Dirichlet series
| L(s) = 1 | + 8·3-s − 4·4-s + 8·5-s + 6·7-s + 2·8-s + 36·9-s + 12·11-s − 32·12-s + 4·13-s + 64·15-s + 4·16-s + 20·17-s + 4·19-s − 32·20-s + 48·21-s + 16·24-s + 36·25-s + 120·27-s − 24·28-s + 2·31-s − 8·32-s + 96·33-s + 48·35-s − 144·36-s + 2·37-s + 32·39-s + 16·40-s + ⋯ |
| L(s) = 1 | + 4.61·3-s − 2·4-s + 3.57·5-s + 2.26·7-s + 0.707·8-s + 12·9-s + 3.61·11-s − 9.23·12-s + 1.10·13-s + 16.5·15-s + 16-s + 4.85·17-s + 0.917·19-s − 7.15·20-s + 10.4·21-s + 3.26·24-s + 36/5·25-s + 23.0·27-s − 4.53·28-s + 0.359·31-s − 1.41·32-s + 16.7·33-s + 8.11·35-s − 24·36-s + 0.328·37-s + 5.12·39-s + 2.52·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{8} \cdot 23^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.59771\times 10^{14}\) |
| Root analytic conductor: | \(7.95998\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{8} \cdot 23^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6765.544542\) |
| \(L(\frac12)\) | \(\approx\) | \(6765.544542\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{8} \) |
| 5 | \( ( 1 - T )^{8} \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 + p^{2} T^{2} - p T^{3} + 3 p^{2} T^{4} - p^{3} T^{5} + 15 p T^{6} - p^{4} T^{7} + 9 p^{3} T^{8} - p^{5} T^{9} + 15 p^{3} T^{10} - p^{6} T^{11} + 3 p^{6} T^{12} - p^{6} T^{13} + p^{8} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 6 T + 39 T^{2} - 164 T^{3} + 688 T^{4} - 2336 T^{5} + 7933 T^{6} - 3286 p T^{7} + 65582 T^{8} - 3286 p^{2} T^{9} + 7933 p^{2} T^{10} - 2336 p^{3} T^{11} + 688 p^{4} T^{12} - 164 p^{5} T^{13} + 39 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 12 T + 105 T^{2} - 62 p T^{3} + 3741 T^{4} - 17708 T^{5} + 74752 T^{6} - 25902 p T^{7} + 989598 T^{8} - 25902 p^{2} T^{9} + 74752 p^{2} T^{10} - 17708 p^{3} T^{11} + 3741 p^{4} T^{12} - 62 p^{6} T^{13} + 105 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 4 T + 53 T^{2} - 172 T^{3} + 1512 T^{4} - 4466 T^{5} + 30683 T^{6} - 79154 T^{7} + 456564 T^{8} - 79154 p T^{9} + 30683 p^{2} T^{10} - 4466 p^{3} T^{11} + 1512 p^{4} T^{12} - 172 p^{5} T^{13} + 53 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 20 T + 232 T^{2} - 1894 T^{3} + 12192 T^{4} - 65374 T^{5} + 309006 T^{6} - 1344512 T^{7} + 5620994 T^{8} - 1344512 p T^{9} + 309006 p^{2} T^{10} - 65374 p^{3} T^{11} + 12192 p^{4} T^{12} - 1894 p^{5} T^{13} + 232 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 4 T + 111 T^{2} - 480 T^{3} + 5721 T^{4} - 25392 T^{5} + 183614 T^{6} - 773108 T^{7} + 4108830 T^{8} - 773108 p T^{9} + 183614 p^{2} T^{10} - 25392 p^{3} T^{11} + 5721 p^{4} T^{12} - 480 p^{5} T^{13} + 111 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 112 T^{2} + 8 p T^{3} + 6388 T^{4} + 21304 T^{5} + 256352 T^{6} + 1035520 T^{7} + 8000150 T^{8} + 1035520 p T^{9} + 256352 p^{2} T^{10} + 21304 p^{3} T^{11} + 6388 p^{4} T^{12} + 8 p^{6} T^{13} + 112 p^{6} T^{14} + p^{8} T^{16} \) | |
| 31 | \( 1 - 2 T + 139 T^{2} - 102 T^{3} + 9105 T^{4} + 3864 T^{5} + 386718 T^{6} + 475036 T^{7} + 12925354 T^{8} + 475036 p T^{9} + 386718 p^{2} T^{10} + 3864 p^{3} T^{11} + 9105 p^{4} T^{12} - 102 p^{5} T^{13} + 139 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 2 T + 133 T^{2} - 30 T^{3} + 10452 T^{4} + 672 T^{5} + 598323 T^{6} + 159736 T^{7} + 677140 p T^{8} + 159736 p T^{9} + 598323 p^{2} T^{10} + 672 p^{3} T^{11} + 10452 p^{4} T^{12} - 30 p^{5} T^{13} + 133 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 28 T + 619 T^{2} - 9254 T^{3} + 118311 T^{4} - 1216460 T^{5} + 11062734 T^{6} - 85237042 T^{7} + 588991490 T^{8} - 85237042 p T^{9} + 11062734 p^{2} T^{10} - 1216460 p^{3} T^{11} + 118311 p^{4} T^{12} - 9254 p^{5} T^{13} + 619 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 4 T + 235 T^{2} - 1150 T^{3} + 25238 T^{4} - 142252 T^{5} + 1698849 T^{6} - 9950950 T^{7} + 83299176 T^{8} - 9950950 p T^{9} + 1698849 p^{2} T^{10} - 142252 p^{3} T^{11} + 25238 p^{4} T^{12} - 1150 p^{5} T^{13} + 235 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 12 T + 172 T^{2} + 1924 T^{3} + 18340 T^{4} + 162532 T^{5} + 1327412 T^{6} + 9979372 T^{7} + 69806966 T^{8} + 9979372 p T^{9} + 1327412 p^{2} T^{10} + 162532 p^{3} T^{11} + 18340 p^{4} T^{12} + 1924 p^{5} T^{13} + 172 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 6 T + 262 T^{2} - 1364 T^{3} + 33156 T^{4} - 151640 T^{5} + 2769912 T^{6} - 11092198 T^{7} + 169525818 T^{8} - 11092198 p T^{9} + 2769912 p^{2} T^{10} - 151640 p^{3} T^{11} + 33156 p^{4} T^{12} - 1364 p^{5} T^{13} + 262 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 2 T + 238 T^{2} - 94 T^{3} + 25320 T^{4} + 39506 T^{5} + 1659558 T^{6} + 6088798 T^{7} + 93432902 T^{8} + 6088798 p T^{9} + 1659558 p^{2} T^{10} + 39506 p^{3} T^{11} + 25320 p^{4} T^{12} - 94 p^{5} T^{13} + 238 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 32 T + 692 T^{2} - 10428 T^{3} + 130514 T^{4} - 1343444 T^{5} + 12471428 T^{6} - 103910748 T^{7} + 837799767 T^{8} - 103910748 p T^{9} + 12471428 p^{2} T^{10} - 1343444 p^{3} T^{11} + 130514 p^{4} T^{12} - 10428 p^{5} T^{13} + 692 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 32 T + 849 T^{2} - 15624 T^{3} + 245954 T^{4} - 3200360 T^{5} + 36436375 T^{6} - 360086432 T^{7} + 3144809882 T^{8} - 360086432 p T^{9} + 36436375 p^{2} T^{10} - 3200360 p^{3} T^{11} + 245954 p^{4} T^{12} - 15624 p^{5} T^{13} + 849 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 2 T + 157 T^{2} - 452 T^{3} + 16577 T^{4} - 111412 T^{5} + 1371128 T^{6} - 11715886 T^{7} + 87116122 T^{8} - 11715886 p T^{9} + 1371128 p^{2} T^{10} - 111412 p^{3} T^{11} + 16577 p^{4} T^{12} - 452 p^{5} T^{13} + 157 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 2 T + 322 T^{2} + 246 T^{3} + 48684 T^{4} - 4470 T^{5} + 4882422 T^{6} - 2679490 T^{7} + 387294454 T^{8} - 2679490 p T^{9} + 4882422 p^{2} T^{10} - 4470 p^{3} T^{11} + 48684 p^{4} T^{12} + 246 p^{5} T^{13} + 322 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 36 T + 927 T^{2} + 17860 T^{3} + 289297 T^{4} + 3970024 T^{5} + 47792530 T^{6} + 506211344 T^{7} + 4777285598 T^{8} + 506211344 p T^{9} + 47792530 p^{2} T^{10} + 3970024 p^{3} T^{11} + 289297 p^{4} T^{12} + 17860 p^{5} T^{13} + 927 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 10 T + 138 T^{2} - 112 T^{3} + 17452 T^{4} - 136884 T^{5} + 2664524 T^{6} - 6531366 T^{7} + 146194082 T^{8} - 6531366 p T^{9} + 2664524 p^{2} T^{10} - 136884 p^{3} T^{11} + 17452 p^{4} T^{12} - 112 p^{5} T^{13} + 138 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 42 T + 1154 T^{2} - 22630 T^{3} + 365504 T^{4} - 4982154 T^{5} + 60472690 T^{6} - 655494166 T^{7} + 6500872326 T^{8} - 655494166 p T^{9} + 60472690 p^{2} T^{10} - 4982154 p^{3} T^{11} + 365504 p^{4} T^{12} - 22630 p^{5} T^{13} + 1154 p^{6} T^{14} - 42 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 16 T + 522 T^{2} - 5570 T^{3} + 108122 T^{4} - 801234 T^{5} + 12805060 T^{6} - 71188924 T^{7} + 1225405638 T^{8} - 71188924 p T^{9} + 12805060 p^{2} T^{10} - 801234 p^{3} T^{11} + 108122 p^{4} T^{12} - 5570 p^{5} T^{13} + 522 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.21598884000243264137322243701, −3.13030411548948265165286926945, −2.93282471179533297712544031172, −2.76061798206151410091180656775, −2.72449870833722202906660715137, −2.62468187294715290977774098188, −2.61155798734546493764150112408, −2.25175122119986610775823131090, −2.20352609559880512711449303181, −2.14274710363131191160257924661, −2.05841590858901394270843295606, −2.05807388503308786495019363768, −1.75611723685664540580313630964, −1.71202644302847303231104054620, −1.62985615222501610941152857782, −1.60491628583452280266993466236, −1.48844691913642887206227190578, −1.37603028750643096936766698279, −1.05009439670831447896444044093, −1.02774611201056359424203408121, −0.844684785860145858880450154760, −0.832694785638239461388079267892, −0.75964880610399439581095003410, −0.74635681495676394388303219511, −0.72516235639024298721934167682, 0.72516235639024298721934167682, 0.74635681495676394388303219511, 0.75964880610399439581095003410, 0.832694785638239461388079267892, 0.844684785860145858880450154760, 1.02774611201056359424203408121, 1.05009439670831447896444044093, 1.37603028750643096936766698279, 1.48844691913642887206227190578, 1.60491628583452280266993466236, 1.62985615222501610941152857782, 1.71202644302847303231104054620, 1.75611723685664540580313630964, 2.05807388503308786495019363768, 2.05841590858901394270843295606, 2.14274710363131191160257924661, 2.20352609559880512711449303181, 2.25175122119986610775823131090, 2.61155798734546493764150112408, 2.62468187294715290977774098188, 2.72449870833722202906660715137, 2.76061798206151410091180656775, 2.93282471179533297712544031172, 3.13030411548948265165286926945, 3.21598884000243264137322243701
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.