Dirichlet series
| L(s) = 1 | − 4·2-s − 3·3-s + 6·4-s − 7·5-s + 12·6-s − 7-s + 11·9-s + 28·10-s − 2·11-s − 18·12-s + 2·13-s + 4·14-s + 21·15-s − 15·16-s − 5·17-s − 44·18-s − 11·19-s − 42·20-s + 3·21-s + 8·22-s + 4·23-s + 33·25-s − 8·26-s − 28·27-s − 6·28-s + 4·29-s − 84·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 1.73·3-s + 3·4-s − 3.13·5-s + 4.89·6-s − 0.377·7-s + 11/3·9-s + 8.85·10-s − 0.603·11-s − 5.19·12-s + 0.554·13-s + 1.06·14-s + 5.42·15-s − 3.75·16-s − 1.21·17-s − 10.3·18-s − 2.52·19-s − 9.39·20-s + 0.654·21-s + 1.70·22-s + 0.834·23-s + 33/5·25-s − 1.56·26-s − 5.38·27-s − 1.13·28-s + 0.742·29-s − 15.3·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 7^{8} \cdot 23^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1910.13\) |
| Root analytic conductor: | \(1.60349\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 7^{8} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.06734670321\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06734670321\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 7 | \( 1 + T + 10 T^{2} + 5 T^{3} + 101 T^{4} + 5 p T^{5} + 10 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | |
| 23 | \( ( 1 - T + T^{2} )^{4} \) | |
| good | 3 | \( 1 + p T - 2 T^{2} - 11 T^{3} + 2 p T^{4} + 14 T^{5} - 55 T^{6} + 4 T^{7} + 265 T^{8} + 4 p T^{9} - 55 p^{2} T^{10} + 14 p^{3} T^{11} + 2 p^{5} T^{12} - 11 p^{5} T^{13} - 2 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 7 T + 16 T^{2} + 11 T^{3} + 42 T^{4} + 302 T^{5} + 759 T^{6} + 164 p T^{7} + 629 T^{8} + 164 p^{2} T^{9} + 759 p^{2} T^{10} + 302 p^{3} T^{11} + 42 p^{4} T^{12} + 11 p^{5} T^{13} + 16 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 2 T - 28 T^{2} - 10 T^{3} + 497 T^{4} - 221 T^{5} - 6197 T^{6} + 1119 T^{7} + 62253 T^{8} + 1119 p T^{9} - 6197 p^{2} T^{10} - 221 p^{3} T^{11} + 497 p^{4} T^{12} - 10 p^{5} T^{13} - 28 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( ( 1 - T + 10 T^{2} - 3 p T^{3} + 265 T^{4} - 3 p^{2} T^{5} + 10 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 17 | \( 1 + 5 T - 7 T^{2} + 182 T^{3} + 992 T^{4} - 1409 T^{5} + 16768 T^{6} + 81816 T^{7} - 128295 T^{8} + 81816 p T^{9} + 16768 p^{2} T^{10} - 1409 p^{3} T^{11} + 992 p^{4} T^{12} + 182 p^{5} T^{13} - 7 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 11 T + 58 T^{2} + 5 T^{3} - 1588 T^{4} - 10798 T^{5} - 19773 T^{6} + 111998 T^{7} + 991893 T^{8} + 111998 p T^{9} - 19773 p^{2} T^{10} - 10798 p^{3} T^{11} - 1588 p^{4} T^{12} + 5 p^{5} T^{13} + 58 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( ( 1 - 2 T + 105 T^{2} - 142 T^{3} + 4387 T^{4} - 142 p T^{5} + 105 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 31 | \( 1 + 6 T - 22 T^{2} + 154 T^{3} + 1815 T^{4} - 2971 T^{5} + 46757 T^{6} + 258161 T^{7} - 1367963 T^{8} + 258161 p T^{9} + 46757 p^{2} T^{10} - 2971 p^{3} T^{11} + 1815 p^{4} T^{12} + 154 p^{5} T^{13} - 22 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T - 24 T^{2} - 254 T^{3} + 3537 T^{4} + 10387 T^{5} + 13153 T^{6} - 445375 T^{7} - 2071589 T^{8} - 445375 p T^{9} + 13153 p^{2} T^{10} + 10387 p^{3} T^{11} + 3537 p^{4} T^{12} - 254 p^{5} T^{13} - 24 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( ( 1 - 9 T + 130 T^{2} - 913 T^{3} + 7715 T^{4} - 913 p T^{5} + 130 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 43 | \( ( 1 - 4 T + 88 T^{2} - 231 T^{3} + 4891 T^{4} - 231 p T^{5} + 88 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 11 T - 32 T^{2} - 917 T^{3} - 1542 T^{4} + 25678 T^{5} + 64251 T^{6} + 9500 T^{7} + 2548805 T^{8} + 9500 p T^{9} + 64251 p^{2} T^{10} + 25678 p^{3} T^{11} - 1542 p^{4} T^{12} - 917 p^{5} T^{13} - 32 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + T - 2 p T^{2} + 1003 T^{3} + 7106 T^{4} - 81262 T^{5} + 312937 T^{6} + 3384288 T^{7} - 27436599 T^{8} + 3384288 p T^{9} + 312937 p^{2} T^{10} - 81262 p^{3} T^{11} + 7106 p^{4} T^{12} + 1003 p^{5} T^{13} - 2 p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 12 T - 110 T^{2} - 1038 T^{3} + 17071 T^{4} + 89163 T^{5} - 1328213 T^{6} - 1314213 T^{7} + 101933545 T^{8} - 1314213 p T^{9} - 1328213 p^{2} T^{10} + 89163 p^{3} T^{11} + 17071 p^{4} T^{12} - 1038 p^{5} T^{13} - 110 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 21 T + 227 T^{2} + 2200 T^{3} + 15516 T^{4} + 55319 T^{5} + 113588 T^{6} - 1353730 T^{7} - 25954415 T^{8} - 1353730 p T^{9} + 113588 p^{2} T^{10} + 55319 p^{3} T^{11} + 15516 p^{4} T^{12} + 2200 p^{5} T^{13} + 227 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 3 T - 57 T^{2} + 128 T^{3} - 1826 T^{4} + 12479 T^{5} + 210616 T^{6} - 1277498 T^{7} - 3661345 T^{8} - 1277498 p T^{9} + 210616 p^{2} T^{10} + 12479 p^{3} T^{11} - 1826 p^{4} T^{12} + 128 p^{5} T^{13} - 57 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 - 11 T + 208 T^{2} - 1919 T^{3} + 19265 T^{4} - 1919 p T^{5} + 208 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( 1 - 16 T - 38 T^{2} + 1194 T^{3} + 8479 T^{4} - 97547 T^{5} - 661641 T^{6} + 4541411 T^{7} + 22286675 T^{8} + 4541411 p T^{9} - 661641 p^{2} T^{10} - 97547 p^{3} T^{11} + 8479 p^{4} T^{12} + 1194 p^{5} T^{13} - 38 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 21 T + 23 T^{2} + 1180 T^{3} + 16656 T^{4} - 227185 T^{5} - 619006 T^{6} - 2725552 T^{7} + 210226075 T^{8} - 2725552 p T^{9} - 619006 p^{2} T^{10} - 227185 p^{3} T^{11} + 16656 p^{4} T^{12} + 1180 p^{5} T^{13} + 23 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 - 4 T + 177 T^{2} - 644 T^{3} + 21517 T^{4} - 644 p T^{5} + 177 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 27 T + 236 T^{2} + 551 T^{3} + 2876 T^{4} + 45858 T^{5} - 297743 T^{6} - 2114986 T^{7} + 29395401 T^{8} - 2114986 p T^{9} - 297743 p^{2} T^{10} + 45858 p^{3} T^{11} + 2876 p^{4} T^{12} + 551 p^{5} T^{13} + 236 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( ( 1 - 6 T + 355 T^{2} - 16 p T^{3} + 50029 T^{4} - 16 p^{2} T^{5} + 355 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
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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
−4.97467168129973175537505836739, −4.89816942337439214627295560617, −4.88304133116254841435159823437, −4.87019046319939083824560049358, −4.46180931334775298891975094805, −4.36768867488444447139865524111, −4.34395857203823037259302264529, −4.27807246801149085385609155524, −4.00441012442163824843701266131, −3.94604990999983482193264610405, −3.92733476383879010713317471781, −3.45431407634302859287882390395, −3.38667664162145454717527283792, −3.10531674061861146930969354187, −3.07104739073207130633112884700, −2.68707538065687514314063399988, −2.35886412154525801782841577629, −2.30989076152133357370854176806, −1.83285075804741934406157329409, −1.71388420520114867792770960064, −1.56032278180990819691875708338, −0.972352487748841390901780782432, −0.848211961855845004252168154450, −0.50659055062977806152411770583, −0.35390093347535317312634982734, 0.35390093347535317312634982734, 0.50659055062977806152411770583, 0.848211961855845004252168154450, 0.972352487748841390901780782432, 1.56032278180990819691875708338, 1.71388420520114867792770960064, 1.83285075804741934406157329409, 2.30989076152133357370854176806, 2.35886412154525801782841577629, 2.68707538065687514314063399988, 3.07104739073207130633112884700, 3.10531674061861146930969354187, 3.38667664162145454717527283792, 3.45431407634302859287882390395, 3.92733476383879010713317471781, 3.94604990999983482193264610405, 4.00441012442163824843701266131, 4.27807246801149085385609155524, 4.34395857203823037259302264529, 4.36768867488444447139865524111, 4.46180931334775298891975094805, 4.87019046319939083824560049358, 4.88304133116254841435159823437, 4.89816942337439214627295560617, 4.97467168129973175537505836739
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.