Dirichlet series
| L(s) = 1 | + 40·7-s − 44·13-s + 32·16-s − 46·25-s + 320·31-s + 4·37-s + 256·43-s + 800·49-s + 8·61-s + 88·67-s + 364·73-s − 1.76e3·91-s + 304·97-s + 736·103-s + 1.28e3·112-s − 1.04e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 968·169-s + 173-s + ⋯ |
| L(s) = 1 | + 40/7·7-s − 3.38·13-s + 2·16-s − 1.83·25-s + 10.3·31-s + 4/37·37-s + 5.95·43-s + 16.3·49-s + 8/61·61-s + 1.31·67-s + 4.98·73-s − 19.3·91-s + 3.13·97-s + 7.14·103-s + 80/7·112-s − 8.59·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 5.72·169-s + 0.00578·173-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{64} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.83772\times 10^{16}\) |
| Root analytic conductor: | \(3.32196\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{64} \cdot 5^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(170.3997140\) |
| \(L(\frac12)\) | \(\approx\) | \(170.3997140\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 46 T^{2} - 31 p T^{4} - 28742 T^{6} - 4696 p^{3} T^{8} - 28742 p^{4} T^{10} - 31 p^{9} T^{12} + 46 p^{12} T^{14} + p^{16} T^{16} \) | |
| good | 2 | \( 1 - p^{5} T^{4} + 215 p T^{8} - 41 p^{6} T^{12} - 3599 T^{16} - 41 p^{14} T^{20} + 215 p^{17} T^{24} - p^{29} T^{28} + p^{32} T^{32} \) |
| 7 | \( ( 1 - 20 T + 200 T^{2} - 1672 T^{3} + 1451 p T^{4} - 37264 T^{5} + 111672 T^{6} - 79764 T^{7} - 2482700 T^{8} - 79764 p^{2} T^{9} + 111672 p^{4} T^{10} - 37264 p^{6} T^{11} + 1451 p^{9} T^{12} - 1672 p^{10} T^{13} + 200 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 11 | \( ( 1 + 520 T^{2} + 13040 p T^{4} + 27225448 T^{6} + 3802453438 T^{8} + 27225448 p^{4} T^{10} + 13040 p^{9} T^{12} + 520 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 22 T + 242 T^{2} + 2024 T^{3} - 18202 T^{4} - 315238 T^{5} - 482064 T^{6} + 39413022 T^{7} + 1161118891 T^{8} + 39413022 p^{2} T^{9} - 482064 p^{4} T^{10} - 315238 p^{6} T^{11} - 18202 p^{8} T^{12} + 2024 p^{10} T^{13} + 242 p^{12} T^{14} + 22 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 - 287090 T^{4} + 35878666897 T^{8} - 2768406563136530 T^{12} + \)\(20\!\cdots\!08\)\( T^{16} - 2768406563136530 p^{8} T^{20} + 35878666897 p^{16} T^{24} - 287090 p^{24} T^{28} + p^{32} T^{32} \) | |
| 19 | \( ( 1 - 878 T^{2} + 371413 T^{4} - 182405990 T^{6} + 83587290688 T^{8} - 182405990 p^{4} T^{10} + 371413 p^{8} T^{12} - 878 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 - 1158182 T^{4} + 777505126033 T^{8} - 350169895125016574 T^{12} + \)\(11\!\cdots\!20\)\( T^{16} - 350169895125016574 p^{8} T^{20} + 777505126033 p^{16} T^{24} - 1158182 p^{24} T^{28} + p^{32} T^{32} \) | |
| 29 | \( ( 1 - 2906 T^{2} + 5451841 T^{4} - 6894318578 T^{6} + 6752090922820 T^{8} - 6894318578 p^{4} T^{10} + 5451841 p^{8} T^{12} - 2906 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 80 T + 5626 T^{2} - 241520 T^{3} + 8947186 T^{4} - 241520 p^{2} T^{5} + 5626 p^{4} T^{6} - 80 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 37 | \( ( 1 - 2 T + 2 T^{2} - 41488 T^{3} + 1874993 T^{4} + 101905820 T^{5} + 653065446 T^{6} + 62193454974 T^{7} - 3306296938304 T^{8} + 62193454974 p^{2} T^{9} + 653065446 p^{4} T^{10} + 101905820 p^{6} T^{11} + 1874993 p^{8} T^{12} - 41488 p^{10} T^{13} + 2 p^{12} T^{14} - 2 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 8866 T^{2} + 38220001 T^{4} + 106963893634 T^{6} + 211874447352580 T^{8} + 106963893634 p^{4} T^{10} + 38220001 p^{8} T^{12} + 8866 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 128 T + 8192 T^{2} - 484672 T^{3} + 19178684 T^{4} - 155604544 T^{5} - 19740923904 T^{6} + 2031522937344 T^{7} - 122952302594234 T^{8} + 2031522937344 p^{2} T^{9} - 19740923904 p^{4} T^{10} - 155604544 p^{6} T^{11} + 19178684 p^{8} T^{12} - 484672 p^{10} T^{13} + 8192 p^{12} T^{14} - 128 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 47 | \( 1 - 15838814 T^{4} + 84897466202689 T^{8} + 17725702077609394018 T^{12} - \)\(16\!\cdots\!32\)\( T^{16} + 17725702077609394018 p^{8} T^{20} + 84897466202689 p^{16} T^{24} - 15838814 p^{24} T^{28} + p^{32} T^{32} \) | |
| 53 | \( 1 - 9887558 T^{4} + 77715463879441 T^{8} - \)\(86\!\cdots\!38\)\( T^{12} + \)\(11\!\cdots\!44\)\( T^{16} - \)\(86\!\cdots\!38\)\( p^{8} T^{20} + 77715463879441 p^{16} T^{24} - 9887558 p^{24} T^{28} + p^{32} T^{32} \) | |
| 59 | \( ( 1 - 11474 T^{2} + 69173809 T^{4} - 311922002642 T^{6} + 1169491962965956 T^{8} - 311922002642 p^{4} T^{10} + 69173809 p^{8} T^{12} - 11474 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 2 T + 3775 T^{2} - 38594 T^{3} + 672106 T^{4} - 38594 p^{2} T^{5} + 3775 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 67 | \( ( 1 - 44 T + 968 T^{2} - 842116 T^{3} + 36447056 T^{4} + 359889860 T^{5} + 303463774680 T^{6} - 16149526272660 T^{7} - 331789027751330 T^{8} - 16149526272660 p^{2} T^{9} + 303463774680 p^{4} T^{10} + 359889860 p^{6} T^{11} + 36447056 p^{8} T^{12} - 842116 p^{10} T^{13} + 968 p^{12} T^{14} - 44 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 24244 T^{2} + 320951596 T^{4} + 2725411579420 T^{6} + 16329097048557910 T^{8} + 2725411579420 p^{4} T^{10} + 320951596 p^{8} T^{12} + 24244 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 182 T + 16562 T^{2} - 1341352 T^{3} + 158528129 T^{4} - 16985096476 T^{5} + 1365357280086 T^{6} - 100005577663278 T^{7} + 7255244674113184 T^{8} - 100005577663278 p^{2} T^{9} + 1365357280086 p^{4} T^{10} - 16985096476 p^{6} T^{11} + 158528129 p^{8} T^{12} - 1341352 p^{10} T^{13} + 16562 p^{12} T^{14} - 182 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 24620 T^{2} + 254815228 T^{4} - 1536542774276 T^{6} + 8276199074151670 T^{8} - 1536542774276 p^{4} T^{10} + 254815228 p^{8} T^{12} - 24620 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 83 | \( 1 + 99727936 T^{4} + 9020368131608572 T^{8} + \)\(54\!\cdots\!40\)\( T^{12} + \)\(31\!\cdots\!06\)\( T^{16} + \)\(54\!\cdots\!40\)\( p^{8} T^{20} + 9020368131608572 p^{16} T^{24} + 99727936 p^{24} T^{28} + p^{32} T^{32} \) | |
| 89 | \( ( 1 - 37742 T^{2} + 768946177 T^{4} - 10135389474110 T^{6} + 94882097979750052 T^{8} - 10135389474110 p^{4} T^{10} + 768946177 p^{8} T^{12} - 37742 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 152 T + 11552 T^{2} - 835816 T^{3} - 35181040 T^{4} + 7973620136 T^{5} - 456284693664 T^{6} + 39843483087192 T^{7} - 2956492411793378 T^{8} + 39843483087192 p^{2} T^{9} - 456284693664 p^{4} T^{10} + 7973620136 p^{6} T^{11} - 35181040 p^{8} T^{12} - 835816 p^{10} T^{13} + 11552 p^{12} T^{14} - 152 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
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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
−2.72614356497619300960919139756, −2.59356267793465391677435786490, −2.57257123149080281372950620429, −2.53337578011472565974973638678, −2.40716495194493678668343117892, −2.26184617875281651345208730810, −2.19825934509886808581872747117, −2.17592610265029992249735512645, −2.11541996074735626449425123653, −2.11083906828937059240056005447, −2.03296026415036993689093100063, −1.99957489777353988798841800632, −1.62318013914638854887966657450, −1.59261881613131668175511098104, −1.28426959938822004009626765995, −1.17117014731957850624347405170, −1.09554098315837761099557712382, −1.09326627206871328541584284706, −1.01884897897469706079676568379, −0.920226604053573774163597421288, −0.893798655412328903346709409215, −0.837656052395920540823167060189, −0.54812884599864316351864346759, −0.30102080823318750274862993689, −0.29000788318820273913061228029, 0.29000788318820273913061228029, 0.30102080823318750274862993689, 0.54812884599864316351864346759, 0.837656052395920540823167060189, 0.893798655412328903346709409215, 0.920226604053573774163597421288, 1.01884897897469706079676568379, 1.09326627206871328541584284706, 1.09554098315837761099557712382, 1.17117014731957850624347405170, 1.28426959938822004009626765995, 1.59261881613131668175511098104, 1.62318013914638854887966657450, 1.99957489777353988798841800632, 2.03296026415036993689093100063, 2.11083906828937059240056005447, 2.11541996074735626449425123653, 2.17592610265029992249735512645, 2.19825934509886808581872747117, 2.26184617875281651345208730810, 2.40716495194493678668343117892, 2.53337578011472565974973638678, 2.57257123149080281372950620429, 2.59356267793465391677435786490, 2.72614356497619300960919139756
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.