Dirichlet series
| L(s) = 1 | + 4·2-s + 2·3-s + 6·4-s + 2·5-s + 8·6-s − 4·7-s + 6·9-s + 8·10-s + 4·11-s + 12·12-s + 18·13-s − 16·14-s + 4·15-s − 15·16-s − 6·17-s + 24·18-s + 12·20-s − 8·21-s + 16·22-s + 10·23-s + 9·25-s + 72·26-s + 16·27-s − 24·28-s − 2·29-s + 16·30-s − 52·31-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 3·4-s + 0.894·5-s + 3.26·6-s − 1.51·7-s + 2·9-s + 2.52·10-s + 1.20·11-s + 3.46·12-s + 4.99·13-s − 4.27·14-s + 1.03·15-s − 3.75·16-s − 1.45·17-s + 5.65·18-s + 2.68·20-s − 1.74·21-s + 3.41·22-s + 2.08·23-s + 9/5·25-s + 14.1·26-s + 3.07·27-s − 4.53·28-s − 0.371·29-s + 2.92·30-s − 9.33·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.22043\times 10^{6}\) |
| Root analytic conductor: | \(2.40108\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 19^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(36.76617404\) |
| \(L(\frac12)\) | \(\approx\) | \(36.76617404\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{4} \) |
| 19 | \( 1 \) | |
| good | 3 | \( 1 - 2 T - 2 T^{2} + 10 T^{4} + 2 T^{5} + 8 p T^{6} - 10 p T^{7} - 101 T^{8} - 10 p^{2} T^{9} + 8 p^{3} T^{10} + 2 p^{3} T^{11} + 10 p^{4} T^{12} - 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 2 T - p T^{2} - 22 T^{3} + 64 T^{4} + 112 T^{5} + 57 p T^{6} - 838 T^{7} - 1229 T^{8} - 838 p T^{9} + 57 p^{3} T^{10} + 112 p^{3} T^{11} + 64 p^{4} T^{12} - 22 p^{5} T^{13} - p^{7} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( ( 1 + 2 T + 22 T^{2} + 30 T^{3} + 206 T^{4} + 30 p T^{5} + 22 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( ( 1 - 2 T + 18 T^{2} - 54 T^{3} + 230 T^{4} - 54 p T^{5} + 18 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( 1 - 18 T + 163 T^{2} - 1030 T^{3} + 400 p T^{4} - 21792 T^{5} + 78869 T^{6} - 270590 T^{7} + 954099 T^{8} - 270590 p T^{9} + 78869 p^{2} T^{10} - 21792 p^{3} T^{11} + 400 p^{5} T^{12} - 1030 p^{5} T^{13} + 163 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 6 T - 23 T^{2} - 110 T^{3} + 40 p T^{4} + 724 T^{5} - 18869 T^{6} - 6720 T^{7} + 334479 T^{8} - 6720 p T^{9} - 18869 p^{2} T^{10} + 724 p^{3} T^{11} + 40 p^{5} T^{12} - 110 p^{5} T^{13} - 23 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 10 T + 58 T^{2} - 360 T^{3} + 1810 T^{4} - 8870 T^{5} + 52968 T^{6} - 275510 T^{7} + 1319419 T^{8} - 275510 p T^{9} + 52968 p^{2} T^{10} - 8870 p^{3} T^{11} + 1810 p^{4} T^{12} - 360 p^{5} T^{13} + 58 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 2 T - 81 T^{2} + 6 T^{3} + 3832 T^{4} - 3568 T^{5} - 129527 T^{6} + 52266 T^{7} + 3643051 T^{8} + 52266 p T^{9} - 129527 p^{2} T^{10} - 3568 p^{3} T^{11} + 3832 p^{4} T^{12} + 6 p^{5} T^{13} - 81 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( ( 1 + 26 T + 370 T^{2} + 3414 T^{3} + 22454 T^{4} + 3414 p T^{5} + 370 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 4 T + 129 T^{2} + 398 T^{3} + 6789 T^{4} + 398 p T^{5} + 129 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 + 12 T - 39 T^{2} - 452 T^{3} + 6316 T^{4} + 25736 T^{5} - 348341 T^{6} - 408076 T^{7} + 16207687 T^{8} - 408076 p T^{9} - 348341 p^{2} T^{10} + 25736 p^{3} T^{11} + 6316 p^{4} T^{12} - 452 p^{5} T^{13} - 39 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 10 T - 82 T^{2} + 560 T^{3} + 9370 T^{4} - 30670 T^{5} - 579992 T^{6} + 359370 T^{7} + 30861339 T^{8} + 359370 p T^{9} - 579992 p^{2} T^{10} - 30670 p^{3} T^{11} + 9370 p^{4} T^{12} + 560 p^{5} T^{13} - 82 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 12 T - 48 T^{2} + 856 T^{3} + 4198 T^{4} - 48676 T^{5} - 178016 T^{6} + 524948 T^{7} + 14112403 T^{8} + 524948 p T^{9} - 178016 p^{2} T^{10} - 48676 p^{3} T^{11} + 4198 p^{4} T^{12} + 856 p^{5} T^{13} - 48 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 8 T - 117 T^{2} + 900 T^{3} + 9620 T^{4} - 54732 T^{5} - 591211 T^{6} + 1124590 T^{7} + 35126499 T^{8} + 1124590 p T^{9} - 591211 p^{2} T^{10} - 54732 p^{3} T^{11} + 9620 p^{4} T^{12} + 900 p^{5} T^{13} - 117 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 8 T - 176 T^{2} - 736 T^{3} + 25782 T^{4} + 57968 T^{5} - 2278912 T^{6} - 1136776 T^{7} + 158799731 T^{8} - 1136776 p T^{9} - 2278912 p^{2} T^{10} + 57968 p^{3} T^{11} + 25782 p^{4} T^{12} - 736 p^{5} T^{13} - 176 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 129 T^{2} + 300 T^{3} + 120 p T^{4} - 28500 T^{5} - 219891 T^{6} + 962550 T^{7} + 9741539 T^{8} + 962550 p T^{9} - 219891 p^{2} T^{10} - 28500 p^{3} T^{11} + 120 p^{5} T^{12} + 300 p^{5} T^{13} - 129 p^{6} T^{14} + p^{8} T^{16} \) | |
| 67 | \( 1 - 10 T - 138 T^{2} + 1360 T^{3} + 15050 T^{4} - 112470 T^{5} - 1068008 T^{6} + 3134810 T^{7} + 75569499 T^{8} + 3134810 p T^{9} - 1068008 p^{2} T^{10} - 112470 p^{3} T^{11} + 15050 p^{4} T^{12} + 1360 p^{5} T^{13} - 138 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 184 T^{2} + 720 T^{3} + 17830 T^{4} - 91800 T^{5} - 964096 T^{6} + 3636720 T^{7} + 51551299 T^{8} + 3636720 p T^{9} - 964096 p^{2} T^{10} - 91800 p^{3} T^{11} + 17830 p^{4} T^{12} + 720 p^{5} T^{13} - 184 p^{6} T^{14} + p^{8} T^{16} \) | |
| 73 | \( 1 - 14 T - 147 T^{2} + 1318 T^{3} + 35528 T^{4} - 173092 T^{5} - 3581409 T^{6} + 2002724 T^{7} + 356599063 T^{8} + 2002724 p T^{9} - 3581409 p^{2} T^{10} - 173092 p^{3} T^{11} + 35528 p^{4} T^{12} + 1318 p^{5} T^{13} - 147 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 22 T + 74 T^{2} - 56 T^{3} + 21922 T^{4} - 98442 T^{5} - 2089112 T^{6} + 9976614 T^{7} + 65507451 T^{8} + 9976614 p T^{9} - 2089112 p^{2} T^{10} - 98442 p^{3} T^{11} + 21922 p^{4} T^{12} - 56 p^{5} T^{13} + 74 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 12 T + 136 T^{2} + 156 T^{3} + 1914 T^{4} + 156 p T^{5} + 136 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 16 T - 71 T^{2} - 1864 T^{3} + 8956 T^{4} + 157408 T^{5} - 757749 T^{6} - 7597372 T^{7} + 28710247 T^{8} - 7597372 p T^{9} - 757749 p^{2} T^{10} + 157408 p^{3} T^{11} + 8956 p^{4} T^{12} - 1864 p^{5} T^{13} - 71 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 28 T + 297 T^{2} - 1236 T^{3} - 4112 T^{4} + 169336 T^{5} - 2195481 T^{6} + 12285432 T^{7} - 36943377 T^{8} + 12285432 p T^{9} - 2195481 p^{2} T^{10} + 169336 p^{3} T^{11} - 4112 p^{4} T^{12} - 1236 p^{5} T^{13} + 297 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.24248629997669673172384784304, −4.22797302718528874745357430341, −4.21424318359123275379265760124, −3.98167861634377888319434181292, −3.97895459892587577048609887508, −3.90520899979445098405627888533, −3.65630169495947149407858674479, −3.48476402849334215660787269892, −3.43217127087048107854052970296, −3.37815071265262314896191331859, −3.35762292552573694488109964729, −3.22486513871458333603141495639, −3.03607563691157815910481447397, −2.90745297262201743047609901476, −2.54666346005281100741003237578, −2.46476363596100134616944574226, −2.17090470355986532715872659854, −1.82814389955525570777619100991, −1.81563354144600064224392784827, −1.66270910217665653416022880891, −1.45770859958325146078106448008, −1.43802281331933286445902747949, −1.22915890725299477097686468502, −0.58816523188898089264080613950, −0.39997121756231062620752483480, 0.39997121756231062620752483480, 0.58816523188898089264080613950, 1.22915890725299477097686468502, 1.43802281331933286445902747949, 1.45770859958325146078106448008, 1.66270910217665653416022880891, 1.81563354144600064224392784827, 1.82814389955525570777619100991, 2.17090470355986532715872659854, 2.46476363596100134616944574226, 2.54666346005281100741003237578, 2.90745297262201743047609901476, 3.03607563691157815910481447397, 3.22486513871458333603141495639, 3.35762292552573694488109964729, 3.37815071265262314896191331859, 3.43217127087048107854052970296, 3.48476402849334215660787269892, 3.65630169495947149407858674479, 3.90520899979445098405627888533, 3.97895459892587577048609887508, 3.98167861634377888319434181292, 4.21424318359123275379265760124, 4.22797302718528874745357430341, 4.24248629997669673172384784304
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.