Dirichlet series
| L(s) = 1 | + 5·2-s + 9·3-s + 9·4-s − 4·5-s + 45·6-s + 5·7-s + 8·8-s + 45·9-s − 20·10-s − 11-s + 81·12-s + 13-s + 25·14-s − 36·15-s + 15·16-s + 2·17-s + 225·18-s + 9·19-s − 36·20-s + 45·21-s − 5·22-s − 4·23-s + 72·24-s − 14·25-s + 5·26-s + 165·27-s + 45·28-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 5.19·3-s + 9/2·4-s − 1.78·5-s + 18.3·6-s + 1.88·7-s + 2.82·8-s + 15·9-s − 6.32·10-s − 0.301·11-s + 23.3·12-s + 0.277·13-s + 6.68·14-s − 9.29·15-s + 15/4·16-s + 0.485·17-s + 53.0·18-s + 2.06·19-s − 8.04·20-s + 9.81·21-s − 1.06·22-s − 0.834·23-s + 14.6·24-s − 2.79·25-s + 0.980·26-s + 31.7·27-s + 8.50·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(3^{9} \cdot 29^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.46700\times 10^{11}\) |
| Root analytic conductor: | \(4.48845\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 3^{9} \cdot 29^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(627.5357647\) |
| \(L(\frac12)\) | \(\approx\) | \(627.5357647\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{9} \) |
| 29 | \( 1 \) | |
| good | 2 | \( 1 - 5 T + p^{4} T^{2} - 43 T^{3} + 3 p^{5} T^{4} - 187 T^{5} + 335 T^{6} - 137 p^{2} T^{7} + 105 p^{3} T^{8} - 1225 T^{9} + 105 p^{4} T^{10} - 137 p^{4} T^{11} + 335 p^{3} T^{12} - 187 p^{4} T^{13} + 3 p^{10} T^{14} - 43 p^{6} T^{15} + p^{11} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + 4 T + 6 p T^{2} + 92 T^{3} + 433 T^{4} + 1113 T^{5} + 4008 T^{6} + 8849 T^{7} + 26633 T^{8} + 51301 T^{9} + 26633 p T^{10} + 8849 p^{2} T^{11} + 4008 p^{3} T^{12} + 1113 p^{4} T^{13} + 433 p^{5} T^{14} + 92 p^{6} T^{15} + 6 p^{8} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 7 | \( 1 - 5 T + 47 T^{2} - 172 T^{3} + 137 p T^{4} - 2893 T^{5} + 12304 T^{6} - 4576 p T^{7} + 113343 T^{8} - 258348 T^{9} + 113343 p T^{10} - 4576 p^{3} T^{11} + 12304 p^{3} T^{12} - 2893 p^{4} T^{13} + 137 p^{6} T^{14} - 172 p^{6} T^{15} + 47 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 + T + 49 T^{2} + 1099 T^{4} - 1345 T^{5} + 15750 T^{6} - 39934 T^{7} + 180715 T^{8} - 581228 T^{9} + 180715 p T^{10} - 39934 p^{2} T^{11} + 15750 p^{3} T^{12} - 1345 p^{4} T^{13} + 1099 p^{5} T^{14} + 49 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 - T + 30 T^{2} - 149 T^{3} + 716 T^{4} - 2996 T^{5} + 20065 T^{6} - 53796 T^{7} + 270263 T^{8} - 1030329 T^{9} + 270263 p T^{10} - 53796 p^{2} T^{11} + 20065 p^{3} T^{12} - 2996 p^{4} T^{13} + 716 p^{5} T^{14} - 149 p^{6} T^{15} + 30 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 2 T + 90 T^{2} - 154 T^{3} + 4290 T^{4} - 6733 T^{5} + 135098 T^{6} - 189810 T^{7} + 3077115 T^{8} - 3827129 T^{9} + 3077115 p T^{10} - 189810 p^{2} T^{11} + 135098 p^{3} T^{12} - 6733 p^{4} T^{13} + 4290 p^{5} T^{14} - 154 p^{6} T^{15} + 90 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 - 9 T + 92 T^{2} - 594 T^{3} + 4533 T^{4} - 24609 T^{5} + 146900 T^{6} - 687176 T^{7} + 3585054 T^{8} - 14915472 T^{9} + 3585054 p T^{10} - 687176 p^{2} T^{11} + 146900 p^{3} T^{12} - 24609 p^{4} T^{13} + 4533 p^{5} T^{14} - 594 p^{6} T^{15} + 92 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 4 T + 88 T^{2} + 262 T^{3} + 3461 T^{4} + 8620 T^{5} + 92680 T^{6} + 254144 T^{7} + 2259442 T^{8} + 6737116 T^{9} + 2259442 p T^{10} + 254144 p^{2} T^{11} + 92680 p^{3} T^{12} + 8620 p^{4} T^{13} + 3461 p^{5} T^{14} + 262 p^{6} T^{15} + 88 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 8 T + 223 T^{2} - 1452 T^{3} + 22492 T^{4} - 124237 T^{5} + 1395045 T^{6} - 6653364 T^{7} + 59826447 T^{8} - 245448446 T^{9} + 59826447 p T^{10} - 6653364 p^{2} T^{11} + 1395045 p^{3} T^{12} - 124237 p^{4} T^{13} + 22492 p^{5} T^{14} - 1452 p^{6} T^{15} + 223 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 27 T + 544 T^{2} - 7852 T^{3} + 96462 T^{4} - 991317 T^{5} + 9025832 T^{6} - 71977894 T^{7} + 517158242 T^{8} - 3306525525 T^{9} + 517158242 p T^{10} - 71977894 p^{2} T^{11} + 9025832 p^{3} T^{12} - 991317 p^{4} T^{13} + 96462 p^{5} T^{14} - 7852 p^{6} T^{15} + 544 p^{7} T^{16} - 27 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 12 T + 226 T^{2} - 2136 T^{3} + 24552 T^{4} - 199425 T^{5} + 1767092 T^{6} - 12614876 T^{7} + 94224175 T^{8} - 594125521 T^{9} + 94224175 p T^{10} - 12614876 p^{2} T^{11} + 1767092 p^{3} T^{12} - 199425 p^{4} T^{13} + 24552 p^{5} T^{14} - 2136 p^{6} T^{15} + 226 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 16 T + 349 T^{2} - 4222 T^{3} + 53464 T^{4} - 515381 T^{5} + 4875605 T^{6} - 38702340 T^{7} + 298086737 T^{8} - 1984494250 T^{9} + 298086737 p T^{10} - 38702340 p^{2} T^{11} + 4875605 p^{3} T^{12} - 515381 p^{4} T^{13} + 53464 p^{5} T^{14} - 4222 p^{6} T^{15} + 349 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 8 T + 380 T^{2} + 2565 T^{3} + 65031 T^{4} + 372743 T^{5} + 6647480 T^{6} + 32363927 T^{7} + 451278644 T^{8} + 1849968562 T^{9} + 451278644 p T^{10} + 32363927 p^{2} T^{11} + 6647480 p^{3} T^{12} + 372743 p^{4} T^{13} + 65031 p^{5} T^{14} + 2565 p^{6} T^{15} + 380 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 8 T + 273 T^{2} + 2044 T^{3} + 36211 T^{4} + 260714 T^{5} + 3153661 T^{6} + 21870087 T^{7} + 207103767 T^{8} + 1338205431 T^{9} + 207103767 p T^{10} + 21870087 p^{2} T^{11} + 3153661 p^{3} T^{12} + 260714 p^{4} T^{13} + 36211 p^{5} T^{14} + 2044 p^{6} T^{15} + 273 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 16 T + 393 T^{2} + 3630 T^{3} + 49303 T^{4} + 216050 T^{5} + 1990154 T^{6} - 10149150 T^{7} - 52334721 T^{8} - 1597546588 T^{9} - 52334721 p T^{10} - 10149150 p^{2} T^{11} + 1990154 p^{3} T^{12} + 216050 p^{4} T^{13} + 49303 p^{5} T^{14} + 3630 p^{6} T^{15} + 393 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 21 T + 593 T^{2} - 9039 T^{3} + 147416 T^{4} - 1763827 T^{5} + 20995514 T^{6} - 203924756 T^{7} + 1916422141 T^{8} - 15273534255 T^{9} + 1916422141 p T^{10} - 203924756 p^{2} T^{11} + 20995514 p^{3} T^{12} - 1763827 p^{4} T^{13} + 147416 p^{5} T^{14} - 9039 p^{6} T^{15} + 593 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 3 T + 379 T^{2} - 1384 T^{3} + 72420 T^{4} - 274006 T^{5} + 9106607 T^{6} - 32870880 T^{7} + 821083453 T^{8} - 2656323014 T^{9} + 821083453 p T^{10} - 32870880 p^{2} T^{11} + 9106607 p^{3} T^{12} - 274006 p^{4} T^{13} + 72420 p^{5} T^{14} - 1384 p^{6} T^{15} + 379 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 33 T + 817 T^{2} + 13790 T^{3} + 199530 T^{4} + 2418510 T^{5} + 27125871 T^{6} + 276226520 T^{7} + 2646386681 T^{8} + 23124430206 T^{9} + 2646386681 p T^{10} + 276226520 p^{2} T^{11} + 27125871 p^{3} T^{12} + 2418510 p^{4} T^{13} + 199530 p^{5} T^{14} + 13790 p^{6} T^{15} + 817 p^{7} T^{16} + 33 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 3 T + 369 T^{2} - 774 T^{3} + 68199 T^{4} - 91024 T^{5} + 8591261 T^{6} - 8183155 T^{7} + 814407150 T^{8} - 656168425 T^{9} + 814407150 p T^{10} - 8183155 p^{2} T^{11} + 8591261 p^{3} T^{12} - 91024 p^{4} T^{13} + 68199 p^{5} T^{14} - 774 p^{6} T^{15} + 369 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 3 T + 439 T^{2} - 408 T^{3} + 88556 T^{4} + 87082 T^{5} + 11391461 T^{6} + 27467106 T^{7} + 1100412555 T^{8} + 3084319398 T^{9} + 1100412555 p T^{10} + 27467106 p^{2} T^{11} + 11391461 p^{3} T^{12} + 87082 p^{4} T^{13} + 88556 p^{5} T^{14} - 408 p^{6} T^{15} + 439 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 13 T + 543 T^{2} - 7006 T^{3} + 147206 T^{4} - 1731416 T^{5} + 25372343 T^{6} - 259949004 T^{7} + 2994947111 T^{8} - 26073244458 T^{9} + 2994947111 p T^{10} - 259949004 p^{2} T^{11} + 25372343 p^{3} T^{12} - 1731416 p^{4} T^{13} + 147206 p^{5} T^{14} - 7006 p^{6} T^{15} + 543 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 6 T + 228 T^{2} - 913 T^{3} + 24716 T^{4} + 30276 T^{5} + 2218155 T^{6} + 9025240 T^{7} + 212571166 T^{8} + 639205597 T^{9} + 212571166 p T^{10} + 9025240 p^{2} T^{11} + 2218155 p^{3} T^{12} + 30276 p^{4} T^{13} + 24716 p^{5} T^{14} - 913 p^{6} T^{15} + 228 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 4 T + 518 T^{2} - 2545 T^{3} + 130982 T^{4} - 678282 T^{5} + 21974185 T^{6} - 107665712 T^{7} + 2747966160 T^{8} - 12054120693 T^{9} + 2747966160 p T^{10} - 107665712 p^{2} T^{11} + 21974185 p^{3} T^{12} - 678282 p^{4} T^{13} + 130982 p^{5} T^{14} - 2545 p^{6} T^{15} + 518 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.43448914141439926461674312816, −3.40230527357239877693083830086, −3.24817154899435048336131539653, −3.14682224521119589944545929300, −3.14610368748903989360520842296, −3.02201255042069963681506184636, −2.90097920997988230652501350914, −2.68188503314939395436645158944, −2.56615714909296442090323605982, −2.50901362565075512605969087619, −2.48011741041165369734746377531, −2.24123475052888654306924849141, −2.15217765472860767494570991290, −2.05643331633323160761695515692, −1.83479030865817921238710486721, −1.79528675127940766775274433485, −1.56704000455200897424971009654, −1.54636845532489597001089895428, −1.49190814597990825638163341905, −1.19977192444543495548080064216, −1.11711739091424323293047672640, −0.951545780565455695204273454301, −0.818623466191123082196960760735, −0.58088187010590330080946710141, −0.20132559531369548418947833989, 0.20132559531369548418947833989, 0.58088187010590330080946710141, 0.818623466191123082196960760735, 0.951545780565455695204273454301, 1.11711739091424323293047672640, 1.19977192444543495548080064216, 1.49190814597990825638163341905, 1.54636845532489597001089895428, 1.56704000455200897424971009654, 1.79528675127940766775274433485, 1.83479030865817921238710486721, 2.05643331633323160761695515692, 2.15217765472860767494570991290, 2.24123475052888654306924849141, 2.48011741041165369734746377531, 2.50901362565075512605969087619, 2.56615714909296442090323605982, 2.68188503314939395436645158944, 2.90097920997988230652501350914, 3.02201255042069963681506184636, 3.14610368748903989360520842296, 3.14682224521119589944545929300, 3.24817154899435048336131539653, 3.40230527357239877693083830086, 3.43448914141439926461674312816
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.