Dirichlet series
| L(s) = 1 | − 6·2-s − 6·3-s + 18·4-s + 9·5-s + 36·6-s + 3·7-s − 48·8-s + 9·9-s − 54·10-s − 87·11-s − 108·12-s + 171·13-s − 18·14-s − 54·15-s + 99·16-s + 36·17-s − 54·18-s + 324·19-s + 162·20-s − 18·21-s + 522·22-s − 84·23-s + 288·24-s + 297·25-s − 1.02e3·26-s + 54·28-s + 393·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 9/4·4-s + 0.804·5-s + 2.44·6-s + 0.161·7-s − 2.12·8-s + 1/3·9-s − 1.70·10-s − 2.38·11-s − 2.59·12-s + 3.64·13-s − 0.343·14-s − 0.929·15-s + 1.54·16-s + 0.513·17-s − 0.707·18-s + 3.91·19-s + 1.81·20-s − 0.187·21-s + 5.05·22-s − 0.761·23-s + 2.44·24-s + 2.37·25-s − 7.73·26-s + 0.364·28-s + 2.51·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(206.557\) |
| Root analytic conductor: | \(1.39537\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 11^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.6785242097\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6785242097\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | \( 1 + 87 T + 403 p T^{2} + 1899 p^{2} T^{3} + 7500 p^{3} T^{4} + 1899 p^{5} T^{5} + 403 p^{7} T^{6} + 87 p^{9} T^{7} + p^{12} T^{8} \) | |
| good | 2 | \( 1 + 3 p T + 9 p T^{2} + 3 p^{4} T^{3} + 153 T^{4} + 9 p^{5} T^{5} + 371 T^{6} + 1419 T^{7} + 6393 T^{8} + 1419 p^{3} T^{9} + 371 p^{6} T^{10} + 9 p^{14} T^{11} + 153 p^{12} T^{12} + 3 p^{19} T^{13} + 9 p^{19} T^{14} + 3 p^{22} T^{15} + p^{24} T^{16} \) |
| 5 | \( 1 - 9 T - 216 T^{2} - 108 T^{3} + 28704 T^{4} + 66129 T^{5} + 3870971 T^{6} - 23622552 T^{7} - 735099864 T^{8} - 23622552 p^{3} T^{9} + 3870971 p^{6} T^{10} + 66129 p^{9} T^{11} + 28704 p^{12} T^{12} - 108 p^{15} T^{13} - 216 p^{18} T^{14} - 9 p^{21} T^{15} + p^{24} T^{16} \) | |
| 7 | \( 1 - 3 T + 645 T^{2} - 8102 T^{3} + 318354 T^{4} - 5087519 T^{5} + 22962172 p T^{6} - 1769175174 T^{7} + 69157603561 T^{8} - 1769175174 p^{3} T^{9} + 22962172 p^{7} T^{10} - 5087519 p^{9} T^{11} + 318354 p^{12} T^{12} - 8102 p^{15} T^{13} + 645 p^{18} T^{14} - 3 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 - 171 T + 15261 T^{2} - 1179218 T^{3} + 78389016 T^{4} - 4452059033 T^{5} + 251390254120 T^{6} - 13190914668630 T^{7} + 625464463948177 T^{8} - 13190914668630 p^{3} T^{9} + 251390254120 p^{6} T^{10} - 4452059033 p^{9} T^{11} + 78389016 p^{12} T^{12} - 1179218 p^{15} T^{13} + 15261 p^{18} T^{14} - 171 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 - 36 T - 4704 T^{2} + 183996 T^{3} + 26983893 T^{4} - 1508244396 T^{5} - 94799670436 T^{6} + 159155191584 p T^{7} + 216503731307013 T^{8} + 159155191584 p^{4} T^{9} - 94799670436 p^{6} T^{10} - 1508244396 p^{9} T^{11} + 26983893 p^{12} T^{12} + 183996 p^{15} T^{13} - 4704 p^{18} T^{14} - 36 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 - 324 T + 42702 T^{2} - 2758988 T^{3} + 70535517 T^{4} + 3205929520 T^{5} - 922691677142 T^{6} + 150427813848684 T^{7} - 15705053233704905 T^{8} + 150427813848684 p^{3} T^{9} - 922691677142 p^{6} T^{10} + 3205929520 p^{9} T^{11} + 70535517 p^{12} T^{12} - 2758988 p^{15} T^{13} + 42702 p^{18} T^{14} - 324 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( ( 1 + 42 T + 28427 T^{2} - 413280 T^{3} + 349199196 T^{4} - 413280 p^{3} T^{5} + 28427 p^{6} T^{6} + 42 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 29 | \( 1 - 393 T + 51762 T^{2} + 2821128 T^{3} - 2373469998 T^{4} + 460592296485 T^{5} - 25589160757657 T^{6} - 7422933439897836 T^{7} + 1994632640290085280 T^{8} - 7422933439897836 p^{3} T^{9} - 25589160757657 p^{6} T^{10} + 460592296485 p^{9} T^{11} - 2373469998 p^{12} T^{12} + 2821128 p^{15} T^{13} + 51762 p^{18} T^{14} - 393 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 - 15 T - 48960 T^{2} + 4414990 T^{3} + 1983425964 T^{4} - 36049577855 T^{5} - 67992515413865 T^{6} + 334846364616300 T^{7} + 2537520588739268776 T^{8} + 334846364616300 p^{3} T^{9} - 67992515413865 p^{6} T^{10} - 36049577855 p^{9} T^{11} + 1983425964 p^{12} T^{12} + 4414990 p^{15} T^{13} - 48960 p^{18} T^{14} - 15 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 + 747 T + 208089 T^{2} + 18431284 T^{3} - 1384403574 T^{4} + 152950360711 T^{5} - 32557198551620 T^{6} - 149480116479179160 T^{7} - 55527587568079442063 T^{8} - 149480116479179160 p^{3} T^{9} - 32557198551620 p^{6} T^{10} + 152950360711 p^{9} T^{11} - 1384403574 p^{12} T^{12} + 18431284 p^{15} T^{13} + 208089 p^{18} T^{14} + 747 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 159 T - 208197 T^{2} + 540786 p T^{3} + 14361268182 T^{4} - 959553734445 T^{5} + 186051111389942 T^{6} + 9057919749889008 T^{7} - 68887839310262606715 T^{8} + 9057919749889008 p^{3} T^{9} + 186051111389942 p^{6} T^{10} - 959553734445 p^{9} T^{11} + 14361268182 p^{12} T^{12} + 540786 p^{16} T^{13} - 208197 p^{18} T^{14} - 159 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( ( 1 + 322 T + 181243 T^{2} + 48993496 T^{3} + 21697488148 T^{4} + 48993496 p^{3} T^{5} + 181243 p^{6} T^{6} + 322 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 47 | \( 1 + 351 T - 13296 T^{2} + 44932770 T^{3} + 25775818116 T^{4} + 3450345166263 T^{5} + 42832258537345 p T^{6} + 899717908387248252 T^{7} + \)\(20\!\cdots\!32\)\( T^{8} + 899717908387248252 p^{3} T^{9} + 42832258537345 p^{7} T^{10} + 3450345166263 p^{9} T^{11} + 25775818116 p^{12} T^{12} + 44932770 p^{15} T^{13} - 13296 p^{18} T^{14} + 351 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 + 531 T - 438 p T^{2} + 55947168 T^{3} + 86011920246 T^{4} + 24095935682613 T^{5} + 567789066963755 T^{6} + 3215216614335806880 T^{7} + \)\(27\!\cdots\!92\)\( T^{8} + 3215216614335806880 p^{3} T^{9} + 567789066963755 p^{6} T^{10} + 24095935682613 p^{9} T^{11} + 86011920246 p^{12} T^{12} + 55947168 p^{15} T^{13} - 438 p^{19} T^{14} + 531 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 + 1002 T + 688980 T^{2} + 485993670 T^{3} + 355241156445 T^{4} + 202661193907746 T^{5} + 100593726036552992 T^{6} + 53249849322765499920 T^{7} + \)\(26\!\cdots\!05\)\( T^{8} + 53249849322765499920 p^{3} T^{9} + 100593726036552992 p^{6} T^{10} + 202661193907746 p^{9} T^{11} + 355241156445 p^{12} T^{12} + 485993670 p^{15} T^{13} + 688980 p^{18} T^{14} + 1002 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 - 1449 T + 559458 T^{2} - 1297814 T^{3} + 56591949732 T^{4} - 46750854974495 T^{5} - 2095488968582993 T^{6} - 15938076497384457402 T^{7} + \)\(18\!\cdots\!00\)\( T^{8} - 15938076497384457402 p^{3} T^{9} - 2095488968582993 p^{6} T^{10} - 46750854974495 p^{9} T^{11} + 56591949732 p^{12} T^{12} - 1297814 p^{15} T^{13} + 559458 p^{18} T^{14} - 1449 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( ( 1 + 259 T + 1117027 T^{2} + 221089363 T^{3} + 492802725568 T^{4} + 221089363 p^{3} T^{5} + 1117027 p^{6} T^{6} + 259 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 71 | \( 1 - 429 T + 427302 T^{2} + 117188664 T^{3} - 140939575254 T^{4} + 134614525047165 T^{5} + 3634249555303 p T^{6} - 49916639131846946880 T^{7} + \)\(41\!\cdots\!52\)\( T^{8} - 49916639131846946880 p^{3} T^{9} + 3634249555303 p^{7} T^{10} + 134614525047165 p^{9} T^{11} - 140939575254 p^{12} T^{12} + 117188664 p^{15} T^{13} + 427302 p^{18} T^{14} - 429 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 - 2547 T + 2079912 T^{2} + 411582106 T^{3} - 2098827853302 T^{4} + 1679415683125579 T^{5} - 313174419721114481 T^{6} - \)\(56\!\cdots\!82\)\( T^{7} + \)\(58\!\cdots\!08\)\( T^{8} - \)\(56\!\cdots\!82\)\( p^{3} T^{9} - 313174419721114481 p^{6} T^{10} + 1679415683125579 p^{9} T^{11} - 2098827853302 p^{12} T^{12} + 411582106 p^{15} T^{13} + 2079912 p^{18} T^{14} - 2547 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 - 2805 T + 2454909 T^{2} + 477147760 T^{3} - 2482607356860 T^{4} + 1889298883532485 T^{5} - 272522469312909914 T^{6} - \)\(74\!\cdots\!60\)\( T^{7} + \)\(81\!\cdots\!89\)\( T^{8} - \)\(74\!\cdots\!60\)\( p^{3} T^{9} - 272522469312909914 p^{6} T^{10} + 1889298883532485 p^{9} T^{11} - 2482607356860 p^{12} T^{12} + 477147760 p^{15} T^{13} + 2454909 p^{18} T^{14} - 2805 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 + 2553 T + 3276861 T^{2} + 3557232930 T^{3} + 3857530610346 T^{4} + 3425756556730389 T^{5} + 2523855999787432520 T^{6} + \)\(20\!\cdots\!94\)\( T^{7} + \)\(16\!\cdots\!37\)\( T^{8} + \)\(20\!\cdots\!94\)\( p^{3} T^{9} + 2523855999787432520 p^{6} T^{10} + 3425756556730389 p^{9} T^{11} + 3857530610346 p^{12} T^{12} + 3557232930 p^{15} T^{13} + 3276861 p^{18} T^{14} + 2553 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( ( 1 - 894 T + 1324211 T^{2} - 258930180 T^{3} + 627382282200 T^{4} - 258930180 p^{3} T^{5} + 1324211 p^{6} T^{6} - 894 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 97 | \( 1 - 9 T - 1418094 T^{2} - 1082718536 T^{3} + 1552923605058 T^{4} + 351454828977301 T^{5} - 769417621790276381 T^{6} - \)\(34\!\cdots\!08\)\( T^{7} + \)\(13\!\cdots\!08\)\( T^{8} - \)\(34\!\cdots\!08\)\( p^{3} T^{9} - 769417621790276381 p^{6} T^{10} + 351454828977301 p^{9} T^{11} + 1552923605058 p^{12} T^{12} - 1082718536 p^{15} T^{13} - 1418094 p^{18} T^{14} - 9 p^{21} T^{15} + p^{24} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.955955324313110903886441370561, −7.88257078388935275236840308086, −7.21943912358481746428080179781, −7.14566249398237995335834626232, −6.99300540714576538332452770204, −6.52766807652638573261451184614, −6.48940066816493698980875849404, −6.48380120995525336123793502370, −6.16271692677866093089773720631, −5.88661804657719469367261424182, −5.54716308241448878575945122750, −5.40469474319773766701555659949, −5.21118236109139042911201762805, −4.95055038512853184718436171382, −4.85075632742316730143632564392, −4.71564007572031154349394851724, −3.67324718881870581718367017412, −3.61890354833987843907564749361, −3.13410729031416747220649615240, −3.09430483491226042306453162522, −2.91338334474154452240175044150, −1.83653963426987260236137631995, −1.63861996280661918398598882024, −0.790914546188214919336574994937, −0.78068424904051998672772859366, 0.78068424904051998672772859366, 0.790914546188214919336574994937, 1.63861996280661918398598882024, 1.83653963426987260236137631995, 2.91338334474154452240175044150, 3.09430483491226042306453162522, 3.13410729031416747220649615240, 3.61890354833987843907564749361, 3.67324718881870581718367017412, 4.71564007572031154349394851724, 4.85075632742316730143632564392, 4.95055038512853184718436171382, 5.21118236109139042911201762805, 5.40469474319773766701555659949, 5.54716308241448878575945122750, 5.88661804657719469367261424182, 6.16271692677866093089773720631, 6.48380120995525336123793502370, 6.48940066816493698980875849404, 6.52766807652638573261451184614, 6.99300540714576538332452770204, 7.14566249398237995335834626232, 7.21943912358481746428080179781, 7.88257078388935275236840308086, 7.955955324313110903886441370561
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.