Dirichlet series
| L(s) = 1 | − 10·9-s − 4·17-s + 18·25-s − 4·27-s + 44·29-s − 12·43-s − 8·49-s + 8·53-s − 8·61-s + 8·79-s + 29·81-s + 32·101-s + 12·103-s − 20·107-s + 16·113-s + 62·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 40·153-s + 157-s + 163-s + 167-s + 173-s + ⋯ |
| L(s) = 1 | − 3.33·9-s − 0.970·17-s + 18/5·25-s − 0.769·27-s + 8.17·29-s − 1.82·43-s − 8/7·49-s + 1.09·53-s − 1.02·61-s + 0.900·79-s + 29/9·81-s + 3.18·101-s + 1.18·103-s − 1.93·107-s + 1.50·113-s + 5.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 3.23·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{16} \cdot 13^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{16} \cdot 13^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 7^{16} \cdot 13^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.72643\times 10^{25}\) |
| Root analytic conductor: | \(6.14696\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 7^{16} \cdot 13^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(15.94996283\) |
| \(L(\frac12)\) | \(\approx\) | \(15.94996283\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{8} \) | |
| 13 | \( 1 \) | |
| good | 3 | \( ( 1 + 5 T^{2} + 2 T^{3} + 23 T^{4} - 10 T^{5} + 71 T^{6} - 44 T^{7} + 172 T^{8} - 44 p T^{9} + 71 p^{2} T^{10} - 10 p^{3} T^{11} + 23 p^{4} T^{12} + 2 p^{5} T^{13} + 5 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 5 | \( 1 - 18 T^{2} + 33 p T^{4} - 1192 T^{6} + 1654 p T^{8} - 2014 p^{2} T^{10} + 264568 T^{12} - 1385626 T^{14} + 7206581 T^{16} - 1385626 p^{2} T^{18} + 264568 p^{4} T^{20} - 2014 p^{8} T^{22} + 1654 p^{9} T^{24} - 1192 p^{10} T^{26} + 33 p^{13} T^{28} - 18 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 - 62 T^{2} + 2383 T^{4} - 65804 T^{6} + 1448479 T^{8} - 26383816 T^{10} + 410052117 T^{12} - 5513866654 T^{14} + 64817564296 T^{16} - 5513866654 p^{2} T^{18} + 410052117 p^{4} T^{20} - 26383816 p^{6} T^{22} + 1448479 p^{8} T^{24} - 65804 p^{10} T^{26} + 2383 p^{12} T^{28} - 62 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 2 T + 60 T^{2} + 56 T^{3} + 1818 T^{4} + 938 T^{5} + 2576 p T^{6} + 26202 T^{7} + 858915 T^{8} + 26202 p T^{9} + 2576 p^{3} T^{10} + 938 p^{3} T^{11} + 1818 p^{4} T^{12} + 56 p^{5} T^{13} + 60 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 126 T^{2} + 7967 T^{4} - 328336 T^{6} + 9763215 T^{8} - 219839492 T^{10} + 3896050157 T^{12} - 59376356222 T^{14} + 983287620232 T^{16} - 59376356222 p^{2} T^{18} + 3896050157 p^{4} T^{20} - 219839492 p^{6} T^{22} + 9763215 p^{8} T^{24} - 328336 p^{10} T^{26} + 7967 p^{12} T^{28} - 126 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 4 p T^{2} + 32 T^{3} + 4136 T^{4} + 480 T^{5} + 128308 T^{6} - 30016 T^{7} + 3195918 T^{8} - 30016 p T^{9} + 128308 p^{2} T^{10} + 480 p^{3} T^{11} + 4136 p^{4} T^{12} + 32 p^{5} T^{13} + 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 22 T + 349 T^{2} - 3968 T^{3} + 38262 T^{4} - 309218 T^{5} + 2224848 T^{6} - 14043982 T^{7} + 80270429 T^{8} - 14043982 p T^{9} + 2224848 p^{2} T^{10} - 309218 p^{3} T^{11} + 38262 p^{4} T^{12} - 3968 p^{5} T^{13} + 349 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( 1 - 112 T^{2} + 10054 T^{4} - 663312 T^{6} + 37238817 T^{8} - 1770900480 T^{10} + 74155545630 T^{12} - 2733161113024 T^{14} + 89896366998916 T^{16} - 2733161113024 p^{2} T^{18} + 74155545630 p^{4} T^{20} - 1770900480 p^{6} T^{22} + 37238817 p^{8} T^{24} - 663312 p^{10} T^{26} + 10054 p^{12} T^{28} - 112 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 - 348 T^{2} + 55750 T^{4} - 5334880 T^{6} + 326466609 T^{8} - 12059796984 T^{10} + 147240616278 T^{12} + 10676178288852 T^{14} - 682301076408764 T^{16} + 10676178288852 p^{2} T^{18} + 147240616278 p^{4} T^{20} - 12059796984 p^{6} T^{22} + 326466609 p^{8} T^{24} - 5334880 p^{10} T^{26} + 55750 p^{12} T^{28} - 348 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( 1 - 116 T^{2} + 5958 T^{4} - 183552 T^{6} + 5462513 T^{8} - 160988552 T^{10} + 6920309974 T^{12} - 668156741060 T^{14} + 39622037835044 T^{16} - 668156741060 p^{2} T^{18} + 6920309974 p^{4} T^{20} - 160988552 p^{6} T^{22} + 5462513 p^{8} T^{24} - 183552 p^{10} T^{26} + 5958 p^{12} T^{28} - 116 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 + 6 T + 75 T^{2} + 268 T^{3} + 3727 T^{4} + 20392 T^{5} + 250249 T^{6} + 909938 T^{7} + 10384508 T^{8} + 909938 p T^{9} + 250249 p^{2} T^{10} + 20392 p^{3} T^{11} + 3727 p^{4} T^{12} + 268 p^{5} T^{13} + 75 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 448 T^{2} + 100038 T^{4} - 315720 p T^{6} + 1641254969 T^{8} - 143811301408 T^{10} + 10339158247606 T^{12} - 13246282381400 p T^{14} + 31743997447328660 T^{16} - 13246282381400 p^{3} T^{18} + 10339158247606 p^{4} T^{20} - 143811301408 p^{6} T^{22} + 1641254969 p^{8} T^{24} - 315720 p^{11} T^{26} + 100038 p^{12} T^{28} - 448 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 4 T + 224 T^{2} - 800 T^{3} + 25774 T^{4} - 79860 T^{5} + 2022616 T^{6} - 5508148 T^{7} + 120632807 T^{8} - 5508148 p T^{9} + 2022616 p^{2} T^{10} - 79860 p^{3} T^{11} + 25774 p^{4} T^{12} - 800 p^{5} T^{13} + 224 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 - 680 T^{2} + 227470 T^{4} - 49591472 T^{6} + 7873411241 T^{8} - 963951514400 T^{10} + 94045513598166 T^{12} - 7452960385827656 T^{14} + 484654913392810452 T^{16} - 7452960385827656 p^{2} T^{18} + 94045513598166 p^{4} T^{20} - 963951514400 p^{6} T^{22} + 7873411241 p^{8} T^{24} - 49591472 p^{10} T^{26} + 227470 p^{12} T^{28} - 680 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 4 T + 226 T^{2} + 864 T^{3} + 24273 T^{4} + 109056 T^{5} + 1742754 T^{6} + 9653740 T^{7} + 108166516 T^{8} + 9653740 p T^{9} + 1742754 p^{2} T^{10} + 109056 p^{3} T^{11} + 24273 p^{4} T^{12} + 864 p^{5} T^{13} + 226 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 636 T^{2} + 201838 T^{4} - 42572224 T^{6} + 6701115489 T^{8} - 836404133304 T^{10} + 85713795072342 T^{12} - 7358793835031916 T^{14} + 534925707276599524 T^{16} - 7358793835031916 p^{2} T^{18} + 85713795072342 p^{4} T^{20} - 836404133304 p^{6} T^{22} + 6701115489 p^{8} T^{24} - 42572224 p^{10} T^{26} + 201838 p^{12} T^{28} - 636 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 - 320 T^{2} + 61528 T^{4} - 8001344 T^{6} + 755666780 T^{8} - 51308246720 T^{10} + 2249354991720 T^{12} - 38859832180928 T^{14} - 1192129125034170 T^{16} - 38859832180928 p^{2} T^{18} + 2249354991720 p^{4} T^{20} - 51308246720 p^{6} T^{22} + 755666780 p^{8} T^{24} - 8001344 p^{10} T^{26} + 61528 p^{12} T^{28} - 320 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 772 T^{2} + 282118 T^{4} - 64658544 T^{6} + 10419834513 T^{8} - 1263553418760 T^{10} + 122174565931446 T^{12} - 10061786521834564 T^{14} + 756301601251325860 T^{16} - 10061786521834564 p^{2} T^{18} + 122174565931446 p^{4} T^{20} - 1263553418760 p^{6} T^{22} + 10419834513 p^{8} T^{24} - 64658544 p^{10} T^{26} + 282118 p^{12} T^{28} - 772 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 4 T + 304 T^{2} - 1156 T^{3} + 50272 T^{4} - 208292 T^{5} + 5836752 T^{6} - 23572964 T^{7} + 515508478 T^{8} - 23572964 p T^{9} + 5836752 p^{2} T^{10} - 208292 p^{3} T^{11} + 50272 p^{4} T^{12} - 1156 p^{5} T^{13} + 304 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 672 T^{2} + 221670 T^{4} - 47919008 T^{6} + 7631842561 T^{8} - 959103793088 T^{10} + 100521201925918 T^{12} - 9279560368991872 T^{14} + 792373557138192548 T^{16} - 9279560368991872 p^{2} T^{18} + 100521201925918 p^{4} T^{20} - 959103793088 p^{6} T^{22} + 7631842561 p^{8} T^{24} - 47919008 p^{10} T^{26} + 221670 p^{12} T^{28} - 672 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 - 918 T^{2} + 397743 T^{4} - 108084392 T^{6} + 20699681191 T^{8} - 2994871483724 T^{10} + 347274153483757 T^{12} - 34506218580789910 T^{14} + 3150777374201176952 T^{16} - 34506218580789910 p^{2} T^{18} + 347274153483757 p^{4} T^{20} - 2994871483724 p^{6} T^{22} + 20699681191 p^{8} T^{24} - 108084392 p^{10} T^{26} + 397743 p^{12} T^{28} - 918 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 - 754 T^{2} + 291663 T^{4} - 75917584 T^{6} + 14838722415 T^{8} - 2321543382604 T^{10} + 304481996216653 T^{12} - 34752766858161266 T^{14} + 3546726608920010344 T^{16} - 34752766858161266 p^{2} T^{18} + 304481996216653 p^{4} T^{20} - 2321543382604 p^{6} T^{22} + 14838722415 p^{8} T^{24} - 75917584 p^{10} T^{26} + 291663 p^{12} T^{28} - 754 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−1.95067600950685436685736322898, −1.94745277679299836498939914974, −1.80225110072130960808220382850, −1.79557558256969655480373035610, −1.76282753914571325891857826640, −1.72816206445115593472958707433, −1.56809062095288780670498944473, −1.41505738173023489250215027169, −1.34192192377389136765721403972, −1.32530464343064680322807877219, −1.28787445465462228220106613937, −1.18656482950223277969859999021, −1.10998312757624947405084360378, −1.06367377592841536765818046377, −1.01671700933554016377652800529, −0.76407858948884052793142028874, −0.73886639848652566918606491520, −0.69737685811817522001143864588, −0.68689115513874057587174128738, −0.61992305001248106256363655565, −0.45760474117859880152562752226, −0.40980423092886500616129478233, −0.39416703092203424141795327325, −0.16238168348727500233620762862, −0.11552890404840550831708274963, 0.11552890404840550831708274963, 0.16238168348727500233620762862, 0.39416703092203424141795327325, 0.40980423092886500616129478233, 0.45760474117859880152562752226, 0.61992305001248106256363655565, 0.68689115513874057587174128738, 0.69737685811817522001143864588, 0.73886639848652566918606491520, 0.76407858948884052793142028874, 1.01671700933554016377652800529, 1.06367377592841536765818046377, 1.10998312757624947405084360378, 1.18656482950223277969859999021, 1.28787445465462228220106613937, 1.32530464343064680322807877219, 1.34192192377389136765721403972, 1.41505738173023489250215027169, 1.56809062095288780670498944473, 1.72816206445115593472958707433, 1.76282753914571325891857826640, 1.79557558256969655480373035610, 1.80225110072130960808220382850, 1.94745277679299836498939914974, 1.95067600950685436685736322898
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.