Dirichlet series
| L(s) = 1 | + 4-s − 8·7-s + 4·8-s + 16-s + 8·17-s + 28·25-s − 8·28-s + 16·31-s + 8·32-s − 16·41-s − 24·47-s − 12·49-s − 32·56-s + 11·64-s + 8·68-s − 48·71-s − 48·79-s + 16·89-s + 32·97-s + 28·100-s − 8·112-s − 64·119-s + 92·121-s + 16·124-s + 127-s + 12·128-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.02·7-s + 1.41·8-s + 1/4·16-s + 1.94·17-s + 28/5·25-s − 1.51·28-s + 2.87·31-s + 1.41·32-s − 2.49·41-s − 3.50·47-s − 1.71·49-s − 4.27·56-s + 11/8·64-s + 0.970·68-s − 5.69·71-s − 5.40·79-s + 1.69·89-s + 3.24·97-s + 14/5·100-s − 0.755·112-s − 5.86·119-s + 8.36·121-s + 1.43·124-s + 0.0887·127-s + 1.06·128-s + 0.0873·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 3^{32} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.10971\times 10^{16}\) |
| Root analytic conductor: | \(3.30507\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 3^{32} \cdot 19^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.377149617\) |
| \(L(\frac12)\) | \(\approx\) | \(7.377149617\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T^{2} - p^{2} T^{3} + 3 p T^{6} + p^{3} T^{8} + 3 p^{3} T^{10} - p^{7} T^{13} - p^{6} T^{14} + p^{8} T^{16} \) |
| 3 | \( 1 \) | |
| 19 | \( ( 1 + T^{2} )^{8} \) | |
| good | 5 | \( 1 - 28 T^{2} + 438 T^{4} - 4848 T^{6} + 42529 T^{8} - 62616 p T^{10} + 2009318 T^{12} - 11529948 T^{14} + 60231524 T^{16} - 11529948 p^{2} T^{18} + 2009318 p^{4} T^{20} - 62616 p^{7} T^{22} + 42529 p^{8} T^{24} - 4848 p^{10} T^{26} + 438 p^{12} T^{28} - 28 p^{14} T^{30} + p^{16} T^{32} \) |
| 7 | \( ( 1 + 4 T + 30 T^{2} + 68 T^{3} + 360 T^{4} + 584 T^{5} + 3284 T^{6} + 4880 T^{7} + 26161 T^{8} + 4880 p T^{9} + 3284 p^{2} T^{10} + 584 p^{3} T^{11} + 360 p^{4} T^{12} + 68 p^{5} T^{13} + 30 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 11 | \( 1 - 92 T^{2} + 4326 T^{4} - 138936 T^{6} + 3407825 T^{8} - 67421256 T^{10} + 1109052726 T^{12} - 15437378244 T^{14} + 183500018436 T^{16} - 15437378244 p^{2} T^{18} + 1109052726 p^{4} T^{20} - 67421256 p^{6} T^{22} + 3407825 p^{8} T^{24} - 138936 p^{10} T^{26} + 4326 p^{12} T^{28} - 92 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 - 96 T^{2} + 4590 T^{4} - 145832 T^{6} + 3460001 T^{8} - 65542128 T^{10} + 1045890990 T^{12} - 14852004936 T^{14} + 197390117924 T^{16} - 14852004936 p^{2} T^{18} + 1045890990 p^{4} T^{20} - 65542128 p^{6} T^{22} + 3460001 p^{8} T^{24} - 145832 p^{10} T^{26} + 4590 p^{12} T^{28} - 96 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 - 4 T + 60 T^{2} - 216 T^{3} + 1706 T^{4} - 4476 T^{5} + 30416 T^{6} - 51164 T^{7} + 480003 T^{8} - 51164 p T^{9} + 30416 p^{2} T^{10} - 4476 p^{3} T^{11} + 1706 p^{4} T^{12} - 216 p^{5} T^{13} + 60 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 + 60 T^{2} - 76 T^{3} + 109 p T^{4} - 4156 T^{5} + 75872 T^{6} - 152912 T^{7} + 1893888 T^{8} - 152912 p T^{9} + 75872 p^{2} T^{10} - 4156 p^{3} T^{11} + 109 p^{5} T^{12} - 76 p^{5} T^{13} + 60 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 216 T^{2} + 21422 T^{4} - 1343568 T^{6} + 63937425 T^{8} - 2636294320 T^{10} + 99364061438 T^{12} - 3391411721192 T^{14} + 103804339922532 T^{16} - 3391411721192 p^{2} T^{18} + 99364061438 p^{4} T^{20} - 2636294320 p^{6} T^{22} + 63937425 p^{8} T^{24} - 1343568 p^{10} T^{26} + 21422 p^{12} T^{28} - 216 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 - 8 T + 196 T^{2} - 1432 T^{3} + 17900 T^{4} - 117272 T^{5} + 997916 T^{6} - 5686664 T^{7} + 37305574 T^{8} - 5686664 p T^{9} + 997916 p^{2} T^{10} - 117272 p^{3} T^{11} + 17900 p^{4} T^{12} - 1432 p^{5} T^{13} + 196 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 448 T^{2} + 97096 T^{4} - 13538240 T^{6} + 1361799228 T^{8} - 104985163840 T^{10} + 6427935625976 T^{12} - 319300247763520 T^{14} + 13017076317741510 T^{16} - 319300247763520 p^{2} T^{18} + 6427935625976 p^{4} T^{20} - 104985163840 p^{6} T^{22} + 1361799228 p^{8} T^{24} - 13538240 p^{10} T^{26} + 97096 p^{12} T^{28} - 448 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 + 8 T + 200 T^{2} + 1464 T^{3} + 20020 T^{4} + 127208 T^{5} + 1294840 T^{6} + 7234008 T^{7} + 60783446 T^{8} + 7234008 p T^{9} + 1294840 p^{2} T^{10} + 127208 p^{3} T^{11} + 20020 p^{4} T^{12} + 1464 p^{5} T^{13} + 200 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 484 T^{2} + 114486 T^{4} - 17626832 T^{6} + 1981003681 T^{8} - 172489189384 T^{10} + 12041902947430 T^{12} - 687681116820964 T^{14} + 32467415307528932 T^{16} - 687681116820964 p^{2} T^{18} + 12041902947430 p^{4} T^{20} - 172489189384 p^{6} T^{22} + 1981003681 p^{8} T^{24} - 17626832 p^{10} T^{26} + 114486 p^{12} T^{28} - 484 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 + 12 T + 230 T^{2} + 1740 T^{3} + 17497 T^{4} + 72208 T^{5} + 426938 T^{6} - 505824 T^{7} + 864076 T^{8} - 505824 p T^{9} + 426938 p^{2} T^{10} + 72208 p^{3} T^{11} + 17497 p^{4} T^{12} + 1740 p^{5} T^{13} + 230 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 568 T^{2} + 160894 T^{4} - 30026272 T^{6} + 4120038209 T^{8} - 439876082224 T^{10} + 37748747628942 T^{12} - 2653751888034360 T^{14} + 154373495002173956 T^{16} - 2653751888034360 p^{2} T^{18} + 37748747628942 p^{4} T^{20} - 439876082224 p^{6} T^{22} + 4120038209 p^{8} T^{24} - 30026272 p^{10} T^{26} + 160894 p^{12} T^{28} - 568 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 548 T^{2} + 149614 T^{4} - 27223768 T^{6} + 3714016473 T^{8} - 403655596264 T^{10} + 36158344547598 T^{12} - 2721990080866300 T^{14} + 173995217146714804 T^{16} - 2721990080866300 p^{2} T^{18} + 36158344547598 p^{4} T^{20} - 403655596264 p^{6} T^{22} + 3714016473 p^{8} T^{24} - 27223768 p^{10} T^{26} + 149614 p^{12} T^{28} - 548 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 - 4 p T^{2} + 29622 T^{4} - 1899400 T^{6} + 41603905 T^{8} + 2312073928 T^{10} - 36116466618 T^{12} - 26518643483884 T^{14} + 2619416083527844 T^{16} - 26518643483884 p^{2} T^{18} - 36116466618 p^{4} T^{20} + 2312073928 p^{6} T^{22} + 41603905 p^{8} T^{24} - 1899400 p^{10} T^{26} + 29622 p^{12} T^{28} - 4 p^{15} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 - 588 T^{2} + 169518 T^{4} - 32357584 T^{6} + 4649137449 T^{8} - 538935344360 T^{10} + 52394328139710 T^{12} - 4360644811515548 T^{14} + 313763277516961716 T^{16} - 4360644811515548 p^{2} T^{18} + 52394328139710 p^{4} T^{20} - 538935344360 p^{6} T^{22} + 4649137449 p^{8} T^{24} - 32357584 p^{10} T^{26} + 169518 p^{12} T^{28} - 588 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 24 T + 580 T^{2} + 8776 T^{3} + 132836 T^{4} + 1549960 T^{5} + 17835132 T^{6} + 167522904 T^{7} + 1548536566 T^{8} + 167522904 p T^{9} + 17835132 p^{2} T^{10} + 1549960 p^{3} T^{11} + 132836 p^{4} T^{12} + 8776 p^{5} T^{13} + 580 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 268 T^{2} - 560 T^{3} + 37634 T^{4} - 139728 T^{5} + 3640384 T^{6} - 16876272 T^{7} + 288747435 T^{8} - 16876272 p T^{9} + 3640384 p^{2} T^{10} - 139728 p^{3} T^{11} + 37634 p^{4} T^{12} - 560 p^{5} T^{13} + 268 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 24 T + 696 T^{2} + 11288 T^{3} + 192468 T^{4} + 2366392 T^{5} + 29698376 T^{6} + 291409976 T^{7} + 2898297046 T^{8} + 291409976 p T^{9} + 29698376 p^{2} T^{10} + 2366392 p^{3} T^{11} + 192468 p^{4} T^{12} + 11288 p^{5} T^{13} + 696 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 888 T^{2} + 388120 T^{4} - 111209896 T^{6} + 23425602588 T^{8} - 3849513927800 T^{10} + 510595309926440 T^{12} - 55757709197515496 T^{14} + 5065547522744531590 T^{16} - 55757709197515496 p^{2} T^{18} + 510595309926440 p^{4} T^{20} - 3849513927800 p^{6} T^{22} + 23425602588 p^{8} T^{24} - 111209896 p^{10} T^{26} + 388120 p^{12} T^{28} - 888 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 8 T + 360 T^{2} - 1944 T^{3} + 59412 T^{4} - 285448 T^{5} + 7491096 T^{6} - 38771288 T^{7} + 773903254 T^{8} - 38771288 p T^{9} + 7491096 p^{2} T^{10} - 285448 p^{3} T^{11} + 59412 p^{4} T^{12} - 1944 p^{5} T^{13} + 360 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 16 T + 484 T^{2} - 7360 T^{3} + 115940 T^{4} - 1500384 T^{5} + 18527132 T^{6} - 192518480 T^{7} + 2118921782 T^{8} - 192518480 p T^{9} + 18527132 p^{2} T^{10} - 1500384 p^{3} T^{11} + 115940 p^{4} T^{12} - 7360 p^{5} T^{13} + 484 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.64607578416362919196676170721, −2.28981155275871014513652356007, −2.22387704398023193956622997598, −2.21809078224836177669057946298, −2.17756739138004484838564464119, −2.14086453610188718864807539033, −2.05594796207227322655878695765, −1.85685422373047762021813650035, −1.61930536457939138500575552862, −1.57837701464188591506430509037, −1.56330546690476315276359325051, −1.55321057336220789923495501003, −1.38028559332456452677021464260, −1.37483643411427005758312948590, −1.36976354048041330381686726233, −1.36542423641713222719527079572, −1.12526529699934575517107707488, −1.07155030322978374490111578425, −0.984067801459624489227430130069, −0.71237077893449917506646099428, −0.60952465340774954094887451172, −0.54642839162971465889655606489, −0.33811799447060360882304225124, −0.25521068562842073471120673113, −0.17334888763053095267683352167, 0.17334888763053095267683352167, 0.25521068562842073471120673113, 0.33811799447060360882304225124, 0.54642839162971465889655606489, 0.60952465340774954094887451172, 0.71237077893449917506646099428, 0.984067801459624489227430130069, 1.07155030322978374490111578425, 1.12526529699934575517107707488, 1.36542423641713222719527079572, 1.36976354048041330381686726233, 1.37483643411427005758312948590, 1.38028559332456452677021464260, 1.55321057336220789923495501003, 1.56330546690476315276359325051, 1.57837701464188591506430509037, 1.61930536457939138500575552862, 1.85685422373047762021813650035, 2.05594796207227322655878695765, 2.14086453610188718864807539033, 2.17756739138004484838564464119, 2.21809078224836177669057946298, 2.22387704398023193956622997598, 2.28981155275871014513652356007, 2.64607578416362919196676170721
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.