Properties

Label 18-1334e9-1.1-c1e9-0-0
Degree $18$
Conductor $1.338\times 10^{28}$
Sign $1$
Analytic cond. $1.76561\times 10^{9}$
Root an. cond. $3.26374$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·2-s − 3·3-s + 45·4-s + 5·5-s + 27·6-s + 6·7-s − 165·8-s − 2·9-s − 45·10-s − 3·11-s − 135·12-s + 13·13-s − 54·14-s − 15·15-s + 495·16-s + 2·17-s + 18·18-s + 16·19-s + 225·20-s − 18·21-s + 27·22-s − 9·23-s + 495·24-s − 117·26-s + 11·27-s + 270·28-s + 9·29-s + ⋯
L(s)  = 1  − 6.36·2-s − 1.73·3-s + 45/2·4-s + 2.23·5-s + 11.0·6-s + 2.26·7-s − 58.3·8-s − 2/3·9-s − 14.2·10-s − 0.904·11-s − 38.9·12-s + 3.60·13-s − 14.4·14-s − 3.87·15-s + 123.·16-s + 0.485·17-s + 4.24·18-s + 3.67·19-s + 50.3·20-s − 3.92·21-s + 5.75·22-s − 1.87·23-s + 101.·24-s − 22.9·26-s + 2.11·27-s + 51.0·28-s + 1.67·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 23^{9} \cdot 29^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 23^{9} \cdot 29^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2^{9} \cdot 23^{9} \cdot 29^{9}\)
Sign: $1$
Analytic conductor: \(1.76561\times 10^{9}\)
Root analytic conductor: \(3.26374\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 2^{9} \cdot 23^{9} \cdot 29^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7068932088\)
\(L(\frac12)\) \(\approx\) \(0.7068932088\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{9} \)
23 \( ( 1 + T )^{9} \)
29 \( ( 1 - T )^{9} \)
good3 \( 1 + p T + 11 T^{2} + 28 T^{3} + 25 p T^{4} + 173 T^{5} + 389 T^{6} + 28 p^{3} T^{7} + 496 p T^{8} + 2548 T^{9} + 496 p^{2} T^{10} + 28 p^{5} T^{11} + 389 p^{3} T^{12} + 173 p^{4} T^{13} + 25 p^{6} T^{14} + 28 p^{6} T^{15} + 11 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} \)
5 \( 1 - p T + p^{2} T^{2} - 86 T^{3} + 59 p T^{4} - 859 T^{5} + 473 p T^{6} - 5918 T^{7} + 14528 T^{8} - 1318 p^{2} T^{9} + 14528 p T^{10} - 5918 p^{2} T^{11} + 473 p^{4} T^{12} - 859 p^{4} T^{13} + 59 p^{6} T^{14} - 86 p^{6} T^{15} + p^{9} T^{16} - p^{9} T^{17} + p^{9} T^{18} \)
7 \( 1 - 6 T + 31 T^{2} - 94 T^{3} + 48 p T^{4} - 146 p T^{5} + 3832 T^{6} - 10778 T^{7} + 4594 p T^{8} - 77512 T^{9} + 4594 p^{2} T^{10} - 10778 p^{2} T^{11} + 3832 p^{3} T^{12} - 146 p^{5} T^{13} + 48 p^{6} T^{14} - 94 p^{6} T^{15} + 31 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 + 3 T + 37 T^{2} + 126 T^{3} + 577 T^{4} + 1719 T^{5} + 4731 T^{6} + 602 p T^{7} + 18286 T^{8} - 22508 T^{9} + 18286 p T^{10} + 602 p^{3} T^{11} + 4731 p^{3} T^{12} + 1719 p^{4} T^{13} + 577 p^{5} T^{14} + 126 p^{6} T^{15} + 37 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 - p T + 109 T^{2} - 50 p T^{3} + 3357 T^{4} - 15877 T^{5} + 71985 T^{6} - 23330 p T^{7} + 1190396 T^{8} - 4389572 T^{9} + 1190396 p T^{10} - 23330 p^{3} T^{11} + 71985 p^{3} T^{12} - 15877 p^{4} T^{13} + 3357 p^{5} T^{14} - 50 p^{7} T^{15} + 109 p^{7} T^{16} - p^{9} T^{17} + p^{9} T^{18} \)
17 \( 1 - 2 T + 97 T^{2} - 174 T^{3} + 4452 T^{4} - 6558 T^{5} + 131096 T^{6} - 150558 T^{7} + 2838822 T^{8} - 2703620 T^{9} + 2838822 p T^{10} - 150558 p^{2} T^{11} + 131096 p^{3} T^{12} - 6558 p^{4} T^{13} + 4452 p^{5} T^{14} - 174 p^{6} T^{15} + 97 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 - 16 T + 11 p T^{2} - 1952 T^{3} + 16166 T^{4} - 112012 T^{5} + 705990 T^{6} - 3901776 T^{7} + 19854828 T^{8} - 90063464 T^{9} + 19854828 p T^{10} - 3901776 p^{2} T^{11} + 705990 p^{3} T^{12} - 112012 p^{4} T^{13} + 16166 p^{5} T^{14} - 1952 p^{6} T^{15} + 11 p^{8} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 - 15 T + 313 T^{2} - 3326 T^{3} + 40089 T^{4} - 331215 T^{5} + 2914023 T^{6} - 19478454 T^{7} + 135323190 T^{8} - 741499932 T^{9} + 135323190 p T^{10} - 19478454 p^{2} T^{11} + 2914023 p^{3} T^{12} - 331215 p^{4} T^{13} + 40089 p^{5} T^{14} - 3326 p^{6} T^{15} + 313 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 - 12 T + 271 T^{2} - 2502 T^{3} + 32442 T^{4} - 248006 T^{5} + 2386294 T^{6} - 15596530 T^{7} + 121941700 T^{8} - 683631308 T^{9} + 121941700 p T^{10} - 15596530 p^{2} T^{11} + 2386294 p^{3} T^{12} - 248006 p^{4} T^{13} + 32442 p^{5} T^{14} - 2502 p^{6} T^{15} + 271 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 + 6 T + 259 T^{2} + 1478 T^{3} + 32194 T^{4} + 169138 T^{5} + 2547830 T^{6} + 11978290 T^{7} + 142380332 T^{8} + 583364320 T^{9} + 142380332 p T^{10} + 11978290 p^{2} T^{11} + 2547830 p^{3} T^{12} + 169138 p^{4} T^{13} + 32194 p^{5} T^{14} + 1478 p^{6} T^{15} + 259 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 + 3 T + 269 T^{2} + 866 T^{3} + 35229 T^{4} + 114987 T^{5} + 2940599 T^{6} + 9168522 T^{7} + 172808102 T^{8} + 480596364 T^{9} + 172808102 p T^{10} + 9168522 p^{2} T^{11} + 2940599 p^{3} T^{12} + 114987 p^{4} T^{13} + 35229 p^{5} T^{14} + 866 p^{6} T^{15} + 269 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 19 T + 391 T^{2} + 4308 T^{3} + 50121 T^{4} + 375761 T^{5} + 3196719 T^{6} + 17901896 T^{7} + 138865804 T^{8} + 732050528 T^{9} + 138865804 p T^{10} + 17901896 p^{2} T^{11} + 3196719 p^{3} T^{12} + 375761 p^{4} T^{13} + 50121 p^{5} T^{14} + 4308 p^{6} T^{15} + 391 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 - 5 T + 269 T^{2} - 1766 T^{3} + 37203 T^{4} - 268243 T^{5} + 3494573 T^{6} - 24631038 T^{7} + 243710552 T^{8} - 1547771246 T^{9} + 243710552 p T^{10} - 24631038 p^{2} T^{11} + 3494573 p^{3} T^{12} - 268243 p^{4} T^{13} + 37203 p^{5} T^{14} - 1766 p^{6} T^{15} + 269 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 - 12 T + 431 T^{2} - 4192 T^{3} + 85592 T^{4} - 699208 T^{5} + 10451456 T^{6} - 72630336 T^{7} + 869808554 T^{8} - 5133152024 T^{9} + 869808554 p T^{10} - 72630336 p^{2} T^{11} + 10451456 p^{3} T^{12} - 699208 p^{4} T^{13} + 85592 p^{5} T^{14} - 4192 p^{6} T^{15} + 431 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 - 12 T + 267 T^{2} - 2610 T^{3} + 34938 T^{4} - 287950 T^{5} + 3340710 T^{6} - 25580678 T^{7} + 261563392 T^{8} - 1830660796 T^{9} + 261563392 p T^{10} - 25580678 p^{2} T^{11} + 3340710 p^{3} T^{12} - 287950 p^{4} T^{13} + 34938 p^{5} T^{14} - 2610 p^{6} T^{15} + 267 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 + 6 T + 393 T^{2} + 1548 T^{3} + 72958 T^{4} + 200942 T^{5} + 8938670 T^{6} + 19319844 T^{7} + 803048344 T^{8} + 1473475132 T^{9} + 803048344 p T^{10} + 19319844 p^{2} T^{11} + 8938670 p^{3} T^{12} + 200942 p^{4} T^{13} + 72958 p^{5} T^{14} + 1548 p^{6} T^{15} + 393 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 - 12 T + 395 T^{2} - 3224 T^{3} + 70436 T^{4} - 457072 T^{5} + 8500132 T^{6} - 46805928 T^{7} + 771431042 T^{8} - 3709596552 T^{9} + 771431042 p T^{10} - 46805928 p^{2} T^{11} + 8500132 p^{3} T^{12} - 457072 p^{4} T^{13} + 70436 p^{5} T^{14} - 3224 p^{6} T^{15} + 395 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 303 T^{2} - 956 T^{3} + 46706 T^{4} - 279674 T^{5} + 5132534 T^{6} - 38612476 T^{7} + 458502672 T^{8} - 3402421596 T^{9} + 458502672 p T^{10} - 38612476 p^{2} T^{11} + 5132534 p^{3} T^{12} - 279674 p^{4} T^{13} + 46706 p^{5} T^{14} - 956 p^{6} T^{15} + 303 p^{7} T^{16} + p^{9} T^{18} \)
79 \( 1 - 29 T + 577 T^{2} - 9460 T^{3} + 139007 T^{4} - 1793165 T^{5} + 21307287 T^{6} - 230758496 T^{7} + 2302111126 T^{8} - 21130106270 T^{9} + 2302111126 p T^{10} - 230758496 p^{2} T^{11} + 21307287 p^{3} T^{12} - 1793165 p^{4} T^{13} + 139007 p^{5} T^{14} - 9460 p^{6} T^{15} + 577 p^{7} T^{16} - 29 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 - 24 T + 475 T^{2} - 4804 T^{3} + 49662 T^{4} - 306068 T^{5} + 4128962 T^{6} - 38111812 T^{7} + 576500562 T^{8} - 4653067240 T^{9} + 576500562 p T^{10} - 38111812 p^{2} T^{11} + 4128962 p^{3} T^{12} - 306068 p^{4} T^{13} + 49662 p^{5} T^{14} - 4804 p^{6} T^{15} + 475 p^{7} T^{16} - 24 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 + 2 T + 645 T^{2} + 840 T^{3} + 193020 T^{4} + 156166 T^{5} + 35495688 T^{6} + 18041240 T^{7} + 4450721690 T^{8} + 1670626772 T^{9} + 4450721690 p T^{10} + 18041240 p^{2} T^{11} + 35495688 p^{3} T^{12} + 156166 p^{4} T^{13} + 193020 p^{5} T^{14} + 840 p^{6} T^{15} + 645 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 - 12 T + 601 T^{2} - 7052 T^{3} + 171102 T^{4} - 1915356 T^{5} + 30899318 T^{6} - 320809996 T^{7} + 3986066034 T^{8} - 36942397616 T^{9} + 3986066034 p T^{10} - 320809996 p^{2} T^{11} + 30899318 p^{3} T^{12} - 1915356 p^{4} T^{13} + 171102 p^{5} T^{14} - 7052 p^{6} T^{15} + 601 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \)
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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

−3.40970456709120643040606032091, −3.35098233686215685089388782659, −3.32509732493313467973532903195, −3.30921023006251045254538193903, −3.22494184019183549246381555996, −3.21307249614394447937254989670, −3.01137774704123708665246071223, −2.46324198653980051704074156127, −2.42721940127987544203852861645, −2.39235582799229503076181717904, −2.28506029421221763419676384026, −2.15798894444054046436560488654, −2.12401939565870333318809052285, −2.10196634266879139529369326590, −1.85638528208037249683311629996, −1.58462708458992996973490060953, −1.56709946558897551870475970795, −1.25283925883067969225724054143, −1.18772605783289451063603258964, −1.09545359167212322957761179958, −1.01687236777822328548813323506, −0.969965419987869845313287018566, −0.52755455114326347778427261107, −0.45034989007500523889645252373, −0.39521974607632473808026692609, 0.39521974607632473808026692609, 0.45034989007500523889645252373, 0.52755455114326347778427261107, 0.969965419987869845313287018566, 1.01687236777822328548813323506, 1.09545359167212322957761179958, 1.18772605783289451063603258964, 1.25283925883067969225724054143, 1.56709946558897551870475970795, 1.58462708458992996973490060953, 1.85638528208037249683311629996, 2.10196634266879139529369326590, 2.12401939565870333318809052285, 2.15798894444054046436560488654, 2.28506029421221763419676384026, 2.39235582799229503076181717904, 2.42721940127987544203852861645, 2.46324198653980051704074156127, 3.01137774704123708665246071223, 3.21307249614394447937254989670, 3.22494184019183549246381555996, 3.30921023006251045254538193903, 3.32509732493313467973532903195, 3.35098233686215685089388782659, 3.40970456709120643040606032091

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.