Dirichlet series
| L(s) = 1 | − 3·3-s − 6·4-s − 9·5-s + 4·8-s − 6·9-s + 18·12-s + 3·13-s + 27·15-s + 12·16-s − 9·17-s + 54·20-s − 12·24-s + 45·25-s + 28·27-s − 15·29-s − 30·31-s − 27·32-s + 36·36-s − 30·37-s − 9·39-s − 36·40-s − 18·41-s − 6·43-s + 54·45-s + 21·47-s − 36·48-s − 30·49-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 3·4-s − 4.02·5-s + 1.41·8-s − 2·9-s + 5.19·12-s + 0.832·13-s + 6.97·15-s + 3·16-s − 2.18·17-s + 12.0·20-s − 2.44·24-s + 9·25-s + 5.38·27-s − 2.78·29-s − 5.38·31-s − 4.77·32-s + 6·36-s − 4.93·37-s − 1.44·39-s − 5.69·40-s − 2.81·41-s − 0.914·43-s + 8.04·45-s + 3.06·47-s − 5.19·48-s − 4.28·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{9} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{9} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(5^{9} \cdot 19^{18}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.68403\times 10^{10}\) |
| Root analytic conductor: | \(3.79644\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 5^{9} \cdot 19^{18} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | 5 | \( ( 1 + T )^{9} \) |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + 3 p T^{2} - p^{2} T^{3} + 3 p^{3} T^{4} - 21 T^{5} + 9 p^{3} T^{6} - 75 T^{7} + 171 T^{8} - 177 T^{9} + 171 p T^{10} - 75 p^{2} T^{11} + 9 p^{6} T^{12} - 21 p^{4} T^{13} + 3 p^{8} T^{14} - p^{8} T^{15} + 3 p^{8} T^{16} + p^{9} T^{18} \) |
| 3 | \( 1 + p T + 5 p T^{2} + 35 T^{3} + 37 p T^{4} + 79 p T^{5} + 193 p T^{6} + 373 p T^{7} + 749 p T^{8} + 3871 T^{9} + 749 p^{2} T^{10} + 373 p^{3} T^{11} + 193 p^{4} T^{12} + 79 p^{5} T^{13} + 37 p^{6} T^{14} + 35 p^{6} T^{15} + 5 p^{8} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) | |
| 7 | \( 1 + 30 T^{2} + 10 T^{3} + 474 T^{4} + 192 T^{5} + 5389 T^{6} + 285 p T^{7} + 6801 p T^{8} + 319 p^{2} T^{9} + 6801 p^{2} T^{10} + 285 p^{3} T^{11} + 5389 p^{3} T^{12} + 192 p^{4} T^{13} + 474 p^{5} T^{14} + 10 p^{6} T^{15} + 30 p^{7} T^{16} + p^{9} T^{18} \) | |
| 11 | \( 1 + 63 T^{2} - 40 T^{3} + 1866 T^{4} - 2319 T^{5} + 3211 p T^{6} - 58584 T^{7} + 492000 T^{8} - 834149 T^{9} + 492000 p T^{10} - 58584 p^{2} T^{11} + 3211 p^{4} T^{12} - 2319 p^{4} T^{13} + 1866 p^{5} T^{14} - 40 p^{6} T^{15} + 63 p^{7} T^{16} + p^{9} T^{18} \) | |
| 13 | \( 1 - 3 T + 60 T^{2} - 108 T^{3} + 1821 T^{4} - 2454 T^{5} + 40032 T^{6} - 44769 T^{7} + 669741 T^{8} - 647665 T^{9} + 669741 p T^{10} - 44769 p^{2} T^{11} + 40032 p^{3} T^{12} - 2454 p^{4} T^{13} + 1821 p^{5} T^{14} - 108 p^{6} T^{15} + 60 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 9 T + 141 T^{2} + 968 T^{3} + 8505 T^{4} + 46953 T^{5} + 300109 T^{6} + 1382454 T^{7} + 7113252 T^{8} + 27872793 T^{9} + 7113252 p T^{10} + 1382454 p^{2} T^{11} + 300109 p^{3} T^{12} + 46953 p^{4} T^{13} + 8505 p^{5} T^{14} + 968 p^{6} T^{15} + 141 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 93 T^{2} + 170 T^{3} + 4152 T^{4} + 16017 T^{5} + 5556 p T^{6} + 721299 T^{7} + 3260967 T^{8} + 20307351 T^{9} + 3260967 p T^{10} + 721299 p^{2} T^{11} + 5556 p^{4} T^{12} + 16017 p^{4} T^{13} + 4152 p^{5} T^{14} + 170 p^{6} T^{15} + 93 p^{7} T^{16} + p^{9} T^{18} \) | |
| 29 | \( 1 + 15 T + 207 T^{2} + 1961 T^{3} + 17040 T^{4} + 122643 T^{5} + 832285 T^{6} + 4988805 T^{7} + 29400456 T^{8} + 158202393 T^{9} + 29400456 p T^{10} + 4988805 p^{2} T^{11} + 832285 p^{3} T^{12} + 122643 p^{4} T^{13} + 17040 p^{5} T^{14} + 1961 p^{6} T^{15} + 207 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 30 T + 564 T^{2} + 7480 T^{3} + 79374 T^{4} + 695745 T^{5} + 5298840 T^{6} + 35767185 T^{7} + 221521584 T^{8} + 1271602303 T^{9} + 221521584 p T^{10} + 35767185 p^{2} T^{11} + 5298840 p^{3} T^{12} + 695745 p^{4} T^{13} + 79374 p^{5} T^{14} + 7480 p^{6} T^{15} + 564 p^{7} T^{16} + 30 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 30 T + 600 T^{2} + 8522 T^{3} + 98484 T^{4} + 943845 T^{5} + 7904815 T^{6} + 1585047 p T^{7} + 399084690 T^{8} + 2506518869 T^{9} + 399084690 p T^{10} + 1585047 p^{3} T^{11} + 7904815 p^{3} T^{12} + 943845 p^{4} T^{13} + 98484 p^{5} T^{14} + 8522 p^{6} T^{15} + 600 p^{7} T^{16} + 30 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 18 T + 351 T^{2} + 3497 T^{3} + 36516 T^{4} + 213015 T^{5} + 33967 p T^{6} + 2472450 T^{7} + 9125322 T^{8} - 132745475 T^{9} + 9125322 p T^{10} + 2472450 p^{2} T^{11} + 33967 p^{4} T^{12} + 213015 p^{4} T^{13} + 36516 p^{5} T^{14} + 3497 p^{6} T^{15} + 351 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 6 T + 288 T^{2} + 1474 T^{3} + 38502 T^{4} + 166296 T^{5} + 3190276 T^{6} + 11690208 T^{7} + 185153736 T^{8} + 583746205 T^{9} + 185153736 p T^{10} + 11690208 p^{2} T^{11} + 3190276 p^{3} T^{12} + 166296 p^{4} T^{13} + 38502 p^{5} T^{14} + 1474 p^{6} T^{15} + 288 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 21 T + 363 T^{2} - 4203 T^{3} + 46251 T^{4} - 413523 T^{5} + 77413 p T^{6} - 27504027 T^{7} + 209788005 T^{8} - 1420501763 T^{9} + 209788005 p T^{10} - 27504027 p^{2} T^{11} + 77413 p^{4} T^{12} - 413523 p^{4} T^{13} + 46251 p^{5} T^{14} - 4203 p^{6} T^{15} + 363 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 9 T + 330 T^{2} - 2810 T^{3} + 55461 T^{4} - 421788 T^{5} + 5919750 T^{6} - 39597783 T^{7} + 438400545 T^{8} - 47510049 p T^{9} + 438400545 p T^{10} - 39597783 p^{2} T^{11} + 5919750 p^{3} T^{12} - 421788 p^{4} T^{13} + 55461 p^{5} T^{14} - 2810 p^{6} T^{15} + 330 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 27 T + 669 T^{2} + 11256 T^{3} + 171393 T^{4} + 2134647 T^{5} + 24312614 T^{6} + 239475900 T^{7} + 36822012 p T^{8} + 17366476051 T^{9} + 36822012 p^{2} T^{10} + 239475900 p^{2} T^{11} + 24312614 p^{3} T^{12} + 2134647 p^{4} T^{13} + 171393 p^{5} T^{14} + 11256 p^{6} T^{15} + 669 p^{7} T^{16} + 27 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 12 T + 330 T^{2} - 3534 T^{3} + 55302 T^{4} - 532932 T^{5} + 6274883 T^{6} - 52798755 T^{7} + 514842033 T^{8} - 3745507321 T^{9} + 514842033 p T^{10} - 52798755 p^{2} T^{11} + 6274883 p^{3} T^{12} - 532932 p^{4} T^{13} + 55302 p^{5} T^{14} - 3534 p^{6} T^{15} + 330 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 36 T + 786 T^{2} + 12125 T^{3} + 153165 T^{4} + 1670532 T^{5} + 16876290 T^{6} + 160030224 T^{7} + 1448180877 T^{8} + 12228577499 T^{9} + 1448180877 p T^{10} + 160030224 p^{2} T^{11} + 16876290 p^{3} T^{12} + 1670532 p^{4} T^{13} + 153165 p^{5} T^{14} + 12125 p^{6} T^{15} + 786 p^{7} T^{16} + 36 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 6 T + 462 T^{2} - 3172 T^{3} + 103323 T^{4} - 716955 T^{5} + 14792891 T^{6} - 94240119 T^{7} + 1475179173 T^{8} - 8119496483 T^{9} + 1475179173 p T^{10} - 94240119 p^{2} T^{11} + 14792891 p^{3} T^{12} - 716955 p^{4} T^{13} + 103323 p^{5} T^{14} - 3172 p^{6} T^{15} + 462 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 9 T + 390 T^{2} + 3040 T^{3} + 71094 T^{4} + 465699 T^{5} + 8187870 T^{6} + 45826335 T^{7} + 706947231 T^{8} + 3580256437 T^{9} + 706947231 p T^{10} + 45826335 p^{2} T^{11} + 8187870 p^{3} T^{12} + 465699 p^{4} T^{13} + 71094 p^{5} T^{14} + 3040 p^{6} T^{15} + 390 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 45 T + 1338 T^{2} + 28725 T^{3} + 508509 T^{4} + 7567899 T^{5} + 98988848 T^{6} + 1142795487 T^{7} + 11884603458 T^{8} + 110936558185 T^{9} + 11884603458 p T^{10} + 1142795487 p^{2} T^{11} + 98988848 p^{3} T^{12} + 7567899 p^{4} T^{13} + 508509 p^{5} T^{14} + 28725 p^{6} T^{15} + 1338 p^{7} T^{16} + 45 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 576 T^{2} - 507 T^{3} + 151803 T^{4} - 247698 T^{5} + 24576246 T^{6} - 50790690 T^{7} + 2768966469 T^{8} - 5598273343 T^{9} + 2768966469 p T^{10} - 50790690 p^{2} T^{11} + 24576246 p^{3} T^{12} - 247698 p^{4} T^{13} + 151803 p^{5} T^{14} - 507 p^{6} T^{15} + 576 p^{7} T^{16} + p^{9} T^{18} \) | |
| 89 | \( 1 - 9 T + 327 T^{2} - 4225 T^{3} + 72243 T^{4} - 850371 T^{5} + 11822154 T^{6} - 119250132 T^{7} + 1360333200 T^{8} - 12582561855 T^{9} + 1360333200 p T^{10} - 119250132 p^{2} T^{11} + 11822154 p^{3} T^{12} - 850371 p^{4} T^{13} + 72243 p^{5} T^{14} - 4225 p^{6} T^{15} + 327 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 45 T + 1344 T^{2} + 28316 T^{3} + 489126 T^{4} + 7111941 T^{5} + 92246475 T^{6} + 1084413258 T^{7} + 11843233752 T^{8} + 120453864257 T^{9} + 11843233752 p T^{10} + 1084413258 p^{2} T^{11} + 92246475 p^{3} T^{12} + 7111941 p^{4} T^{13} + 489126 p^{5} T^{14} + 28316 p^{6} T^{15} + 1344 p^{7} T^{16} + 45 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.99930527437413014026875760459, −3.93243671242134100034605902567, −3.87286781261623783483957892016, −3.83738665181085116790901100380, −3.79580543036009754375074950172, −3.67356805163624339298449499099, −3.59126644276337815727356952071, −3.48109919372336375440881895897, −3.18442172113361746817033261622, −3.14683752113462164746947464842, −3.10319389933574561654846941743, −2.89902251423658392809866365112, −2.79983746251609878632583005610, −2.79466554842994831230278583188, −2.45023617452399897959320134560, −2.34078253538495758748695285855, −2.28236051977775551760791137972, −2.12192110844482138330789120242, −1.66838672209491873278737338578, −1.58648554449940937693834104293, −1.54346189259478576796323754432, −1.44566158239833429770003512364, −1.42883749958246717501413899108, −1.16557534386890422230536614388, −1.10383329278208963677528337985, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.10383329278208963677528337985, 1.16557534386890422230536614388, 1.42883749958246717501413899108, 1.44566158239833429770003512364, 1.54346189259478576796323754432, 1.58648554449940937693834104293, 1.66838672209491873278737338578, 2.12192110844482138330789120242, 2.28236051977775551760791137972, 2.34078253538495758748695285855, 2.45023617452399897959320134560, 2.79466554842994831230278583188, 2.79983746251609878632583005610, 2.89902251423658392809866365112, 3.10319389933574561654846941743, 3.14683752113462164746947464842, 3.18442172113361746817033261622, 3.48109919372336375440881895897, 3.59126644276337815727356952071, 3.67356805163624339298449499099, 3.79580543036009754375074950172, 3.83738665181085116790901100380, 3.87286781261623783483957892016, 3.93243671242134100034605902567, 3.99930527437413014026875760459
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.