Dirichlet series
| L(s) = 1 | − 4·2-s + 6·3-s + 8·4-s + 4·5-s − 24·6-s + 2·7-s − 18·8-s + 22·9-s − 16·10-s − 2·11-s + 48·12-s + 10·13-s − 8·14-s + 24·15-s + 34·16-s + 2·17-s − 88·18-s − 4·19-s + 32·20-s + 12·21-s + 8·22-s + 4·23-s − 108·24-s + 11·25-s − 40·26-s + 58·27-s + 16·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3.46·3-s + 4·4-s + 1.78·5-s − 9.79·6-s + 0.755·7-s − 6.36·8-s + 22/3·9-s − 5.05·10-s − 0.603·11-s + 13.8·12-s + 2.77·13-s − 2.13·14-s + 6.19·15-s + 17/2·16-s + 0.485·17-s − 20.7·18-s − 0.917·19-s + 7.15·20-s + 2.61·21-s + 1.70·22-s + 0.834·23-s − 22.0·24-s + 11/5·25-s − 7.84·26-s + 11.1·27-s + 3.02·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(11^{8} \cdot 17^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(24.7143\) |
| Root analytic conductor: | \(1.22196\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 11^{8} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.475787152\) |
| \(L(\frac12)\) | \(\approx\) | \(2.475787152\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 34 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) | |
| good | 2 | \( ( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} )^{2} \) |
| 3 | \( 1 - 2 p T + 14 T^{2} - 10 T^{3} - p^{2} T^{4} + 2 T^{5} + 248 T^{7} - 743 T^{8} + 248 p T^{9} + 2 p^{3} T^{11} - p^{6} T^{12} - 10 p^{5} T^{13} + 14 p^{6} T^{14} - 2 p^{8} T^{15} + p^{8} T^{16} \) | |
| 5 | \( 1 - 4 T + p T^{2} - 6 T^{3} + 19 T^{4} - 6 T^{5} + 7 p T^{6} - 184 T^{7} + 376 T^{8} - 184 p T^{9} + 7 p^{3} T^{10} - 6 p^{3} T^{11} + 19 p^{4} T^{12} - 6 p^{5} T^{13} + p^{7} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + 43 T^{4} - 26 T^{5} + 304 T^{6} - 216 p T^{7} + 3033 T^{8} - 216 p^{2} T^{9} + 304 p^{2} T^{10} - 26 p^{3} T^{11} + 43 p^{4} T^{12} - 2 p^{6} T^{13} + 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( ( 1 - 5 T + 2 T^{2} - 25 T^{3} + 259 T^{4} - 25 p T^{5} + 2 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 19 | \( 1 + 4 T - 14 T^{2} - 40 T^{3} + 203 T^{4} - 500 T^{5} - 4 p T^{6} - 3856 T^{7} - 113747 T^{8} - 3856 p T^{9} - 4 p^{3} T^{10} - 500 p^{3} T^{11} + 203 p^{4} T^{12} - 40 p^{5} T^{13} - 14 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 2 T + 82 T^{2} - 112 T^{3} + 2703 T^{4} - 112 p T^{5} + 82 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 5 T^{2} + 1164 T^{4} - 660 T^{5} - 16795 T^{6} - 80190 T^{7} + 1094291 T^{8} - 80190 p T^{9} - 16795 p^{2} T^{10} - 660 p^{3} T^{11} + 1164 p^{4} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \) | |
| 31 | \( 1 + 4 T - 2 T^{2} + 44 T^{3} + 1307 T^{4} + 7060 T^{5} + 1352 T^{6} + 144992 T^{7} + 1532245 T^{8} + 144992 p T^{9} + 1352 p^{2} T^{10} + 7060 p^{3} T^{11} + 1307 p^{4} T^{12} + 44 p^{5} T^{13} - 2 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 10 T + 2 p T^{2} - 310 T^{3} + 427 T^{4} + 2150 T^{5} + 8752 T^{6} - 255780 T^{7} + 2223585 T^{8} - 255780 p T^{9} + 8752 p^{2} T^{10} + 2150 p^{3} T^{11} + 427 p^{4} T^{12} - 310 p^{5} T^{13} + 2 p^{7} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 10 T - 15 T^{2} - 390 T^{3} - 901 T^{4} + 6510 T^{5} + 103535 T^{6} - 210290 T^{7} - 7676904 T^{8} - 210290 p T^{9} + 103535 p^{2} T^{10} + 6510 p^{3} T^{11} - 901 p^{4} T^{12} - 390 p^{5} T^{13} - 15 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 47 | \( 1 - 10 T + 63 T^{2} - 180 T^{3} - 385 T^{4} + 10560 T^{5} - 87667 T^{6} + 866090 T^{7} - 9453336 T^{8} + 866090 p T^{9} - 87667 p^{2} T^{10} + 10560 p^{3} T^{11} - 385 p^{4} T^{12} - 180 p^{5} T^{13} + 63 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 4 T - 2 p T^{2} + 700 T^{3} + 3251 T^{4} - 40292 T^{5} + 165120 T^{6} + 963152 T^{7} - 17026683 T^{8} + 963152 p T^{9} + 165120 p^{2} T^{10} - 40292 p^{3} T^{11} + 3251 p^{4} T^{12} + 700 p^{5} T^{13} - 2 p^{7} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( ( 1 - 3 T - 50 T^{2} + 327 T^{3} + 1969 T^{4} + 327 p T^{5} - 50 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 61 | \( 1 - 30 T + 570 T^{2} - 8630 T^{3} + 107619 T^{4} - 1160150 T^{5} + 11255440 T^{6} - 99108900 T^{7} + 801945121 T^{8} - 99108900 p T^{9} + 11255440 p^{2} T^{10} - 1160150 p^{3} T^{11} + 107619 p^{4} T^{12} - 8630 p^{5} T^{13} + 570 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 10 T + 219 T^{2} + 1820 T^{3} + 20357 T^{4} + 1820 p T^{5} + 219 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 18 T - 2 T^{2} + 2466 T^{3} - 14073 T^{4} - 200010 T^{5} + 2354912 T^{6} + 6874272 T^{7} - 219035335 T^{8} + 6874272 p T^{9} + 2354912 p^{2} T^{10} - 200010 p^{3} T^{11} - 14073 p^{4} T^{12} + 2466 p^{5} T^{13} - 2 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 6 T - 102 T^{2} - 502 T^{3} + 7683 T^{4} + 29078 T^{5} - 370904 T^{6} - 1714236 T^{7} - 3606647 T^{8} - 1714236 p T^{9} - 370904 p^{2} T^{10} + 29078 p^{3} T^{11} + 7683 p^{4} T^{12} - 502 p^{5} T^{13} - 102 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 4 T - 58 T^{2} - 8 p T^{3} - 9053 T^{4} - 103580 T^{5} + 189628 T^{6} + 9453696 T^{7} + 74812485 T^{8} + 9453696 p T^{9} + 189628 p^{2} T^{10} - 103580 p^{3} T^{11} - 9053 p^{4} T^{12} - 8 p^{6} T^{13} - 58 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 20 T + 2 T^{2} - 2120 T^{3} - 6445 T^{4} + 119740 T^{5} + 812412 T^{6} - 848320 T^{7} - 34624491 T^{8} - 848320 p T^{9} + 812412 p^{2} T^{10} + 119740 p^{3} T^{11} - 6445 p^{4} T^{12} - 2120 p^{5} T^{13} + 2 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 24 T + 316 T^{2} + 2280 T^{3} + 17190 T^{4} + 2280 p T^{5} + 316 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 12 T - 74 T^{2} + 2340 T^{3} - 7389 T^{4} - 876 T^{5} - 406180 T^{6} - 8351136 T^{7} + 259303637 T^{8} - 8351136 p T^{9} - 406180 p^{2} T^{10} - 876 p^{3} T^{11} - 7389 p^{4} T^{12} + 2340 p^{5} T^{13} - 74 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.96025591940146402348093784519, −5.72553558868592956206456158417, −5.71987552905335270148556211017, −5.39525909994423972259783779481, −5.35038323610511920920254050193, −4.92849605838415496352917512836, −4.63195359115671035188846274171, −4.57628089575792889903496855583, −4.33581310430583300512281539006, −4.11437206136530350261856851586, −3.93985581942489634653102862324, −3.88464415354035243467297400901, −3.72954212252259519017541071880, −3.23528976162768513002526213156, −3.14196642280455700056520733524, −2.99982782918793499317773798535, −2.89402927455475508291492486354, −2.49923366850725567992912819262, −2.46648393126332152477317168277, −2.31203225111523509896355082328, −1.98790049033873964319730830088, −1.94452181782367182720462367491, −1.37730924163509438414736587896, −1.08621226589157458777001933948, −1.03598939302934267626168419160, 1.03598939302934267626168419160, 1.08621226589157458777001933948, 1.37730924163509438414736587896, 1.94452181782367182720462367491, 1.98790049033873964319730830088, 2.31203225111523509896355082328, 2.46648393126332152477317168277, 2.49923366850725567992912819262, 2.89402927455475508291492486354, 2.99982782918793499317773798535, 3.14196642280455700056520733524, 3.23528976162768513002526213156, 3.72954212252259519017541071880, 3.88464415354035243467297400901, 3.93985581942489634653102862324, 4.11437206136530350261856851586, 4.33581310430583300512281539006, 4.57628089575792889903496855583, 4.63195359115671035188846274171, 4.92849605838415496352917512836, 5.35038323610511920920254050193, 5.39525909994423972259783779481, 5.71987552905335270148556211017, 5.72553558868592956206456158417, 5.96025591940146402348093784519
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.