Dirichlet series
| L(s) = 1 | + 2·4-s − 16-s − 26·25-s + 16·31-s − 12·37-s − 26·49-s + 8·64-s + 96·67-s − 20·73-s + 60·97-s − 52·100-s − 64·103-s + 60·121-s + 32·124-s + 127-s + 131-s + 137-s + 139-s − 24·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 4-s − 1/4·16-s − 5.19·25-s + 2.87·31-s − 1.97·37-s − 3.71·49-s + 64-s + 11.7·67-s − 2.34·73-s + 6.09·97-s − 5.19·100-s − 6.30·103-s + 5.45·121-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.97·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0232593\) |
| Root analytic conductor: | \(0.889111\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.7969100608\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7969100608\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - 60 T^{2} + 1779 T^{4} - 33610 T^{6} + 439341 T^{8} - 33610 p^{2} T^{10} + 1779 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \) | |
| good | 2 | \( 1 - p T^{2} + 5 T^{4} - 5 p^{2} T^{6} + 15 p T^{8} - 53 p T^{10} + 53 p^{2} T^{12} - 175 p T^{14} + 1105 T^{16} - 175 p^{3} T^{18} + 53 p^{6} T^{20} - 53 p^{7} T^{22} + 15 p^{9} T^{24} - 5 p^{12} T^{26} + 5 p^{12} T^{28} - p^{15} T^{30} + p^{16} T^{32} \) |
| 5 | \( 1 + 26 T^{2} + 267 T^{4} + 1158 T^{6} - 69 p T^{8} - 23178 T^{10} - 84123 T^{12} - 137046 T^{14} - 351264 T^{16} - 137046 p^{2} T^{18} - 84123 p^{4} T^{20} - 23178 p^{6} T^{22} - 69 p^{9} T^{24} + 1158 p^{10} T^{26} + 267 p^{12} T^{28} + 26 p^{14} T^{30} + p^{16} T^{32} \) | |
| 7 | \( ( 1 + 13 T^{2} + 20 T^{3} + 80 T^{4} + 260 T^{5} + 783 T^{6} + 310 p T^{7} + 6819 T^{8} + 310 p^{2} T^{9} + 783 p^{2} T^{10} + 260 p^{3} T^{11} + 80 p^{4} T^{12} + 20 p^{5} T^{13} + 13 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 7 T^{2} + 80 T^{3} + 170 T^{4} + 560 T^{5} + 4587 T^{6} + 13570 T^{7} + 40179 T^{8} + 13570 p T^{9} + 4587 p^{2} T^{10} + 560 p^{3} T^{11} + 170 p^{4} T^{12} + 80 p^{5} T^{13} + 7 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 58 T^{2} + 1230 T^{4} + 3270 T^{6} - 669405 T^{8} + 13275246 T^{10} - 49036968 T^{12} - 2836388280 T^{14} + 74928517905 T^{16} - 2836388280 p^{2} T^{18} - 49036968 p^{4} T^{20} + 13275246 p^{6} T^{22} - 669405 p^{8} T^{24} + 3270 p^{10} T^{26} + 1230 p^{12} T^{28} - 58 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 34 T^{2} + 20 T^{3} + 395 T^{4} + 680 T^{5} + 1356 T^{6} + 48280 T^{7} - 47451 T^{8} + 48280 p T^{9} + 1356 p^{2} T^{10} + 680 p^{3} T^{11} + 395 p^{4} T^{12} + 20 p^{5} T^{13} + 34 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 102 T^{2} + 5430 T^{4} - 195472 T^{6} + 5177079 T^{8} - 195472 p^{2} T^{10} + 5430 p^{4} T^{12} - 102 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 + 20 T^{2} + 338 T^{4} + 28080 T^{6} - 44437 T^{8} - 6939980 T^{10} + 33962216 T^{12} - 13624904800 T^{14} - 492015847995 T^{16} - 13624904800 p^{2} T^{18} + 33962216 p^{4} T^{20} - 6939980 p^{6} T^{22} - 44437 p^{8} T^{24} + 28080 p^{10} T^{26} + 338 p^{12} T^{28} + 20 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 - 8 T - 24 T^{2} + 88 T^{3} + 1681 T^{4} - 7064 T^{5} + 54924 T^{6} - 168376 T^{7} - 1019363 T^{8} - 168376 p T^{9} + 54924 p^{2} T^{10} - 7064 p^{3} T^{11} + 1681 p^{4} T^{12} + 88 p^{5} T^{13} - 24 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 6 T - 7 T^{2} - 118 T^{3} - 254 T^{4} + 3396 T^{5} + 36665 T^{6} - 93294 T^{7} - 1522501 T^{8} - 93294 p T^{9} + 36665 p^{2} T^{10} + 3396 p^{3} T^{11} - 254 p^{4} T^{12} - 118 p^{5} T^{13} - 7 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 138 T^{2} + 4581 T^{4} + 242158 T^{6} - 19370244 T^{8} + 1146836 p T^{10} + 27362707279 T^{12} - 302286948126 T^{14} - 26002484153453 T^{16} - 302286948126 p^{2} T^{18} + 27362707279 p^{4} T^{20} + 1146836 p^{7} T^{22} - 19370244 p^{8} T^{24} + 242158 p^{10} T^{26} + 4581 p^{12} T^{28} - 138 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 - 180 T^{2} + 12786 T^{4} - 463360 T^{6} + 14361051 T^{8} - 463360 p^{2} T^{10} + 12786 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 192 T^{2} + 14115 T^{4} + 332470 T^{6} - 12112005 T^{8} - 594955574 T^{10} + 21788151037 T^{12} + 2123156175840 T^{14} + 93108257741080 T^{16} + 2123156175840 p^{2} T^{18} + 21788151037 p^{4} T^{20} - 594955574 p^{6} T^{22} - 12112005 p^{8} T^{24} + 332470 p^{10} T^{26} + 14115 p^{12} T^{28} + 192 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 166 T^{2} + 9174 T^{4} - 111674 T^{6} - 30841109 T^{8} - 525703922 T^{10} + 67446933816 T^{12} + 3591171944168 T^{14} + 110175435537457 T^{16} + 3591171944168 p^{2} T^{18} + 67446933816 p^{4} T^{20} - 525703922 p^{6} T^{22} - 30841109 p^{8} T^{24} - 111674 p^{10} T^{26} + 9174 p^{12} T^{28} + 166 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 + 170 T^{2} + 22518 T^{4} + 2498730 T^{6} + 233218683 T^{8} + 320662230 p T^{10} + 1419551700816 T^{12} + 95367637994400 T^{14} + 5833449282172425 T^{16} + 95367637994400 p^{2} T^{18} + 1419551700816 p^{4} T^{20} + 320662230 p^{7} T^{22} + 233218683 p^{8} T^{24} + 2498730 p^{10} T^{26} + 22518 p^{12} T^{28} + 170 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 83 T^{2} + 420 T^{3} + 3413 T^{4} + 34860 T^{5} + 227941 T^{6} + 69440 p T^{7} + 131300 p T^{8} + 69440 p^{2} T^{9} + 227941 p^{2} T^{10} + 34860 p^{3} T^{11} + 3413 p^{4} T^{12} + 420 p^{5} T^{13} + 83 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 24 T + 429 T^{2} - 5008 T^{3} + 47859 T^{4} - 5008 p T^{5} + 429 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 71 | \( 1 + 338 T^{2} + 50091 T^{4} + 3709542 T^{6} + 63034791 T^{8} - 13931687706 T^{10} - 1397230758891 T^{12} - 60140798780574 T^{14} - 2187888491323968 T^{16} - 60140798780574 p^{2} T^{18} - 1397230758891 p^{4} T^{20} - 13931687706 p^{6} T^{22} + 63034791 p^{8} T^{24} + 3709542 p^{10} T^{26} + 50091 p^{12} T^{28} + 338 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( ( 1 + 10 T + 276 T^{2} + 2170 T^{3} + 24127 T^{4} + 117430 T^{5} - 172242 T^{6} - 4912540 T^{7} - 124201175 T^{8} - 4912540 p T^{9} - 172242 p^{2} T^{10} + 117430 p^{3} T^{11} + 24127 p^{4} T^{12} + 2170 p^{5} T^{13} + 276 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 203 T^{2} - 180 T^{3} + 12338 T^{4} - 36540 T^{5} - 807689 T^{6} - 4199410 T^{7} - 150806725 T^{8} - 4199410 p T^{9} - 807689 p^{2} T^{10} - 36540 p^{3} T^{11} + 12338 p^{4} T^{12} - 180 p^{5} T^{13} + 203 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 208 T^{2} + 15045 T^{4} + 774660 T^{6} - 236047260 T^{8} + 17546406456 T^{10} - 74692835973 T^{12} - 112052982977490 T^{14} + 12843285776364315 T^{16} - 112052982977490 p^{2} T^{18} - 74692835973 p^{4} T^{20} + 17546406456 p^{6} T^{22} - 236047260 p^{8} T^{24} + 774660 p^{10} T^{26} + 15045 p^{12} T^{28} - 208 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 582 T^{2} + 157158 T^{4} - 25689904 T^{6} + 2778331335 T^{8} - 25689904 p^{2} T^{10} + 157158 p^{4} T^{12} - 582 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 30 T + 411 T^{2} - 3150 T^{3} + 11217 T^{4} + 58230 T^{5} - 1794727 T^{6} + 27808950 T^{7} - 324085500 T^{8} + 27808950 p T^{9} - 1794727 p^{2} T^{10} + 58230 p^{3} T^{11} + 11217 p^{4} T^{12} - 3150 p^{5} T^{13} + 411 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.40098637983589955664125670651, −4.32881215120726805005211213193, −4.12774478154439322216338921883, −3.90525435360678872864559383888, −3.80538971274827949688639623189, −3.73667479960224348845956228086, −3.62353736683087917681730009855, −3.58679690427864680825003633188, −3.56752487493087648862174388482, −3.43283514169905169840176564669, −3.35048469645583794992470785526, −3.30445251546400004271626914298, −3.05119460112927957937182536110, −2.82530119061808492440557214047, −2.64018109142552767631757516850, −2.38550035670366967048423833177, −2.37075233636542388048259931911, −2.34669555998390238218662787717, −2.18107314573745517132346405538, −2.03259686727371635080682113062, −2.01661134218978341531482207024, −1.88980897242938690503569313175, −1.37900142979280929225521552717, −1.33073915980037750972244826925, −0.855652997329293444855679185769, 0.855652997329293444855679185769, 1.33073915980037750972244826925, 1.37900142979280929225521552717, 1.88980897242938690503569313175, 2.01661134218978341531482207024, 2.03259686727371635080682113062, 2.18107314573745517132346405538, 2.34669555998390238218662787717, 2.37075233636542388048259931911, 2.38550035670366967048423833177, 2.64018109142552767631757516850, 2.82530119061808492440557214047, 3.05119460112927957937182536110, 3.30445251546400004271626914298, 3.35048469645583794992470785526, 3.43283514169905169840176564669, 3.56752487493087648862174388482, 3.58679690427864680825003633188, 3.62353736683087917681730009855, 3.73667479960224348845956228086, 3.80538971274827949688639623189, 3.90525435360678872864559383888, 4.12774478154439322216338921883, 4.32881215120726805005211213193, 4.40098637983589955664125670651
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.