Dirichlet series
| L(s) = 1 | − 128·4-s + 9.21e3·16-s − 2.38e4·25-s − 4.47e4·37-s + 4.50e4·43-s − 4.91e5·64-s − 3.30e5·67-s − 1.10e5·79-s + 3.05e6·100-s − 9.04e5·109-s + 1.40e6·121-s + 127-s + 131-s + 137-s + 139-s + 5.73e6·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.52e6·169-s − 5.77e6·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4·4-s + 9·16-s − 7.63·25-s − 5.37·37-s + 3.71·43-s − 15·64-s − 9.00·67-s − 1.99·79-s + 30.5·100-s − 7.29·109-s + 8.71·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 21.5·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 14.8·169-s − 14.8·172-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{32} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.57078\times 10^{34}\) |
| Root analytic conductor: | \(11.8936\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [5/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(0.6618234164\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6618234164\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{4} T^{2} )^{8} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( ( 1 + 11936 T^{2} + 89329536 T^{4} + 437199411616 T^{6} + 1607733918098786 T^{8} + 437199411616 p^{10} T^{10} + 89329536 p^{20} T^{12} + 11936 p^{30} T^{14} + p^{40} T^{16} )^{2} \) |
| 11 | \( ( 1 - 701832 T^{2} + 266344204924 T^{4} - 68410065807031032 T^{6} + \)\(12\!\cdots\!94\)\( T^{8} - 68410065807031032 p^{10} T^{10} + 266344204924 p^{20} T^{12} - 701832 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 2760880 T^{2} + 261805798272 p T^{4} - 2448116804271915824 T^{6} + \)\(11\!\cdots\!30\)\( T^{8} - 2448116804271915824 p^{10} T^{10} + 261805798272 p^{21} T^{12} - 2760880 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 4687120 T^{2} + 599277962240 p T^{4} + 13714624740038826256 T^{6} + \)\(17\!\cdots\!46\)\( T^{8} + 13714624740038826256 p^{10} T^{10} + 599277962240 p^{21} T^{12} + 4687120 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 13575288 T^{2} + 4909699781268 p T^{4} - \)\(40\!\cdots\!16\)\( T^{6} + \)\(11\!\cdots\!90\)\( T^{8} - \)\(40\!\cdots\!16\)\( p^{10} T^{10} + 4909699781268 p^{21} T^{12} - 13575288 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 22523144 T^{2} + 181747670376028 T^{4} - \)\(34\!\cdots\!08\)\( T^{6} - \)\(19\!\cdots\!06\)\( T^{8} - \)\(34\!\cdots\!08\)\( p^{10} T^{10} + 181747670376028 p^{20} T^{12} - 22523144 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 95507872 T^{2} + 4437659166390108 T^{4} - \)\(13\!\cdots\!96\)\( T^{6} + \)\(31\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!96\)\( p^{10} T^{10} + 4437659166390108 p^{20} T^{12} - 95507872 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 168552856 T^{2} + 13669184212108956 T^{4} - \)\(69\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!74\)\( T^{8} - \)\(69\!\cdots\!76\)\( p^{10} T^{10} + 13669184212108956 p^{20} T^{12} - 168552856 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 11192 T + 253012672 T^{2} + 2016445918808 T^{3} + 26245272146422834 T^{4} + 2016445918808 p^{5} T^{5} + 253012672 p^{10} T^{6} + 11192 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 41 | \( ( 1 + 596354288 T^{2} + 175362052971160704 T^{4} + \)\(33\!\cdots\!64\)\( T^{6} + \)\(45\!\cdots\!70\)\( T^{8} + \)\(33\!\cdots\!64\)\( p^{10} T^{10} + 175362052971160704 p^{20} T^{12} + 596354288 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 11272 T + 558832044 T^{2} - 4660176835400 T^{3} + 121824708065369558 T^{4} - 4660176835400 p^{5} T^{5} + 558832044 p^{10} T^{6} - 11272 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 47 | \( ( 1 + 1304976728 T^{2} + 836470465047407004 T^{4} + \)\(33\!\cdots\!16\)\( T^{6} + \)\(92\!\cdots\!62\)\( T^{8} + \)\(33\!\cdots\!16\)\( p^{10} T^{10} + 836470465047407004 p^{20} T^{12} + 1304976728 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 53 | \( ( 1 - 646977888 T^{2} + 110123237127180388 T^{4} - \)\(51\!\cdots\!24\)\( T^{6} + \)\(27\!\cdots\!66\)\( T^{8} - \)\(51\!\cdots\!24\)\( p^{10} T^{10} + 110123237127180388 p^{20} T^{12} - 646977888 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 59 | \( ( 1 + 2408721528 T^{2} + 3848035930741198524 T^{4} + \)\(40\!\cdots\!04\)\( T^{6} + \)\(33\!\cdots\!30\)\( T^{8} + \)\(40\!\cdots\!04\)\( p^{10} T^{10} + 3848035930741198524 p^{20} T^{12} + 2408721528 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 4065340432 T^{2} + 8523944720148550080 T^{4} - \)\(11\!\cdots\!16\)\( T^{6} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(11\!\cdots\!16\)\( p^{10} T^{10} + 8523944720148550080 p^{20} T^{12} - 4065340432 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 82736 T + 5267105116 T^{2} + 261649425413936 T^{3} + 9709239449399191702 T^{4} + 261649425413936 p^{5} T^{5} + 5267105116 p^{10} T^{6} + 82736 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 71 | \( ( 1 - 9337899496 T^{2} + 41574749879332606620 T^{4} - \)\(11\!\cdots\!84\)\( T^{6} + \)\(24\!\cdots\!82\)\( T^{8} - \)\(11\!\cdots\!84\)\( p^{10} T^{10} + 41574749879332606620 p^{20} T^{12} - 9337899496 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 13359987168 T^{2} + 83726523504035216832 T^{4} - \)\(31\!\cdots\!44\)\( T^{6} + \)\(80\!\cdots\!02\)\( T^{8} - \)\(31\!\cdots\!44\)\( p^{10} T^{10} + 83726523504035216832 p^{20} T^{12} - 13359987168 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 27720 T + 10249680588 T^{2} + 254590843809960 T^{3} + 44633400589566907430 T^{4} + 254590843809960 p^{5} T^{5} + 10249680588 p^{10} T^{6} + 27720 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 83 | \( ( 1 + 24407385432 T^{2} + \)\(28\!\cdots\!76\)\( T^{4} + \)\(19\!\cdots\!00\)\( T^{6} + \)\(94\!\cdots\!94\)\( T^{8} + \)\(19\!\cdots\!00\)\( p^{10} T^{10} + \)\(28\!\cdots\!76\)\( p^{20} T^{12} + 24407385432 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 25714834992 T^{2} + \)\(35\!\cdots\!96\)\( T^{4} + \)\(32\!\cdots\!56\)\( T^{6} + \)\(21\!\cdots\!86\)\( T^{8} + \)\(32\!\cdots\!56\)\( p^{10} T^{10} + \)\(35\!\cdots\!96\)\( p^{20} T^{12} + 25714834992 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 42559338432 T^{2} + \)\(88\!\cdots\!76\)\( T^{4} - \)\(12\!\cdots\!88\)\( T^{6} + \)\(11\!\cdots\!90\)\( T^{8} - \)\(12\!\cdots\!88\)\( p^{10} T^{10} + \)\(88\!\cdots\!76\)\( p^{20} T^{12} - 42559338432 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
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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
−1.92147314440224418685354009998, −1.73111508764768032097479639366, −1.67370031725591399018520209999, −1.62157382798276842847472823069, −1.59293892001301166000864038369, −1.56057476533987381913038249752, −1.50108487225938361556719064699, −1.49978032644509267154250289222, −1.32608892818626536358797280503, −1.12363797581267672709783204886, −1.11920304787739641123358978106, −1.06249547602665814181668982646, −0.954980966400728623586216207545, −0.75843979795144732223326499181, −0.61689428038983038079262349186, −0.61297543198369416963344841441, −0.60096449449186839440675589680, −0.54684740328724884636634943699, −0.52682704487576477546816528551, −0.43216124155535586820634662027, −0.27475825161678761800682821854, −0.23618893473765333262511666610, −0.17711980661667230394843041469, −0.17139534796024498261442685020, −0.06227551642013602607752612188, 0.06227551642013602607752612188, 0.17139534796024498261442685020, 0.17711980661667230394843041469, 0.23618893473765333262511666610, 0.27475825161678761800682821854, 0.43216124155535586820634662027, 0.52682704487576477546816528551, 0.54684740328724884636634943699, 0.60096449449186839440675589680, 0.61297543198369416963344841441, 0.61689428038983038079262349186, 0.75843979795144732223326499181, 0.954980966400728623586216207545, 1.06249547602665814181668982646, 1.11920304787739641123358978106, 1.12363797581267672709783204886, 1.32608892818626536358797280503, 1.49978032644509267154250289222, 1.50108487225938361556719064699, 1.56057476533987381913038249752, 1.59293892001301166000864038369, 1.62157382798276842847472823069, 1.67370031725591399018520209999, 1.73111508764768032097479639366, 1.92147314440224418685354009998
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.