Dirichlet series
| L(s) = 1 | − 3·5-s − 6·11-s − 6·13-s − 12·17-s + 12·19-s − 3·25-s + 3·29-s + 27·37-s − 18·41-s − 12·43-s − 9·47-s − 9·49-s + 36·53-s + 18·55-s + 6·59-s − 9·61-s + 18·65-s + 6·67-s − 3·71-s − 42·73-s + 12·79-s − 9·83-s + 36·85-s + 27·89-s − 36·95-s − 24·97-s − 24·101-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.80·11-s − 1.66·13-s − 2.91·17-s + 2.75·19-s − 3/5·25-s + 0.557·29-s + 4.43·37-s − 2.81·41-s − 1.82·43-s − 1.31·47-s − 9/7·49-s + 4.94·53-s + 2.42·55-s + 0.781·59-s − 1.15·61-s + 2.23·65-s + 0.733·67-s − 0.356·71-s − 4.91·73-s + 1.35·79-s − 0.987·83-s + 3.90·85-s + 2.86·89-s − 3.69·95-s − 2.43·97-s − 2.38·101-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 37^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.09592\times 10^{12}\) |
| Root analytic conductor: | \(3.26129\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 37^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.331591690\) |
| \(L(\frac12)\) | \(\approx\) | \(5.331591690\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 37 | \( 1 - 27 T + 372 T^{2} - 3645 T^{3} + 29445 T^{4} - 206334 T^{5} + 1305727 T^{6} - 206334 p T^{7} + 29445 p^{2} T^{8} - 3645 p^{3} T^{9} + 372 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 5 | \( 1 + 3 T + 12 T^{2} + 4 p T^{3} + 69 T^{4} + 51 T^{5} + 39 p T^{6} - 474 T^{7} - 27 p T^{8} - 3573 T^{9} + 243 T^{10} - 19089 T^{11} + 799 T^{12} - 19089 p T^{13} + 243 p^{2} T^{14} - 3573 p^{3} T^{15} - 27 p^{5} T^{16} - 474 p^{5} T^{17} + 39 p^{7} T^{18} + 51 p^{7} T^{19} + 69 p^{8} T^{20} + 4 p^{10} T^{21} + 12 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 + 9 T^{2} + 4 T^{3} + 54 T^{4} + 9 T^{5} + 361 T^{6} + 1233 T^{7} + 2601 T^{8} + 14506 T^{9} + 6822 T^{10} + 11898 p T^{11} - 6137 T^{12} + 11898 p^{2} T^{13} + 6822 p^{2} T^{14} + 14506 p^{3} T^{15} + 2601 p^{4} T^{16} + 1233 p^{5} T^{17} + 361 p^{6} T^{18} + 9 p^{7} T^{19} + 54 p^{8} T^{20} + 4 p^{9} T^{21} + 9 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 + 6 T - 21 T^{2} - 140 T^{3} + 420 T^{4} + 1779 T^{5} - 9324 T^{6} - 26649 T^{7} + 121065 T^{8} + 224682 T^{9} - 1610496 T^{10} - 520290 T^{11} + 21950575 T^{12} - 520290 p T^{13} - 1610496 p^{2} T^{14} + 224682 p^{3} T^{15} + 121065 p^{4} T^{16} - 26649 p^{5} T^{17} - 9324 p^{6} T^{18} + 1779 p^{7} T^{19} + 420 p^{8} T^{20} - 140 p^{9} T^{21} - 21 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 6 T - 6 T^{2} - 85 T^{3} + 108 T^{4} + 2214 T^{5} + 3686 T^{6} - 23208 T^{7} - 45408 T^{8} + 361791 T^{9} + 1669224 T^{10} - 487254 T^{11} - 19840909 T^{12} - 487254 p T^{13} + 1669224 p^{2} T^{14} + 361791 p^{3} T^{15} - 45408 p^{4} T^{16} - 23208 p^{5} T^{17} + 3686 p^{6} T^{18} + 2214 p^{7} T^{19} + 108 p^{8} T^{20} - 85 p^{9} T^{21} - 6 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 12 T + 54 T^{2} - 27 T^{3} - 2250 T^{4} - 15234 T^{5} - 37240 T^{6} + 148140 T^{7} + 1813338 T^{8} + 7510887 T^{9} + 6053202 T^{10} - 110409714 T^{11} - 713553657 T^{12} - 110409714 p T^{13} + 6053202 p^{2} T^{14} + 7510887 p^{3} T^{15} + 1813338 p^{4} T^{16} + 148140 p^{5} T^{17} - 37240 p^{6} T^{18} - 15234 p^{7} T^{19} - 2250 p^{8} T^{20} - 27 p^{9} T^{21} + 54 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 12 T + 15 T^{2} + 417 T^{3} - 1788 T^{4} - 6882 T^{5} + 59981 T^{6} + 34821 T^{7} - 1184805 T^{8} + 290922 T^{9} + 19828971 T^{10} - 137250 p T^{11} - 19021539 p T^{12} - 137250 p^{2} T^{13} + 19828971 p^{2} T^{14} + 290922 p^{3} T^{15} - 1184805 p^{4} T^{16} + 34821 p^{5} T^{17} + 59981 p^{6} T^{18} - 6882 p^{7} T^{19} - 1788 p^{8} T^{20} + 417 p^{9} T^{21} + 15 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 75 T^{2} + 220 T^{3} + 3051 T^{4} - 15687 T^{5} - 55152 T^{6} + 661359 T^{7} - 271665 T^{8} - 15457143 T^{9} + 57850158 T^{10} + 162631875 T^{11} - 1804291337 T^{12} + 162631875 p T^{13} + 57850158 p^{2} T^{14} - 15457143 p^{3} T^{15} - 271665 p^{4} T^{16} + 661359 p^{5} T^{17} - 55152 p^{6} T^{18} - 15687 p^{7} T^{19} + 3051 p^{8} T^{20} + 220 p^{9} T^{21} - 75 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( 1 - 3 T - 120 T^{2} + 239 T^{3} + 7869 T^{4} - 9102 T^{5} - 386775 T^{6} + 289515 T^{7} + 15631950 T^{8} - 7161081 T^{9} - 553749141 T^{10} + 78710484 T^{11} + 17292889753 T^{12} + 78710484 p T^{13} - 553749141 p^{2} T^{14} - 7161081 p^{3} T^{15} + 15631950 p^{4} T^{16} + 289515 p^{5} T^{17} - 386775 p^{6} T^{18} - 9102 p^{7} T^{19} + 7869 p^{8} T^{20} + 239 p^{9} T^{21} - 120 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 144 T^{2} - 115 T^{3} + 297 p T^{4} - 10485 T^{5} + 353757 T^{6} - 10485 p T^{7} + 297 p^{3} T^{8} - 115 p^{3} T^{9} + 144 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 18 T + 252 T^{2} + 2538 T^{3} + 25884 T^{4} + 214884 T^{5} + 1753877 T^{6} + 12319101 T^{7} + 90050697 T^{8} + 567842238 T^{9} + 3799055709 T^{10} + 23018170491 T^{11} + 156226587861 T^{12} + 23018170491 p T^{13} + 3799055709 p^{2} T^{14} + 567842238 p^{3} T^{15} + 90050697 p^{4} T^{16} + 12319101 p^{5} T^{17} + 1753877 p^{6} T^{18} + 214884 p^{7} T^{19} + 25884 p^{8} T^{20} + 2538 p^{9} T^{21} + 252 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 6 T + 189 T^{2} + 912 T^{3} + 16434 T^{4} + 66135 T^{5} + 877159 T^{6} + 66135 p T^{7} + 16434 p^{2} T^{8} + 912 p^{3} T^{9} + 189 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 9 T - 132 T^{2} - 1025 T^{3} + 11313 T^{4} + 57162 T^{5} - 859803 T^{6} - 3158979 T^{7} + 49441578 T^{8} + 118838949 T^{9} - 2752098309 T^{10} - 1604807964 T^{11} + 147387989593 T^{12} - 1604807964 p T^{13} - 2752098309 p^{2} T^{14} + 118838949 p^{3} T^{15} + 49441578 p^{4} T^{16} - 3158979 p^{5} T^{17} - 859803 p^{6} T^{18} + 57162 p^{7} T^{19} + 11313 p^{8} T^{20} - 1025 p^{9} T^{21} - 132 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 36 T + 693 T^{2} - 9355 T^{3} + 100008 T^{4} - 908136 T^{5} + 7257735 T^{6} - 51115473 T^{7} + 307641870 T^{8} - 1461870432 T^{9} + 4254055146 T^{10} + 5765217048 T^{11} - 144265331177 T^{12} + 5765217048 p T^{13} + 4254055146 p^{2} T^{14} - 1461870432 p^{3} T^{15} + 307641870 p^{4} T^{16} - 51115473 p^{5} T^{17} + 7257735 p^{6} T^{18} - 908136 p^{7} T^{19} + 100008 p^{8} T^{20} - 9355 p^{9} T^{21} + 693 p^{10} T^{22} - 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 6 T - 243 T^{2} + 1620 T^{3} + 26667 T^{4} - 211974 T^{5} - 1696687 T^{6} + 18242478 T^{7} + 59900220 T^{8} - 1031639868 T^{9} - 421191612 T^{10} + 25575080148 T^{11} - 53314777095 T^{12} + 25575080148 p T^{13} - 421191612 p^{2} T^{14} - 1031639868 p^{3} T^{15} + 59900220 p^{4} T^{16} + 18242478 p^{5} T^{17} - 1696687 p^{6} T^{18} - 211974 p^{7} T^{19} + 26667 p^{8} T^{20} + 1620 p^{9} T^{21} - 243 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 9 T + 63 T^{2} + 1512 T^{3} + 8460 T^{4} + 20322 T^{5} + 832286 T^{6} - 2133 T^{7} - 35869266 T^{8} + 103562496 T^{9} - 3721021155 T^{10} - 38527639956 T^{11} - 34644086529 T^{12} - 38527639956 p T^{13} - 3721021155 p^{2} T^{14} + 103562496 p^{3} T^{15} - 35869266 p^{4} T^{16} - 2133 p^{5} T^{17} + 832286 p^{6} T^{18} + 20322 p^{7} T^{19} + 8460 p^{8} T^{20} + 1512 p^{9} T^{21} + 63 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 6 T + 42 T^{2} + 352 T^{3} - 4710 T^{4} + 60132 T^{5} - 28151 T^{6} - 2398914 T^{7} + 42648813 T^{8} - 264786950 T^{9} + 618884031 T^{10} + 13163120556 T^{11} - 182670695615 T^{12} + 13163120556 p T^{13} + 618884031 p^{2} T^{14} - 264786950 p^{3} T^{15} + 42648813 p^{4} T^{16} - 2398914 p^{5} T^{17} - 28151 p^{6} T^{18} + 60132 p^{7} T^{19} - 4710 p^{8} T^{20} + 352 p^{9} T^{21} + 42 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 3 T + 234 T^{2} + 324 T^{3} + 30429 T^{4} + 11856 T^{5} + 3278126 T^{6} + 3536028 T^{7} + 320286249 T^{8} + 545110992 T^{9} + 25836521040 T^{10} + 40671083907 T^{11} + 1849654958673 T^{12} + 40671083907 p T^{13} + 25836521040 p^{2} T^{14} + 545110992 p^{3} T^{15} + 320286249 p^{4} T^{16} + 3536028 p^{5} T^{17} + 3278126 p^{6} T^{18} + 11856 p^{7} T^{19} + 30429 p^{8} T^{20} + 324 p^{9} T^{21} + 234 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( ( 1 + 21 T + 450 T^{2} + 5604 T^{3} + 74934 T^{4} + 716829 T^{5} + 7180669 T^{6} + 716829 p T^{7} + 74934 p^{2} T^{8} + 5604 p^{3} T^{9} + 450 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( 1 - 12 T + 111 T^{2} - 1541 T^{3} + 25770 T^{4} - 233292 T^{5} + 2394469 T^{6} - 25787493 T^{7} + 323541576 T^{8} - 2771225120 T^{9} + 24175268682 T^{10} - 230896025520 T^{11} + 2519251716379 T^{12} - 230896025520 p T^{13} + 24175268682 p^{2} T^{14} - 2771225120 p^{3} T^{15} + 323541576 p^{4} T^{16} - 25787493 p^{5} T^{17} + 2394469 p^{6} T^{18} - 233292 p^{7} T^{19} + 25770 p^{8} T^{20} - 1541 p^{9} T^{21} + 111 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 9 T + 126 T^{2} + 288 T^{3} + 16695 T^{4} + 67455 T^{5} + 2045531 T^{6} + 5405994 T^{7} + 194979879 T^{8} + 320041935 T^{9} + 19691641971 T^{10} + 52512992901 T^{11} + 1820862610437 T^{12} + 52512992901 p T^{13} + 19691641971 p^{2} T^{14} + 320041935 p^{3} T^{15} + 194979879 p^{4} T^{16} + 5405994 p^{5} T^{17} + 2045531 p^{6} T^{18} + 67455 p^{7} T^{19} + 16695 p^{8} T^{20} + 288 p^{9} T^{21} + 126 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 27 T + 474 T^{2} - 5462 T^{3} + 56682 T^{4} - 462984 T^{5} + 4073421 T^{6} - 27907434 T^{7} + 273342399 T^{8} - 2086071867 T^{9} + 22248589074 T^{10} - 129309282516 T^{11} + 1474612483999 T^{12} - 129309282516 p T^{13} + 22248589074 p^{2} T^{14} - 2086071867 p^{3} T^{15} + 273342399 p^{4} T^{16} - 27907434 p^{5} T^{17} + 4073421 p^{6} T^{18} - 462984 p^{7} T^{19} + 56682 p^{8} T^{20} - 5462 p^{9} T^{21} + 474 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 24 T - 27 T^{2} - 5604 T^{3} - 26343 T^{4} + 707565 T^{5} + 6243050 T^{6} - 44540559 T^{7} - 608791527 T^{8} + 1610221353 T^{9} + 39608297676 T^{10} + 595910511 T^{11} - 2273432889831 T^{12} + 595910511 p T^{13} + 39608297676 p^{2} T^{14} + 1610221353 p^{3} T^{15} - 608791527 p^{4} T^{16} - 44540559 p^{5} T^{17} + 6243050 p^{6} T^{18} + 707565 p^{7} T^{19} - 26343 p^{8} T^{20} - 5604 p^{9} T^{21} - 27 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.01663288763618395481253948157, −3.00870292960195928137093013680, −2.89571224929181699184261975496, −2.73825284204001726834838643683, −2.69689242786504469309807034025, −2.65207756429422874577286319846, −2.48504313888771088549759841384, −2.22751085010694853391328842050, −2.15469252698067948764897033019, −2.14800535826854608563126703548, −2.13818833077844194118821721599, −2.09346541948480485878905665490, −2.08325496240302340487316107365, −1.70202904055719849235631058524, −1.48054664589223682256735506907, −1.43864836104413730373562993830, −1.42715670768071471756659491141, −1.41194631100546574995083847383, −1.19939663238097430398270335919, −0.802461304236057234486921536106, −0.60461037287681279561604386193, −0.56043407373748593917059722103, −0.42640295860556849710568019865, −0.37716516757896937382239744674, −0.32061277125877477902542403863, 0.32061277125877477902542403863, 0.37716516757896937382239744674, 0.42640295860556849710568019865, 0.56043407373748593917059722103, 0.60461037287681279561604386193, 0.802461304236057234486921536106, 1.19939663238097430398270335919, 1.41194631100546574995083847383, 1.42715670768071471756659491141, 1.43864836104413730373562993830, 1.48054664589223682256735506907, 1.70202904055719849235631058524, 2.08325496240302340487316107365, 2.09346541948480485878905665490, 2.13818833077844194118821721599, 2.14800535826854608563126703548, 2.15469252698067948764897033019, 2.22751085010694853391328842050, 2.48504313888771088549759841384, 2.65207756429422874577286319846, 2.69689242786504469309807034025, 2.73825284204001726834838643683, 2.89571224929181699184261975496, 3.00870292960195928137093013680, 3.01663288763618395481253948157
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.