Dirichlet series
| L(s) = 1 | + 2·2-s + 11·3-s − 15·4-s − 3·5-s + 22·6-s + 63·7-s − 30·8-s − 24·9-s − 6·10-s − 8·11-s − 165·12-s + 164·13-s + 126·14-s − 33·15-s + 85·16-s + 153·17-s − 48·18-s + 244·19-s + 45·20-s + 693·21-s − 16·22-s − 14·23-s − 330·24-s − 216·25-s + 328·26-s − 734·27-s − 945·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.11·3-s − 1.87·4-s − 0.268·5-s + 1.49·6-s + 3.40·7-s − 1.32·8-s − 8/9·9-s − 0.189·10-s − 0.219·11-s − 3.96·12-s + 3.49·13-s + 2.40·14-s − 0.568·15-s + 1.32·16-s + 2.18·17-s − 0.628·18-s + 2.94·19-s + 0.503·20-s + 7.20·21-s − 0.155·22-s − 0.126·23-s − 2.80·24-s − 1.72·25-s + 2.47·26-s − 5.23·27-s − 6.37·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{9} \cdot 17^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{9} \cdot 17^{9}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(7^{9} \cdot 17^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.14684\times 10^{7}\) |
| Root analytic conductor: | \(2.64975\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 7^{9} \cdot 17^{9} ,\ ( \ : [3/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(36.23531657\) |
| \(L(\frac12)\) | \(\approx\) | \(36.23531657\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - p T )^{9} \) |
| 17 | \( ( 1 - p T )^{9} \) | |
| good | 2 | \( 1 - p T + 19 T^{2} - 19 p T^{3} + 27 p^{3} T^{4} - 351 T^{5} + 1151 p T^{6} - 3213 T^{7} + 9587 p T^{8} - 7243 p^{2} T^{9} + 9587 p^{4} T^{10} - 3213 p^{6} T^{11} + 1151 p^{10} T^{12} - 351 p^{12} T^{13} + 27 p^{18} T^{14} - 19 p^{19} T^{15} + 19 p^{21} T^{16} - p^{25} T^{17} + p^{27} T^{18} \) |
| 3 | \( 1 - 11 T + 145 T^{2} - 125 p^{2} T^{3} + 9700 T^{4} - 61367 T^{5} + 140359 p T^{6} - 764239 p T^{7} + 4628051 p T^{8} - 68242328 T^{9} + 4628051 p^{4} T^{10} - 764239 p^{7} T^{11} + 140359 p^{10} T^{12} - 61367 p^{12} T^{13} + 9700 p^{15} T^{14} - 125 p^{20} T^{15} + 145 p^{21} T^{16} - 11 p^{24} T^{17} + p^{27} T^{18} \) | |
| 5 | \( 1 + 3 T + 9 p^{2} T^{2} + 623 T^{3} + 39672 T^{4} + 116877 T^{5} + 6610307 T^{6} + 14952629 T^{7} + 946161639 T^{8} + 239676368 p T^{9} + 946161639 p^{3} T^{10} + 14952629 p^{6} T^{11} + 6610307 p^{9} T^{12} + 116877 p^{12} T^{13} + 39672 p^{15} T^{14} + 623 p^{18} T^{15} + 9 p^{23} T^{16} + 3 p^{24} T^{17} + p^{27} T^{18} \) | |
| 11 | \( 1 + 8 T + 2631 T^{2} + 86456 T^{3} + 4645888 T^{4} + 258027536 T^{5} + 9508311832 T^{6} + 385630998024 T^{7} + 19581105335482 T^{8} + 460688921406672 T^{9} + 19581105335482 p^{3} T^{10} + 385630998024 p^{6} T^{11} + 9508311832 p^{9} T^{12} + 258027536 p^{12} T^{13} + 4645888 p^{15} T^{14} + 86456 p^{18} T^{15} + 2631 p^{21} T^{16} + 8 p^{24} T^{17} + p^{27} T^{18} \) | |
| 13 | \( 1 - 164 T + 19173 T^{2} - 1492416 T^{3} + 99821484 T^{4} - 425916976 p T^{5} + 313244794940 T^{6} - 16804871164864 T^{7} + 925352695042902 T^{8} - 3432443000189368 p T^{9} + 925352695042902 p^{3} T^{10} - 16804871164864 p^{6} T^{11} + 313244794940 p^{9} T^{12} - 425916976 p^{13} T^{13} + 99821484 p^{15} T^{14} - 1492416 p^{18} T^{15} + 19173 p^{21} T^{16} - 164 p^{24} T^{17} + p^{27} T^{18} \) | |
| 19 | \( 1 - 244 T + 58891 T^{2} - 8756712 T^{3} + 1278853644 T^{4} - 141230080224 T^{5} + 15697823463108 T^{6} - 74897758566984 p T^{7} + 134586246989207238 T^{8} - 10799954769647632088 T^{9} + 134586246989207238 p^{3} T^{10} - 74897758566984 p^{7} T^{11} + 15697823463108 p^{9} T^{12} - 141230080224 p^{12} T^{13} + 1278853644 p^{15} T^{14} - 8756712 p^{18} T^{15} + 58891 p^{21} T^{16} - 244 p^{24} T^{17} + p^{27} T^{18} \) | |
| 23 | \( 1 + 14 T + 59855 T^{2} + 2404224 T^{3} + 1726165956 T^{4} + 111482933032 T^{5} + 33104078744908 T^{6} + 2698266433693952 T^{7} + 489585167650044206 T^{8} + 40646172682961775380 T^{9} + 489585167650044206 p^{3} T^{10} + 2698266433693952 p^{6} T^{11} + 33104078744908 p^{9} T^{12} + 111482933032 p^{12} T^{13} + 1726165956 p^{15} T^{14} + 2404224 p^{18} T^{15} + 59855 p^{21} T^{16} + 14 p^{24} T^{17} + p^{27} T^{18} \) | |
| 29 | \( 1 + 234 T + 137537 T^{2} + 27396768 T^{3} + 8844610880 T^{4} + 1576807540120 T^{5} + 378483324413896 T^{6} + 61014649118697696 T^{7} + 12154279805993445610 T^{8} + 59639382978517785836 p T^{9} + 12154279805993445610 p^{3} T^{10} + 61014649118697696 p^{6} T^{11} + 378483324413896 p^{9} T^{12} + 1576807540120 p^{12} T^{13} + 8844610880 p^{15} T^{14} + 27396768 p^{18} T^{15} + 137537 p^{21} T^{16} + 234 p^{24} T^{17} + p^{27} T^{18} \) | |
| 31 | \( 1 - 555 T + 195081 T^{2} - 40938455 T^{3} + 8086754432 T^{4} - 1589916641721 T^{5} + 389540000200341 T^{6} - 73527742498210593 T^{7} + 12742920086207249289 T^{8} - \)\(19\!\cdots\!72\)\( T^{9} + 12742920086207249289 p^{3} T^{10} - 73527742498210593 p^{6} T^{11} + 389540000200341 p^{9} T^{12} - 1589916641721 p^{12} T^{13} + 8086754432 p^{15} T^{14} - 40938455 p^{18} T^{15} + 195081 p^{21} T^{16} - 555 p^{24} T^{17} + p^{27} T^{18} \) | |
| 37 | \( 1 + 364 T + 172825 T^{2} + 52471776 T^{3} + 18467680592 T^{4} + 4383541388080 T^{5} + 1207060654185752 T^{6} + 269052710749270560 T^{7} + 66637637880948542890 T^{8} + \)\(13\!\cdots\!72\)\( T^{9} + 66637637880948542890 p^{3} T^{10} + 269052710749270560 p^{6} T^{11} + 1207060654185752 p^{9} T^{12} + 4383541388080 p^{12} T^{13} + 18467680592 p^{15} T^{14} + 52471776 p^{18} T^{15} + 172825 p^{21} T^{16} + 364 p^{24} T^{17} + p^{27} T^{18} \) | |
| 41 | \( 1 + 45 T + 209743 T^{2} - 201509 p T^{3} + 29490899716 T^{4} - 2216600282659 T^{5} + 2920419013359197 T^{6} - 350503435898344129 T^{7} + \)\(23\!\cdots\!47\)\( T^{8} - \)\(28\!\cdots\!24\)\( T^{9} + \)\(23\!\cdots\!47\)\( p^{3} T^{10} - 350503435898344129 p^{6} T^{11} + 2920419013359197 p^{9} T^{12} - 2216600282659 p^{12} T^{13} + 29490899716 p^{15} T^{14} - 201509 p^{19} T^{15} + 209743 p^{21} T^{16} + 45 p^{24} T^{17} + p^{27} T^{18} \) | |
| 43 | \( 1 + 135 T + 502467 T^{2} + 2193777 p T^{3} + 118297747296 T^{4} + 27013066879005 T^{5} + 17585139903694223 T^{6} + 4313895909707560529 T^{7} + \)\(18\!\cdots\!89\)\( T^{8} + \)\(42\!\cdots\!88\)\( T^{9} + \)\(18\!\cdots\!89\)\( p^{3} T^{10} + 4313895909707560529 p^{6} T^{11} + 17585139903694223 p^{9} T^{12} + 27013066879005 p^{12} T^{13} + 118297747296 p^{15} T^{14} + 2193777 p^{19} T^{15} + 502467 p^{21} T^{16} + 135 p^{24} T^{17} + p^{27} T^{18} \) | |
| 47 | \( 1 + 172 T + 756143 T^{2} + 127627816 T^{3} + 268472184700 T^{4} + 42314987148352 T^{5} + 58680675425196868 T^{6} + 8284022767869333208 T^{7} + \)\(86\!\cdots\!02\)\( T^{8} + \)\(10\!\cdots\!28\)\( T^{9} + \)\(86\!\cdots\!02\)\( p^{3} T^{10} + 8284022767869333208 p^{6} T^{11} + 58680675425196868 p^{9} T^{12} + 42314987148352 p^{12} T^{13} + 268472184700 p^{15} T^{14} + 127627816 p^{18} T^{15} + 756143 p^{21} T^{16} + 172 p^{24} T^{17} + p^{27} T^{18} \) | |
| 53 | \( 1 - 101 T + 455439 T^{2} - 138153787 T^{3} + 134183529744 T^{4} - 42576840456497 T^{5} + 33829884752720833 T^{6} - 9413943542608083927 T^{7} + \)\(59\!\cdots\!11\)\( T^{8} - \)\(17\!\cdots\!04\)\( T^{9} + \)\(59\!\cdots\!11\)\( p^{3} T^{10} - 9413943542608083927 p^{6} T^{11} + 33829884752720833 p^{9} T^{12} - 42576840456497 p^{12} T^{13} + 134183529744 p^{15} T^{14} - 138153787 p^{18} T^{15} + 455439 p^{21} T^{16} - 101 p^{24} T^{17} + p^{27} T^{18} \) | |
| 59 | \( 1 - 280 T + 909551 T^{2} - 207394704 T^{3} + 479029952488 T^{4} - 1653010441920 p T^{5} + 172508506126727584 T^{6} - 30465673226695874288 T^{7} + \)\(46\!\cdots\!34\)\( T^{8} - \)\(73\!\cdots\!32\)\( T^{9} + \)\(46\!\cdots\!34\)\( p^{3} T^{10} - 30465673226695874288 p^{6} T^{11} + 172508506126727584 p^{9} T^{12} - 1653010441920 p^{13} T^{13} + 479029952488 p^{15} T^{14} - 207394704 p^{18} T^{15} + 909551 p^{21} T^{16} - 280 p^{24} T^{17} + p^{27} T^{18} \) | |
| 61 | \( 1 - 639 T + 1204805 T^{2} - 685887987 T^{3} + 720223755280 T^{4} - 395735715767469 T^{5} + 291642385087181411 T^{6} - \)\(15\!\cdots\!25\)\( T^{7} + \)\(86\!\cdots\!27\)\( T^{8} - \)\(40\!\cdots\!36\)\( T^{9} + \)\(86\!\cdots\!27\)\( p^{3} T^{10} - \)\(15\!\cdots\!25\)\( p^{6} T^{11} + 291642385087181411 p^{9} T^{12} - 395735715767469 p^{12} T^{13} + 720223755280 p^{15} T^{14} - 685887987 p^{18} T^{15} + 1204805 p^{21} T^{16} - 639 p^{24} T^{17} + p^{27} T^{18} \) | |
| 67 | \( 1 - 35 T + 1643073 T^{2} - 23684919 T^{3} + 1284260653044 T^{4} - 15806361282969 T^{5} + 664226594711006077 T^{6} - 14880156563049751573 T^{7} + \)\(25\!\cdots\!41\)\( T^{8} - \)\(68\!\cdots\!48\)\( T^{9} + \)\(25\!\cdots\!41\)\( p^{3} T^{10} - 14880156563049751573 p^{6} T^{11} + 664226594711006077 p^{9} T^{12} - 15806361282969 p^{12} T^{13} + 1284260653044 p^{15} T^{14} - 23684919 p^{18} T^{15} + 1643073 p^{21} T^{16} - 35 p^{24} T^{17} + p^{27} T^{18} \) | |
| 71 | \( 1 + 1616 T + 3558291 T^{2} + 4105335560 T^{3} + 5240700857688 T^{4} + 4689975179390032 T^{5} + 4381311738644494832 T^{6} + \)\(31\!\cdots\!56\)\( T^{7} + \)\(23\!\cdots\!22\)\( T^{8} + \)\(13\!\cdots\!76\)\( T^{9} + \)\(23\!\cdots\!22\)\( p^{3} T^{10} + \)\(31\!\cdots\!56\)\( p^{6} T^{11} + 4381311738644494832 p^{9} T^{12} + 4689975179390032 p^{12} T^{13} + 5240700857688 p^{15} T^{14} + 4105335560 p^{18} T^{15} + 3558291 p^{21} T^{16} + 1616 p^{24} T^{17} + p^{27} T^{18} \) | |
| 73 | \( 1 - 1049 T + 2962095 T^{2} - 2580739491 T^{3} + 3957911630060 T^{4} - 2958043480378901 T^{5} + 3218372784686736637 T^{6} - \)\(20\!\cdots\!31\)\( T^{7} + \)\(17\!\cdots\!75\)\( T^{8} - \)\(97\!\cdots\!76\)\( T^{9} + \)\(17\!\cdots\!75\)\( p^{3} T^{10} - \)\(20\!\cdots\!31\)\( p^{6} T^{11} + 3218372784686736637 p^{9} T^{12} - 2958043480378901 p^{12} T^{13} + 3957911630060 p^{15} T^{14} - 2580739491 p^{18} T^{15} + 2962095 p^{21} T^{16} - 1049 p^{24} T^{17} + p^{27} T^{18} \) | |
| 79 | \( 1 - 2304 T + 3998927 T^{2} - 64332120 p T^{3} + 5545160554012 T^{4} - 5159198937149808 T^{5} + 4425215144057061284 T^{6} - \)\(34\!\cdots\!84\)\( T^{7} + \)\(25\!\cdots\!78\)\( T^{8} - \)\(18\!\cdots\!48\)\( T^{9} + \)\(25\!\cdots\!78\)\( p^{3} T^{10} - \)\(34\!\cdots\!84\)\( p^{6} T^{11} + 4425215144057061284 p^{9} T^{12} - 5159198937149808 p^{12} T^{13} + 5545160554012 p^{15} T^{14} - 64332120 p^{19} T^{15} + 3998927 p^{21} T^{16} - 2304 p^{24} T^{17} + p^{27} T^{18} \) | |
| 83 | \( 1 - 2508 T + 6283035 T^{2} - 10014959928 T^{3} + 15095387724380 T^{4} - 17972546410319040 T^{5} + 20195040057578666132 T^{6} - \)\(19\!\cdots\!88\)\( T^{7} + \)\(17\!\cdots\!74\)\( T^{8} - \)\(13\!\cdots\!92\)\( T^{9} + \)\(17\!\cdots\!74\)\( p^{3} T^{10} - \)\(19\!\cdots\!88\)\( p^{6} T^{11} + 20195040057578666132 p^{9} T^{12} - 17972546410319040 p^{12} T^{13} + 15095387724380 p^{15} T^{14} - 10014959928 p^{18} T^{15} + 6283035 p^{21} T^{16} - 2508 p^{24} T^{17} + p^{27} T^{18} \) | |
| 89 | \( 1 - 2762 T + 7104533 T^{2} - 11487020912 T^{3} + 17296692527840 T^{4} - 20203038096126584 T^{5} + 22633446507113232200 T^{6} - \)\(21\!\cdots\!40\)\( T^{7} + \)\(20\!\cdots\!62\)\( T^{8} - \)\(16\!\cdots\!56\)\( T^{9} + \)\(20\!\cdots\!62\)\( p^{3} T^{10} - \)\(21\!\cdots\!40\)\( p^{6} T^{11} + 22633446507113232200 p^{9} T^{12} - 20203038096126584 p^{12} T^{13} + 17296692527840 p^{15} T^{14} - 11487020912 p^{18} T^{15} + 7104533 p^{21} T^{16} - 2762 p^{24} T^{17} + p^{27} T^{18} \) | |
| 97 | \( 1 - 3107 T + 8280575 T^{2} - 15313931817 T^{3} + 26098513452732 T^{4} - 36834273906623427 T^{5} + 48726046896068557545 T^{6} - \)\(56\!\cdots\!21\)\( T^{7} + \)\(61\!\cdots\!39\)\( T^{8} - \)\(62\!\cdots\!04\)\( p T^{9} + \)\(61\!\cdots\!39\)\( p^{3} T^{10} - \)\(56\!\cdots\!21\)\( p^{6} T^{11} + 48726046896068557545 p^{9} T^{12} - 36834273906623427 p^{12} T^{13} + 26098513452732 p^{15} T^{14} - 15313931817 p^{18} T^{15} + 8280575 p^{21} T^{16} - 3107 p^{24} T^{17} + p^{27} T^{18} \) | |
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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
−5.11416879798894994565757920246, −5.03228427201292864797662572211, −5.01170050505041734572118304779, −4.77134260976956282860112892443, −4.74056086565849536828522524687, −4.70178614526185273724658896411, −4.20809667798585514751793488517, −3.93577279452100139835594608735, −3.76922793935275457603266660180, −3.64547074849579019770663114965, −3.54568411298690253109115832231, −3.51951577532816389959312990694, −3.44804848700139479210332730532, −3.33827297202347876480502641478, −2.89260393186160560044492603578, −2.63221884268312863606032404417, −2.50883368420805445454568357721, −2.22300531362943512228173521293, −1.91350705201481158987435085256, −1.83069355736653053178418956734, −1.33353272187663302917383706765, −1.23579772631249405156352991532, −0.988951160857284496286882075767, −0.57649031153392002392685903267, −0.53743239395995487779825838597, 0.53743239395995487779825838597, 0.57649031153392002392685903267, 0.988951160857284496286882075767, 1.23579772631249405156352991532, 1.33353272187663302917383706765, 1.83069355736653053178418956734, 1.91350705201481158987435085256, 2.22300531362943512228173521293, 2.50883368420805445454568357721, 2.63221884268312863606032404417, 2.89260393186160560044492603578, 3.33827297202347876480502641478, 3.44804848700139479210332730532, 3.51951577532816389959312990694, 3.54568411298690253109115832231, 3.64547074849579019770663114965, 3.76922793935275457603266660180, 3.93577279452100139835594608735, 4.20809667798585514751793488517, 4.70178614526185273724658896411, 4.74056086565849536828522524687, 4.77134260976956282860112892443, 5.01170050505041734572118304779, 5.03228427201292864797662572211, 5.11416879798894994565757920246
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.