Dirichlet series
| L(s) = 1 | + 5-s + 5·7-s + 9·9-s − 9·11-s − 4·17-s − 8·19-s − 25·23-s + 13·25-s + 2·27-s + 12·29-s + 5·35-s − 13·37-s − 32·41-s + 34·43-s + 9·45-s + 24·47-s + 6·49-s − 2·53-s − 9·55-s − 2·59-s + 13·61-s + 45·63-s − 2·67-s + 20·71-s − 5·73-s − 45·77-s − 16·79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.88·7-s + 3·9-s − 2.71·11-s − 0.970·17-s − 1.83·19-s − 5.21·23-s + 13/5·25-s + 0.384·27-s + 2.22·29-s + 0.845·35-s − 2.13·37-s − 4.99·41-s + 5.18·43-s + 1.34·45-s + 3.50·47-s + 6/7·49-s − 0.274·53-s − 1.21·55-s − 0.260·59-s + 1.66·61-s + 5.66·63-s − 0.244·67-s + 2.37·71-s − 0.585·73-s − 5.12·77-s − 1.80·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 7^{16} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.37054\times 10^{14}\) |
| Root analytic conductor: | \(2.91480\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 7^{16} \cdot 19^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2367532092\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2367532092\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - 5 T + 19 T^{2} - 25 T^{3} + 18 T^{4} + 198 T^{5} - 40 p T^{6} + 780 T^{7} + 1213 T^{8} + 780 p T^{9} - 40 p^{3} T^{10} + 198 p^{3} T^{11} + 18 p^{4} T^{12} - 25 p^{5} T^{13} + 19 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( ( 1 + T + T^{2} )^{8} \) | |
| good | 3 | \( 1 - p^{2} T^{2} - 2 T^{3} + 11 p T^{4} + 20 T^{5} - 25 T^{6} - 104 T^{7} - 76 p T^{8} + 260 T^{9} + 332 p T^{10} + 50 T^{11} - 1000 T^{12} - 2582 T^{13} - 6742 T^{14} + 4660 T^{15} + 35611 T^{16} + 4660 p T^{17} - 6742 p^{2} T^{18} - 2582 p^{3} T^{19} - 1000 p^{4} T^{20} + 50 p^{5} T^{21} + 332 p^{7} T^{22} + 260 p^{7} T^{23} - 76 p^{9} T^{24} - 104 p^{9} T^{25} - 25 p^{10} T^{26} + 20 p^{11} T^{27} + 11 p^{13} T^{28} - 2 p^{13} T^{29} - p^{16} T^{30} + p^{16} T^{32} \) |
| 5 | \( 1 - T - 12 T^{2} + 39 T^{3} + 31 T^{4} - 362 T^{5} + 674 T^{6} + 1148 T^{7} - 6899 T^{8} + 6859 T^{9} + 27426 T^{10} - 85271 T^{11} + 14398 T^{12} + 415869 T^{13} - 794642 T^{14} - 825677 T^{15} + 5379166 T^{16} - 825677 p T^{17} - 794642 p^{2} T^{18} + 415869 p^{3} T^{19} + 14398 p^{4} T^{20} - 85271 p^{5} T^{21} + 27426 p^{6} T^{22} + 6859 p^{7} T^{23} - 6899 p^{8} T^{24} + 1148 p^{9} T^{25} + 674 p^{10} T^{26} - 362 p^{11} T^{27} + 31 p^{12} T^{28} + 39 p^{13} T^{29} - 12 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 9 T - T^{2} - 180 T^{3} + 16 p T^{4} + 3912 T^{5} + 1711 T^{6} - 18033 T^{7} + 52049 T^{8} + 300 T^{9} + 124672 T^{10} + 8119548 T^{11} + 573594 p T^{12} - 68478846 T^{13} + 120011802 T^{14} + 671223132 T^{15} - 130877408 T^{16} + 671223132 p T^{17} + 120011802 p^{2} T^{18} - 68478846 p^{3} T^{19} + 573594 p^{5} T^{20} + 8119548 p^{5} T^{21} + 124672 p^{6} T^{22} + 300 p^{7} T^{23} + 52049 p^{8} T^{24} - 18033 p^{9} T^{25} + 1711 p^{10} T^{26} + 3912 p^{11} T^{27} + 16 p^{13} T^{28} - 180 p^{13} T^{29} - p^{14} T^{30} + 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( ( 1 + 63 T^{2} + T^{3} + 1844 T^{4} + 265 T^{5} + 34701 T^{6} + 9002 T^{7} + 498638 T^{8} + 9002 p T^{9} + 34701 p^{2} T^{10} + 265 p^{3} T^{11} + 1844 p^{4} T^{12} + p^{5} T^{13} + 63 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 + 4 T - 42 T^{2} - 138 T^{3} + 567 T^{4} + 508 T^{5} - 2497 T^{6} + 11554 T^{7} + 55655 T^{8} + 339362 T^{9} - 4466945 T^{10} - 5502422 T^{11} + 125665338 T^{12} - 66838266 T^{13} - 1082815903 T^{14} + 1061169242 T^{15} - 962578164 T^{16} + 1061169242 p T^{17} - 1082815903 p^{2} T^{18} - 66838266 p^{3} T^{19} + 125665338 p^{4} T^{20} - 5502422 p^{5} T^{21} - 4466945 p^{6} T^{22} + 339362 p^{7} T^{23} + 55655 p^{8} T^{24} + 11554 p^{9} T^{25} - 2497 p^{10} T^{26} + 508 p^{11} T^{27} + 567 p^{12} T^{28} - 138 p^{13} T^{29} - 42 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 25 T + 212 T^{2} + 591 T^{3} + 2226 T^{4} + 45987 T^{5} + 211433 T^{6} - 292906 T^{7} + 3178429 T^{8} + 59600505 T^{9} + 58973946 T^{10} - 434411205 T^{11} + 7370838420 T^{12} + 37013437161 T^{13} - 74839503639 T^{14} + 216689542366 T^{15} + 6152855825848 T^{16} + 216689542366 p T^{17} - 74839503639 p^{2} T^{18} + 37013437161 p^{3} T^{19} + 7370838420 p^{4} T^{20} - 434411205 p^{5} T^{21} + 58973946 p^{6} T^{22} + 59600505 p^{7} T^{23} + 3178429 p^{8} T^{24} - 292906 p^{9} T^{25} + 211433 p^{10} T^{26} + 45987 p^{11} T^{27} + 2226 p^{12} T^{28} + 591 p^{13} T^{29} + 212 p^{14} T^{30} + 25 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 6 T + 127 T^{2} - 889 T^{3} + 8908 T^{4} - 58681 T^{5} + 443085 T^{6} - 2388316 T^{7} + 15502590 T^{8} - 2388316 p T^{9} + 443085 p^{2} T^{10} - 58681 p^{3} T^{11} + 8908 p^{4} T^{12} - 889 p^{5} T^{13} + 127 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( 1 - 148 T^{2} - 232 T^{3} + 11652 T^{4} + 34764 T^{5} - 587848 T^{6} - 2943080 T^{7} + 19607114 T^{8} + 166684064 T^{9} - 311851580 T^{10} - 6726237076 T^{11} - 8307195696 T^{12} + 182539110156 T^{13} + 824466120740 T^{14} - 2278578346752 T^{15} - 32846844872877 T^{16} - 2278578346752 p T^{17} + 824466120740 p^{2} T^{18} + 182539110156 p^{3} T^{19} - 8307195696 p^{4} T^{20} - 6726237076 p^{5} T^{21} - 311851580 p^{6} T^{22} + 166684064 p^{7} T^{23} + 19607114 p^{8} T^{24} - 2943080 p^{9} T^{25} - 587848 p^{10} T^{26} + 34764 p^{11} T^{27} + 11652 p^{12} T^{28} - 232 p^{13} T^{29} - 148 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 + 13 T - 38 T^{2} - 809 T^{3} + 3697 T^{4} + 48634 T^{5} - 167348 T^{6} - 2330030 T^{7} + 2303761 T^{8} + 86986647 T^{9} + 233609274 T^{10} - 2460132963 T^{11} - 21196359094 T^{12} + 66168659153 T^{13} + 1199299166654 T^{14} - 1339601039713 T^{15} - 54802937041654 T^{16} - 1339601039713 p T^{17} + 1199299166654 p^{2} T^{18} + 66168659153 p^{3} T^{19} - 21196359094 p^{4} T^{20} - 2460132963 p^{5} T^{21} + 233609274 p^{6} T^{22} + 86986647 p^{7} T^{23} + 2303761 p^{8} T^{24} - 2330030 p^{9} T^{25} - 167348 p^{10} T^{26} + 48634 p^{11} T^{27} + 3697 p^{12} T^{28} - 809 p^{13} T^{29} - 38 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( ( 1 + 2 T + p T^{2} )^{16} \) | |
| 43 | \( ( 1 - 17 T + 355 T^{2} - 4036 T^{3} + 49646 T^{4} - 429858 T^{5} + 3936957 T^{6} - 642057 p T^{7} + 204548050 T^{8} - 642057 p^{2} T^{9} + 3936957 p^{2} T^{10} - 429858 p^{3} T^{11} + 49646 p^{4} T^{12} - 4036 p^{5} T^{13} + 355 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 24 T + 184 T^{2} - 680 T^{3} + 11326 T^{4} - 139156 T^{5} + 776192 T^{6} - 8247180 T^{7} + 93928337 T^{8} - 468874744 T^{9} + 3198112288 T^{10} - 41741026080 T^{11} + 248022535182 T^{12} - 1312497956804 T^{13} + 15707997927016 T^{14} - 101510970066836 T^{15} + 438876459024964 T^{16} - 101510970066836 p T^{17} + 15707997927016 p^{2} T^{18} - 1312497956804 p^{3} T^{19} + 248022535182 p^{4} T^{20} - 41741026080 p^{5} T^{21} + 3198112288 p^{6} T^{22} - 468874744 p^{7} T^{23} + 93928337 p^{8} T^{24} - 8247180 p^{9} T^{25} + 776192 p^{10} T^{26} - 139156 p^{11} T^{27} + 11326 p^{12} T^{28} - 680 p^{13} T^{29} + 184 p^{14} T^{30} - 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 2 T - 125 T^{2} + 332 T^{3} + 6141 T^{4} - 43510 T^{5} + 3319 p T^{6} + 775978 T^{7} - 28916662 T^{8} + 193875740 T^{9} + 808658152 T^{10} - 14730288262 T^{11} + 20963069712 T^{12} + 345638692850 T^{13} - 1043078945670 T^{14} - 2308883848056 T^{15} - 25032931361831 T^{16} - 2308883848056 p T^{17} - 1043078945670 p^{2} T^{18} + 345638692850 p^{3} T^{19} + 20963069712 p^{4} T^{20} - 14730288262 p^{5} T^{21} + 808658152 p^{6} T^{22} + 193875740 p^{7} T^{23} - 28916662 p^{8} T^{24} + 775978 p^{9} T^{25} + 3319 p^{11} T^{26} - 43510 p^{11} T^{27} + 6141 p^{12} T^{28} + 332 p^{13} T^{29} - 125 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 2 T - 157 T^{2} - 1020 T^{3} + 10395 T^{4} + 133320 T^{5} - 21819 T^{6} - 8910300 T^{7} - 49517268 T^{8} + 217502722 T^{9} + 4070847182 T^{10} + 11443500824 T^{11} - 145330059164 T^{12} - 1281567395380 T^{13} + 800311664396 T^{14} + 35323212863010 T^{15} + 186946986429675 T^{16} + 35323212863010 p T^{17} + 800311664396 p^{2} T^{18} - 1281567395380 p^{3} T^{19} - 145330059164 p^{4} T^{20} + 11443500824 p^{5} T^{21} + 4070847182 p^{6} T^{22} + 217502722 p^{7} T^{23} - 49517268 p^{8} T^{24} - 8910300 p^{9} T^{25} - 21819 p^{10} T^{26} + 133320 p^{11} T^{27} + 10395 p^{12} T^{28} - 1020 p^{13} T^{29} - 157 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 13 T - 161 T^{2} + 2152 T^{3} + 15718 T^{4} - 150152 T^{5} - 1628061 T^{6} + 8312299 T^{7} + 140927037 T^{8} - 632693160 T^{9} - 9039797416 T^{10} + 44587488312 T^{11} + 568218875138 T^{12} - 1919801126770 T^{13} - 44425357372378 T^{14} + 35513545858192 T^{15} + 3167179598029988 T^{16} + 35513545858192 p T^{17} - 44425357372378 p^{2} T^{18} - 1919801126770 p^{3} T^{19} + 568218875138 p^{4} T^{20} + 44587488312 p^{5} T^{21} - 9039797416 p^{6} T^{22} - 632693160 p^{7} T^{23} + 140927037 p^{8} T^{24} + 8312299 p^{9} T^{25} - 1628061 p^{10} T^{26} - 150152 p^{11} T^{27} + 15718 p^{12} T^{28} + 2152 p^{13} T^{29} - 161 p^{14} T^{30} - 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 2 T - 449 T^{2} - 832 T^{3} + 109405 T^{4} + 182132 T^{5} - 18764225 T^{6} - 26265706 T^{7} + 2514510700 T^{8} + 2741907192 T^{9} - 277915010160 T^{10} - 211935106308 T^{11} + 26147461675100 T^{12} + 11507275093288 T^{13} - 2133738902557462 T^{14} - 297154295840216 T^{15} + 152516990986511231 T^{16} - 297154295840216 p T^{17} - 2133738902557462 p^{2} T^{18} + 11507275093288 p^{3} T^{19} + 26147461675100 p^{4} T^{20} - 211935106308 p^{5} T^{21} - 277915010160 p^{6} T^{22} + 2741907192 p^{7} T^{23} + 2514510700 p^{8} T^{24} - 26265706 p^{9} T^{25} - 18764225 p^{10} T^{26} + 182132 p^{11} T^{27} + 109405 p^{12} T^{28} - 832 p^{13} T^{29} - 449 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 10 T + 328 T^{2} - 2494 T^{3} + 51564 T^{4} - 327342 T^{5} + 78728 p T^{6} - 31264890 T^{7} + 457121126 T^{8} - 31264890 p T^{9} + 78728 p^{3} T^{10} - 327342 p^{3} T^{11} + 51564 p^{4} T^{12} - 2494 p^{5} T^{13} + 328 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 5 T - 202 T^{2} - 383 T^{3} + 28308 T^{4} - 47997 T^{5} - 2508439 T^{6} + 16527214 T^{7} + 131537529 T^{8} - 2468849063 T^{9} + 400280790 T^{10} + 232423833505 T^{11} - 988589946380 T^{12} - 204332076939 p T^{13} + 132548407725043 T^{14} + 441002707086946 T^{15} - 11152078119499092 T^{16} + 441002707086946 p T^{17} + 132548407725043 p^{2} T^{18} - 204332076939 p^{4} T^{19} - 988589946380 p^{4} T^{20} + 232423833505 p^{5} T^{21} + 400280790 p^{6} T^{22} - 2468849063 p^{7} T^{23} + 131537529 p^{8} T^{24} + 16527214 p^{9} T^{25} - 2508439 p^{10} T^{26} - 47997 p^{11} T^{27} + 28308 p^{12} T^{28} - 383 p^{13} T^{29} - 202 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 16 T - 156 T^{2} - 4296 T^{3} + 1116 T^{4} + 564732 T^{5} + 2129256 T^{6} - 51660792 T^{7} - 390856358 T^{8} + 3456672384 T^{9} + 46488996428 T^{10} - 138120763428 T^{11} - 4008695100944 T^{12} + 1964249205724 T^{13} + 289813175999820 T^{14} + 21276706838496 T^{15} - 21613172962733549 T^{16} + 21276706838496 p T^{17} + 289813175999820 p^{2} T^{18} + 1964249205724 p^{3} T^{19} - 4008695100944 p^{4} T^{20} - 138120763428 p^{5} T^{21} + 46488996428 p^{6} T^{22} + 3456672384 p^{7} T^{23} - 390856358 p^{8} T^{24} - 51660792 p^{9} T^{25} + 2129256 p^{10} T^{26} + 564732 p^{11} T^{27} + 1116 p^{12} T^{28} - 4296 p^{13} T^{29} - 156 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 43 T + 1290 T^{2} + 27837 T^{3} + 495581 T^{4} + 7318764 T^{5} + 93657586 T^{6} + 1035827238 T^{7} + 10099433844 T^{8} + 1035827238 p T^{9} + 93657586 p^{2} T^{10} + 7318764 p^{3} T^{11} + 495581 p^{4} T^{12} + 27837 p^{5} T^{13} + 1290 p^{6} T^{14} + 43 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 8 T - 384 T^{2} - 3576 T^{3} + 73972 T^{4} + 745276 T^{5} - 8902288 T^{6} - 85899856 T^{7} + 800278234 T^{8} + 5118364864 T^{9} - 66323552016 T^{10} + 902173636 T^{11} + 6204924073072 T^{12} - 23665480238676 T^{13} - 603458886987728 T^{14} + 13518678755936 p T^{15} + 629713203411035 p T^{16} + 13518678755936 p^{2} T^{17} - 603458886987728 p^{2} T^{18} - 23665480238676 p^{3} T^{19} + 6204924073072 p^{4} T^{20} + 902173636 p^{5} T^{21} - 66323552016 p^{6} T^{22} + 5118364864 p^{7} T^{23} + 800278234 p^{8} T^{24} - 85899856 p^{9} T^{25} - 8902288 p^{10} T^{26} + 745276 p^{11} T^{27} + 73972 p^{12} T^{28} - 3576 p^{13} T^{29} - 384 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( ( 1 + 12 T + 608 T^{2} + 5812 T^{3} + 170620 T^{4} + 1355452 T^{5} + 29452384 T^{6} + 196855172 T^{7} + 3436368006 T^{8} + 196855172 p T^{9} + 29452384 p^{2} T^{10} + 1355452 p^{3} T^{11} + 170620 p^{4} T^{12} + 5812 p^{5} T^{13} + 608 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.64178810241277432937885029862, −2.55177856804825837043226157264, −2.35977555697362913961742307571, −2.34753363972022806701986648447, −2.23782482269404314430288415612, −2.16002446331591563265131908488, −2.05295634683614064456578446369, −1.94939660980990689739645523884, −1.87923108333841110756599046221, −1.86228881271476180327410535785, −1.66484721875396313038981104333, −1.59578975997107943496531828170, −1.57325319904256154760127656288, −1.54197381377723814589403868840, −1.41928720539049182698173968072, −1.41575763431099396657306596658, −1.38311848132481886243977367099, −1.05692246413616973351954481502, −0.992980007023406893000022249582, −0.831861757288346735404092970754, −0.77736495477502191428157309036, −0.55199952380012948974070119999, −0.42622832103886750384284499703, −0.19632005506830970450591939028, −0.03652053292998746092628805865, 0.03652053292998746092628805865, 0.19632005506830970450591939028, 0.42622832103886750384284499703, 0.55199952380012948974070119999, 0.77736495477502191428157309036, 0.831861757288346735404092970754, 0.992980007023406893000022249582, 1.05692246413616973351954481502, 1.38311848132481886243977367099, 1.41575763431099396657306596658, 1.41928720539049182698173968072, 1.54197381377723814589403868840, 1.57325319904256154760127656288, 1.59578975997107943496531828170, 1.66484721875396313038981104333, 1.86228881271476180327410535785, 1.87923108333841110756599046221, 1.94939660980990689739645523884, 2.05295634683614064456578446369, 2.16002446331591563265131908488, 2.23782482269404314430288415612, 2.34753363972022806701986648447, 2.35977555697362913961742307571, 2.55177856804825837043226157264, 2.64178810241277432937885029862
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.