Dirichlet series
| L(s) = 1 | − 9·2-s + 45·4-s + 4·5-s − 4·7-s − 165·8-s − 36·10-s + 4·11-s − 8·13-s + 36·14-s + 495·16-s + 17-s − 10·19-s + 180·20-s − 36·22-s + 8·23-s − 16·25-s + 72·26-s − 180·28-s + 4·29-s − 17·31-s − 1.28e3·32-s − 9·34-s − 16·35-s − 11·37-s + 90·38-s − 660·40-s − 16·43-s + ⋯ |
| L(s) = 1 | − 6.36·2-s + 45/2·4-s + 1.78·5-s − 1.51·7-s − 58.3·8-s − 11.3·10-s + 1.20·11-s − 2.21·13-s + 9.62·14-s + 123.·16-s + 0.242·17-s − 2.29·19-s + 40.2·20-s − 7.67·22-s + 1.66·23-s − 3.19·25-s + 14.1·26-s − 34.0·28-s + 0.742·29-s − 3.05·31-s − 227.·32-s − 1.54·34-s − 2.70·35-s − 1.80·37-s + 14.5·38-s − 104.·40-s − 2.43·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{27} \cdot 149^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{27} \cdot 149^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{9} \cdot 3^{27} \cdot 149^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.86515\times 10^{16}\) |
| Root analytic conductor: | \(8.01546\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{9} \cdot 3^{27} \cdot 149^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{9} \) |
| 3 | \( 1 \) | |
| 149 | \( ( 1 - T )^{9} \) | |
| good | 5 | \( 1 - 4 T + 32 T^{2} - 104 T^{3} + 488 T^{4} - 272 p T^{5} + 4759 T^{6} - 2294 p T^{7} + 32617 T^{8} - 67718 T^{9} + 32617 p T^{10} - 2294 p^{3} T^{11} + 4759 p^{3} T^{12} - 272 p^{5} T^{13} + 488 p^{5} T^{14} - 104 p^{6} T^{15} + 32 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 + 4 T + 6 p T^{2} + 141 T^{3} + 794 T^{4} + 2265 T^{5} + 9183 T^{6} + 3301 p T^{7} + 77577 T^{8} + 179045 T^{9} + 77577 p T^{10} + 3301 p^{3} T^{11} + 9183 p^{3} T^{12} + 2265 p^{4} T^{13} + 794 p^{5} T^{14} + 141 p^{6} T^{15} + 6 p^{8} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - 4 T + 68 T^{2} - 284 T^{3} + 2307 T^{4} - 9201 T^{5} + 50752 T^{6} - 180614 T^{7} + 781453 T^{8} - 2385860 T^{9} + 781453 p T^{10} - 180614 p^{2} T^{11} + 50752 p^{3} T^{12} - 9201 p^{4} T^{13} + 2307 p^{5} T^{14} - 284 p^{6} T^{15} + 68 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 + 8 T + 113 T^{2} + 684 T^{3} + 5484 T^{4} + 26690 T^{5} + 155755 T^{6} + 628402 T^{7} + 2920764 T^{8} + 9865738 T^{9} + 2920764 p T^{10} + 628402 p^{2} T^{11} + 155755 p^{3} T^{12} + 26690 p^{4} T^{13} + 5484 p^{5} T^{14} + 684 p^{6} T^{15} + 113 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - T + 93 T^{2} - 19 T^{3} + 4351 T^{4} + 1013 T^{5} + 135092 T^{6} + 59947 T^{7} + 3044452 T^{8} + 1420770 T^{9} + 3044452 p T^{10} + 59947 p^{2} T^{11} + 135092 p^{3} T^{12} + 1013 p^{4} T^{13} + 4351 p^{5} T^{14} - 19 p^{6} T^{15} + 93 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 10 T + 154 T^{2} + 1087 T^{3} + 9648 T^{4} + 54358 T^{5} + 361242 T^{6} + 1729463 T^{7} + 9424332 T^{8} + 38786652 T^{9} + 9424332 p T^{10} + 1729463 p^{2} T^{11} + 361242 p^{3} T^{12} + 54358 p^{4} T^{13} + 9648 p^{5} T^{14} + 1087 p^{6} T^{15} + 154 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 8 T + 177 T^{2} - 1015 T^{3} + 12770 T^{4} - 54339 T^{5} + 524816 T^{6} - 1728021 T^{7} + 14993524 T^{8} - 42219669 T^{9} + 14993524 p T^{10} - 1728021 p^{2} T^{11} + 524816 p^{3} T^{12} - 54339 p^{4} T^{13} + 12770 p^{5} T^{14} - 1015 p^{6} T^{15} + 177 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 4 T + 140 T^{2} - 228 T^{3} + 8814 T^{4} - 1232 T^{5} + 406708 T^{6} + 76890 T^{7} + 15758047 T^{8} + 990911 T^{9} + 15758047 p T^{10} + 76890 p^{2} T^{11} + 406708 p^{3} T^{12} - 1232 p^{4} T^{13} + 8814 p^{5} T^{14} - 228 p^{6} T^{15} + 140 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 17 T + 294 T^{2} + 3288 T^{3} + 35058 T^{4} + 299254 T^{5} + 2415185 T^{6} + 538327 p T^{7} + 109028017 T^{8} + 624140757 T^{9} + 109028017 p T^{10} + 538327 p^{3} T^{11} + 2415185 p^{3} T^{12} + 299254 p^{4} T^{13} + 35058 p^{5} T^{14} + 3288 p^{6} T^{15} + 294 p^{7} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 11 T + 232 T^{2} + 1750 T^{3} + 21288 T^{4} + 115727 T^{5} + 1094681 T^{6} + 4383700 T^{7} + 40610911 T^{8} + 143060408 T^{9} + 40610911 p T^{10} + 4383700 p^{2} T^{11} + 1094681 p^{3} T^{12} + 115727 p^{4} T^{13} + 21288 p^{5} T^{14} + 1750 p^{6} T^{15} + 232 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 219 T^{2} + 187 T^{3} + 24091 T^{4} + 33174 T^{5} + 1783023 T^{6} + 2683062 T^{7} + 2370443 p T^{8} + 134191801 T^{9} + 2370443 p^{2} T^{10} + 2683062 p^{2} T^{11} + 1783023 p^{3} T^{12} + 33174 p^{4} T^{13} + 24091 p^{5} T^{14} + 187 p^{6} T^{15} + 219 p^{7} T^{16} + p^{9} T^{18} \) | |
| 43 | \( 1 + 16 T + 299 T^{2} + 2907 T^{3} + 31033 T^{4} + 203017 T^{5} + 1564829 T^{6} + 6998481 T^{7} + 50274985 T^{8} + 203934498 T^{9} + 50274985 p T^{10} + 6998481 p^{2} T^{11} + 1564829 p^{3} T^{12} + 203017 p^{4} T^{13} + 31033 p^{5} T^{14} + 2907 p^{6} T^{15} + 299 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 7 T + 233 T^{2} - 1150 T^{3} + 24491 T^{4} - 90119 T^{5} + 1667498 T^{6} - 4659039 T^{7} + 87538662 T^{8} - 208700788 T^{9} + 87538662 p T^{10} - 4659039 p^{2} T^{11} + 1667498 p^{3} T^{12} - 90119 p^{4} T^{13} + 24491 p^{5} T^{14} - 1150 p^{6} T^{15} + 233 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 12 T + 251 T^{2} - 2123 T^{3} + 28110 T^{4} - 185955 T^{5} + 38810 p T^{6} - 11875405 T^{7} + 123270464 T^{8} - 660472201 T^{9} + 123270464 p T^{10} - 11875405 p^{2} T^{11} + 38810 p^{4} T^{12} - 185955 p^{4} T^{13} + 28110 p^{5} T^{14} - 2123 p^{6} T^{15} + 251 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 6 T + 286 T^{2} - 1739 T^{3} + 43076 T^{4} - 249230 T^{5} + 4459908 T^{6} - 23858923 T^{7} + 343771964 T^{8} - 1647933976 T^{9} + 343771964 p T^{10} - 23858923 p^{2} T^{11} + 4459908 p^{3} T^{12} - 249230 p^{4} T^{13} + 43076 p^{5} T^{14} - 1739 p^{6} T^{15} + 286 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 13 T + 484 T^{2} + 5224 T^{3} + 107984 T^{4} + 976962 T^{5} + 14476899 T^{6} + 110318977 T^{7} + 1285534845 T^{8} + 8199304700 T^{9} + 1285534845 p T^{10} + 110318977 p^{2} T^{11} + 14476899 p^{3} T^{12} + 976962 p^{4} T^{13} + 107984 p^{5} T^{14} + 5224 p^{6} T^{15} + 484 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 14 T + 404 T^{2} + 4765 T^{3} + 1140 p T^{4} + 793798 T^{5} + 9308190 T^{6} + 86206581 T^{7} + 824351120 T^{8} + 6730075334 T^{9} + 824351120 p T^{10} + 86206581 p^{2} T^{11} + 9308190 p^{3} T^{12} + 793798 p^{4} T^{13} + 1140 p^{6} T^{14} + 4765 p^{6} T^{15} + 404 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 30 T + 690 T^{2} - 12048 T^{3} + 178293 T^{4} - 2311374 T^{5} + 26830140 T^{6} - 282338634 T^{7} + 2711601060 T^{8} - 23874389457 T^{9} + 2711601060 p T^{10} - 282338634 p^{2} T^{11} + 26830140 p^{3} T^{12} - 2311374 p^{4} T^{13} + 178293 p^{5} T^{14} - 12048 p^{6} T^{15} + 690 p^{7} T^{16} - 30 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 12 T + 377 T^{2} + 4929 T^{3} + 78527 T^{4} + 927103 T^{5} + 11214517 T^{6} + 111105985 T^{7} + 1134999931 T^{8} + 9491974800 T^{9} + 1134999931 p T^{10} + 111105985 p^{2} T^{11} + 11214517 p^{3} T^{12} + 927103 p^{4} T^{13} + 78527 p^{5} T^{14} + 4929 p^{6} T^{15} + 377 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 35 T + 974 T^{2} + 19318 T^{3} + 329779 T^{4} + 4739591 T^{5} + 60921526 T^{6} + 691590981 T^{7} + 7153951615 T^{8} + 66417803924 T^{9} + 7153951615 p T^{10} + 691590981 p^{2} T^{11} + 60921526 p^{3} T^{12} + 4739591 p^{4} T^{13} + 329779 p^{5} T^{14} + 19318 p^{6} T^{15} + 974 p^{7} T^{16} + 35 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 5 T + 419 T^{2} - 2365 T^{3} + 86890 T^{4} - 462426 T^{5} + 11913060 T^{6} - 56786404 T^{7} + 1216257923 T^{8} - 5289477148 T^{9} + 1216257923 p T^{10} - 56786404 p^{2} T^{11} + 11913060 p^{3} T^{12} - 462426 p^{4} T^{13} + 86890 p^{5} T^{14} - 2365 p^{6} T^{15} + 419 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 23 T + 858 T^{2} - 15157 T^{3} + 314746 T^{4} - 4452798 T^{5} + 66088684 T^{6} - 764102797 T^{7} + 8849517519 T^{8} - 83953359151 T^{9} + 8849517519 p T^{10} - 764102797 p^{2} T^{11} + 66088684 p^{3} T^{12} - 4452798 p^{4} T^{13} + 314746 p^{5} T^{14} - 15157 p^{6} T^{15} + 858 p^{7} T^{16} - 23 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 21 T + 846 T^{2} + 14729 T^{3} + 324320 T^{4} + 4641521 T^{5} + 73149346 T^{6} + 8870019 p T^{7} + 10636874310 T^{8} + 102739172088 T^{9} + 10636874310 p T^{10} + 8870019 p^{3} T^{11} + 73149346 p^{3} T^{12} + 4641521 p^{4} T^{13} + 324320 p^{5} T^{14} + 14729 p^{6} T^{15} + 846 p^{7} T^{16} + 21 p^{8} T^{17} + p^{9} T^{18} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.11763154553051407841753694511, −3.00130711002894130869342207155, −2.98740636948925897058116724077, −2.98417109403947272724540474785, −2.73670728058330090729192108296, −2.66197754371092572683679583416, −2.56942635536594183142304299184, −2.41940742307949299457775225947, −2.35217301328854416372373416028, −2.15935143708204838098963881119, −2.11922667824200519163661170152, −2.07138981922815814843856302667, −2.04066775418461761249290570363, −2.03524779103973030688833838064, −2.02447755525788029964145097941, −1.90155157901033363859471150659, −1.47988293267986527200619469544, −1.40215608639621367170923072822, −1.39961464549881625754258589737, −1.26288390226152157287481669498, −1.18706015857149831458228299169, −1.18132559230621461705687492317, −1.13089887684212813256058492753, −1.09198171538632844125847321306, −0.835655961346648460693801053990, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.835655961346648460693801053990, 1.09198171538632844125847321306, 1.13089887684212813256058492753, 1.18132559230621461705687492317, 1.18706015857149831458228299169, 1.26288390226152157287481669498, 1.39961464549881625754258589737, 1.40215608639621367170923072822, 1.47988293267986527200619469544, 1.90155157901033363859471150659, 2.02447755525788029964145097941, 2.03524779103973030688833838064, 2.04066775418461761249290570363, 2.07138981922815814843856302667, 2.11922667824200519163661170152, 2.15935143708204838098963881119, 2.35217301328854416372373416028, 2.41940742307949299457775225947, 2.56942635536594183142304299184, 2.66197754371092572683679583416, 2.73670728058330090729192108296, 2.98417109403947272724540474785, 2.98740636948925897058116724077, 3.00130711002894130869342207155, 3.11763154553051407841753694511
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.