 18920 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( ( 1 - 52 T + 450 T^{2} - 532224 T^{3} + 35666162 T^{4} - 868645540 T^{5} + 179864519968 T^{6} - 10977595733044 T^{7} + 261680209248047 T^{8} - 10977595733044 p^{2} T^{9} + 179864519968 p^{4} T^{10} - 868645540 p^{6} T^{11} + 35666162 p^{8} T^{12} - 532224 p^{10} T^{13} + 450 p^{12} T^{14} - 52 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 61 | \( 1 + 16316 T^{2} + 126039736 T^{4} + 653164234872 T^{6} + 2819894062793314 T^{8} + 11641021751267899300 T^{10} + \)\(46\!\cdots\!32\)\( T^{12} + \)\(16\!\cdots\!44\)\( T^{14} + \)\(61\!\cdots\!79\)\( T^{16} + \)\(16\!\cdots\!44\)\( p^{4} T^{18} + \)\(46\!\cdots\!32\)\( p^{8} T^{20} + 11641021751267899300 p^{12} T^{22} + 2819894062793314 p^{16} T^{24} + 653164234872 p^{20} T^{26} + 126039736 p^{24} T^{28} + 16316 p^{28} T^{30} + p^{32} T^{32} \) | |
| 67 | \( ( 1 + 152 T + 880 T^{2} - 696592 T^{3} + 25800514 T^{4} + 6259290472 T^{5} + 135213846720 T^{6} - 1900601864504 T^{7} + 289722470344099 T^{8} - 1900601864504 p^{2} T^{9} + 135213846720 p^{4} T^{10} + 6259290472 p^{6} T^{11} + 25800514 p^{8} T^{12} - 696592 p^{10} T^{13} + 880 p^{12} T^{14} + 152 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 9864 T^{2} + 51888284 T^{4} - 294531431096 T^{6} + 1789421441990854 T^{8} - 294531431096 p^{4} T^{10} + 51888284 p^{8} T^{12} - 9864 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 56 T - 15324 T^{2} - 438256 T^{3} + 161331658 T^{4} + 2027862984 T^{5} - 1181077316848 T^{6} - 5241850793144 T^{7} + 6684367707557907 T^{8} - 5241850793144 p^{2} T^{9} - 1181077316848 p^{4} T^{10} + 2027862984 p^{6} T^{11} + 161331658 p^{8} T^{12} - 438256 p^{10} T^{13} - 15324 p^{12} T^{14} + 56 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( 1 + 24968 T^{2} + 292531236 T^{4} + 25935455120 p T^{6} + 9731167232364042 T^{8} + 514375386606789128 p T^{10} + \)\(28\!\cdots\!16\)\( T^{12} + \)\(28\!\cdots\!32\)\( T^{14} + \)\(21\!\cdots\!95\)\( T^{16} + \)\(28\!\cdots\!32\)\( p^{4} T^{18} + \)\(28\!\cdots\!16\)\( p^{8} T^{20} + 514375386606789128 p^{13} T^{22} + 9731167232364042 p^{16} T^{24} + 25935455120 p^{21} T^{26} + 292531236 p^{24} T^{28} + 24968 p^{28} T^{30} + p^{32} T^{32} \) | |
| 83 | \( ( 1 + 36 T + 16478 T^{2} + 177884 T^{3} + 135298114 T^{4} + 177884 p^{2} T^{5} + 16478 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 89 | \( ( 1 + 256 T + 17284 T^{2} - 187392 T^{3} + 84842826 T^{4} + 25389871104 T^{5} + 1853625901584 T^{6} + 111590371439872 T^{7} + 11164291531924819 T^{8} + 111590371439872 p^{2} T^{9} + 1853625901584 p^{4} T^{10} + 25389871104 p^{6} T^{11} + 84842826 p^{8} T^{12} - 187392 p^{10} T^{13} + 17284 p^{12} T^{14} + 256 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 32 T + 19484 T^{2} + 1437536 T^{3} + 199130566 T^{4} + 1437536 p^{2} T^{5} + 19484 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
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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
−2.89657492904337933689656031560, −2.76029155226033128961430379692, −2.62039827812119464438629732683, −2.54353436039945922888048098152, −2.46441995515019201546666169236, −2.38371514487879422841599697369, −2.16521030568715761176671445527, −2.07182211286507906868676027167, −1.88904864333896710363744293001, −1.74581870902888099411336164509, −1.69619416534394396793341540028, −1.62430718945620080107624488268, −1.61445960986782808735707971370, −1.58796734081654141951433651755, −1.56046746070257300672003763719, −1.44354890438131644880745859580, −1.40318957077855367834269812886, −1.01767608208821484958135721309, −0.990176941780140191936942690206, −0.844874282571956507355586132846, −0.58701683502079943755496186259, −0.42706778319379278154165946130, −0.25094203447849710628670608485, −0.23517362902267529284147408568, −0.23158182873288350232459241849, 0.23158182873288350232459241849, 0.23517362902267529284147408568, 0.25094203447849710628670608485, 0.42706778319379278154165946130, 0.58701683502079943755496186259, 0.844874282571956507355586132846, 0.990176941780140191936942690206, 1.01767608208821484958135721309, 1.40318957077855367834269812886, 1.44354890438131644880745859580, 1.56046746070257300672003763719, 1.58796734081654141951433651755, 1.61445960986782808735707971370, 1.62430718945620080107624488268, 1.69619416534394396793341540028, 1.74581870902888099411336164509, 1.88904864333896710363744293001, 2.07182211286507906868676027167, 2.16521030568715761176671445527, 2.38371514487879422841599697369, 2.46441995515019201546666169236, 2.54353436039945922888048098152, 2.62039827812119464438629732683, 2.76029155226033128961430379692, 2.89657492904337933689656031560
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.