Properties

Label 18-149e9-1.1-c1e9-0-0
Degree $18$
Conductor $3.620\times 10^{19}$
Sign $1$
Analytic cond. $4.77716$
Root an. cond. $1.09076$
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  − 2-s + 6·3-s − 2·4-s − 5-s − 6·6-s + 3·7-s + 8-s + 9·9-s + 10-s + 5·11-s − 12·12-s + 7·13-s − 3·14-s − 6·15-s − 5·17-s − 9·18-s + 30·19-s + 2·20-s + 18·21-s − 5·22-s − 4·23-s + 6·24-s − 19·25-s − 7·26-s − 19·27-s − 6·28-s − 16·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 3.46·3-s − 4-s − 0.447·5-s − 2.44·6-s + 1.13·7-s + 0.353·8-s + 3·9-s + 0.316·10-s + 1.50·11-s − 3.46·12-s + 1.94·13-s − 0.801·14-s − 1.54·15-s − 1.21·17-s − 2.12·18-s + 6.88·19-s + 0.447·20-s + 3.92·21-s − 1.06·22-s − 0.834·23-s + 1.22·24-s − 3.79·25-s − 1.37·26-s − 3.65·27-s − 1.13·28-s − 2.97·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(149^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(149^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(149^{9}\)
Sign: $1$
Analytic conductor: \(4.77716\)
Root analytic conductor: \(1.09076\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 149^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.537855677\)
\(L(\frac12)\) \(\approx\) \(2.537855677\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( ( 1 - T )^{9} \)
good2 \( 1 + T + 3 T^{2} + p^{2} T^{3} + 9 T^{4} + p^{4} T^{5} + 25 T^{6} + 9 p^{2} T^{7} + 31 p T^{8} + 87 T^{9} + 31 p^{2} T^{10} + 9 p^{4} T^{11} + 25 p^{3} T^{12} + p^{8} T^{13} + 9 p^{5} T^{14} + p^{8} T^{15} + 3 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \)
3 \( 1 - 2 p T + p^{3} T^{2} - 89 T^{3} + 257 T^{4} - 647 T^{5} + 1498 T^{6} - 1051 p T^{7} + 686 p^{2} T^{8} - 1231 p^{2} T^{9} + 686 p^{3} T^{10} - 1051 p^{3} T^{11} + 1498 p^{3} T^{12} - 647 p^{4} T^{13} + 257 p^{5} T^{14} - 89 p^{6} T^{15} + p^{10} T^{16} - 2 p^{9} T^{17} + p^{9} T^{18} \)
5 \( 1 + T + 4 p T^{2} + 36 T^{3} + 227 T^{4} + 497 T^{5} + 1896 T^{6} + 829 p T^{7} + 12332 T^{8} + 24129 T^{9} + 12332 p T^{10} + 829 p^{3} T^{11} + 1896 p^{3} T^{12} + 497 p^{4} T^{13} + 227 p^{5} T^{14} + 36 p^{6} T^{15} + 4 p^{8} T^{16} + p^{8} T^{17} + p^{9} T^{18} \)
7 \( 1 - 3 T + 29 T^{2} - 51 T^{3} + 306 T^{4} - 118 T^{5} + 1250 T^{6} + 3779 T^{7} - 828 T^{8} + 43826 T^{9} - 828 p T^{10} + 3779 p^{2} T^{11} + 1250 p^{3} T^{12} - 118 p^{4} T^{13} + 306 p^{5} T^{14} - 51 p^{6} T^{15} + 29 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 - 5 T + 6 p T^{2} - 238 T^{3} + 171 p T^{4} - 5111 T^{5} + 32578 T^{6} - 69365 T^{7} + 416830 T^{8} - 775333 T^{9} + 416830 p T^{10} - 69365 p^{2} T^{11} + 32578 p^{3} T^{12} - 5111 p^{4} T^{13} + 171 p^{6} T^{14} - 238 p^{6} T^{15} + 6 p^{8} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 - 7 T + 89 T^{2} - 451 T^{3} + 3384 T^{4} - 1042 p T^{5} + 78368 T^{6} - 264453 T^{7} + 1308042 T^{8} - 3879134 T^{9} + 1308042 p T^{10} - 264453 p^{2} T^{11} + 78368 p^{3} T^{12} - 1042 p^{5} T^{13} + 3384 p^{5} T^{14} - 451 p^{6} T^{15} + 89 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 + 5 T + 78 T^{2} + 338 T^{3} + 3051 T^{4} + 13047 T^{5} + 83652 T^{6} + 347423 T^{7} + 1752068 T^{8} + 6781247 T^{9} + 1752068 p T^{10} + 347423 p^{2} T^{11} + 83652 p^{3} T^{12} + 13047 p^{4} T^{13} + 3051 p^{5} T^{14} + 338 p^{6} T^{15} + 78 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 - 30 T + 508 T^{2} - 6093 T^{3} + 57049 T^{4} - 439642 T^{5} + 2886345 T^{6} - 16550571 T^{7} + 84399343 T^{8} - 387118832 T^{9} + 84399343 p T^{10} - 16550571 p^{2} T^{11} + 2886345 p^{3} T^{12} - 439642 p^{4} T^{13} + 57049 p^{5} T^{14} - 6093 p^{6} T^{15} + 508 p^{7} T^{16} - 30 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 + 4 T + 119 T^{2} + 601 T^{3} + 7253 T^{4} + 39337 T^{5} + 306920 T^{6} + 1541807 T^{7} + 9621424 T^{8} + 41684241 T^{9} + 9621424 p T^{10} + 1541807 p^{2} T^{11} + 306920 p^{3} T^{12} + 39337 p^{4} T^{13} + 7253 p^{5} T^{14} + 601 p^{6} T^{15} + 119 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 + 16 T + 313 T^{2} + 3315 T^{3} + 37599 T^{4} + 10337 p T^{5} + 2492220 T^{6} + 549309 p T^{7} + 105797854 T^{8} + 558788491 T^{9} + 105797854 p T^{10} + 549309 p^{3} T^{11} + 2492220 p^{3} T^{12} + 10337 p^{5} T^{13} + 37599 p^{5} T^{14} + 3315 p^{6} T^{15} + 313 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 - 22 T + 370 T^{2} - 4465 T^{3} + 1509 p T^{4} - 411006 T^{5} + 3264831 T^{6} - 22866351 T^{7} + 147390741 T^{8} - 852996952 T^{9} + 147390741 p T^{10} - 22866351 p^{2} T^{11} + 3264831 p^{3} T^{12} - 411006 p^{4} T^{13} + 1509 p^{6} T^{14} - 4465 p^{6} T^{15} + 370 p^{7} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 + 7 T + 191 T^{2} + 1244 T^{3} + 18295 T^{4} + 103479 T^{5} + 1154592 T^{6} + 5701419 T^{7} + 53890220 T^{8} + 238289945 T^{9} + 53890220 p T^{10} + 5701419 p^{2} T^{11} + 1154592 p^{3} T^{12} + 103479 p^{4} T^{13} + 18295 p^{5} T^{14} + 1244 p^{6} T^{15} + 191 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 - 6 T + 184 T^{2} - 961 T^{3} + 17121 T^{4} - 74846 T^{5} + 1092043 T^{6} - 4034815 T^{7} + 53571571 T^{8} - 177744280 T^{9} + 53571571 p T^{10} - 4034815 p^{2} T^{11} + 1092043 p^{3} T^{12} - 74846 p^{4} T^{13} + 17121 p^{5} T^{14} - 961 p^{6} T^{15} + 184 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 - 4 T + 185 T^{2} - 953 T^{3} + 16343 T^{4} - 88077 T^{5} + 996174 T^{6} - 4635451 T^{7} + 49538972 T^{8} - 197020923 T^{9} + 49538972 p T^{10} - 4635451 p^{2} T^{11} + 996174 p^{3} T^{12} - 88077 p^{4} T^{13} + 16343 p^{5} T^{14} - 953 p^{6} T^{15} + 185 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 6 T + 150 T^{2} + 663 T^{3} + 11507 T^{4} + 56438 T^{5} + 765199 T^{6} + 3934089 T^{7} + 41369313 T^{8} + 200799944 T^{9} + 41369313 p T^{10} + 3934089 p^{2} T^{11} + 765199 p^{3} T^{12} + 56438 p^{4} T^{13} + 11507 p^{5} T^{14} + 663 p^{6} T^{15} + 150 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 + 2 T + 307 T^{2} - 233 T^{3} + 42067 T^{4} - 127321 T^{5} + 3757184 T^{6} - 16009801 T^{7} + 255637270 T^{8} - 1082732131 T^{9} + 255637270 p T^{10} - 16009801 p^{2} T^{11} + 3757184 p^{3} T^{12} - 127321 p^{4} T^{13} + 42067 p^{5} T^{14} - 233 p^{6} T^{15} + 307 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 - 43 T + 1242 T^{2} - 26006 T^{3} + 441983 T^{4} - 6253163 T^{5} + 76022816 T^{6} - 802237733 T^{7} + 7439787750 T^{8} - 60769624779 T^{9} + 7439787750 p T^{10} - 802237733 p^{2} T^{11} + 76022816 p^{3} T^{12} - 6253163 p^{4} T^{13} + 441983 p^{5} T^{14} - 26006 p^{6} T^{15} + 1242 p^{7} T^{16} - 43 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 - T + 358 T^{2} - 242 T^{3} + 63555 T^{4} - 24819 T^{5} + 7343060 T^{6} - 1705335 T^{7} + 606371540 T^{8} - 104486291 T^{9} + 606371540 p T^{10} - 1705335 p^{2} T^{11} + 7343060 p^{3} T^{12} - 24819 p^{4} T^{13} + 63555 p^{5} T^{14} - 242 p^{6} T^{15} + 358 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 - 33 T + 765 T^{2} - 12835 T^{3} + 178378 T^{4} - 2099230 T^{5} + 22059354 T^{6} - 210166085 T^{7} + 1872870168 T^{8} - 15714227650 T^{9} + 1872870168 p T^{10} - 210166085 p^{2} T^{11} + 22059354 p^{3} T^{12} - 2099230 p^{4} T^{13} + 178378 p^{5} T^{14} - 12835 p^{6} T^{15} + 765 p^{7} T^{16} - 33 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 - 15 T + 584 T^{2} - 7332 T^{3} + 152485 T^{4} - 1629093 T^{5} + 23706530 T^{6} - 215923795 T^{7} + 2440490328 T^{8} - 18722716807 T^{9} + 2440490328 p T^{10} - 215923795 p^{2} T^{11} + 23706530 p^{3} T^{12} - 1629093 p^{4} T^{13} + 152485 p^{5} T^{14} - 7332 p^{6} T^{15} + 584 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 11 T + 512 T^{2} + 4322 T^{3} + 118455 T^{4} + 810481 T^{5} + 16955652 T^{6} + 97125549 T^{7} + 1692260512 T^{8} + 8276868229 T^{9} + 1692260512 p T^{10} + 97125549 p^{2} T^{11} + 16955652 p^{3} T^{12} + 810481 p^{4} T^{13} + 118455 p^{5} T^{14} + 4322 p^{6} T^{15} + 512 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 - T + 162 T^{2} - 459 T^{3} + 27851 T^{4} - 40734 T^{5} + 3095823 T^{6} - 6391317 T^{7} + 299027381 T^{8} - 407317506 T^{9} + 299027381 p T^{10} - 6391317 p^{2} T^{11} + 3095823 p^{3} T^{12} - 40734 p^{4} T^{13} + 27851 p^{5} T^{14} - 459 p^{6} T^{15} + 162 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 + 4 T + 363 T^{2} + 891 T^{3} + 67113 T^{4} + 110131 T^{5} + 8974278 T^{6} + 12905465 T^{7} + 944992970 T^{8} + 1266425935 T^{9} + 944992970 p T^{10} + 12905465 p^{2} T^{11} + 8974278 p^{3} T^{12} + 110131 p^{4} T^{13} + 67113 p^{5} T^{14} + 891 p^{6} T^{15} + 363 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 + 19 T + 764 T^{2} + 12061 T^{3} + 264441 T^{4} + 3464006 T^{5} + 53998379 T^{6} + 587663363 T^{7} + 7150188775 T^{8} + 64348112686 T^{9} + 7150188775 p T^{10} + 587663363 p^{2} T^{11} + 53998379 p^{3} T^{12} + 3464006 p^{4} T^{13} + 264441 p^{5} T^{14} + 12061 p^{6} T^{15} + 764 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 + T + 411 T^{2} + 855 T^{3} + 75762 T^{4} + 319078 T^{5} + 8117198 T^{6} + 65072265 T^{7} + 644771140 T^{8} + 8006395570 T^{9} + 644771140 p T^{10} + 65072265 p^{2} T^{11} + 8117198 p^{3} T^{12} + 319078 p^{4} T^{13} + 75762 p^{5} T^{14} + 855 p^{6} T^{15} + 411 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \)
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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

−5.40145521112950869943737404335, −5.28395059909668945775915579126, −5.26299004028852380059841902230, −5.24371186392165324650429315453, −5.12656048832169010749327539910, −5.12184302437284254906344434441, −4.54583406848384331012347932305, −4.35750975276117889559370098573, −4.13600241439185324358004936455, −3.92604854862271023476549349182, −3.87257680585912608416081449467, −3.70796332719033368946537794199, −3.64712847963526473298885731700, −3.62667725569529418950905313400, −3.46378699876149319479581571165, −2.97063612682992012344910720465, −2.92257086407155040199688626178, −2.92085090668140378732424238600, −2.65909357733922534396551561506, −2.34180953249202073045309440597, −2.20493743850519582913171122691, −1.72940039954678950438220148878, −1.67130341469649222738482468721, −1.27961562707660271932926578039, −0.823408360967730622042714189513, 0.823408360967730622042714189513, 1.27961562707660271932926578039, 1.67130341469649222738482468721, 1.72940039954678950438220148878, 2.20493743850519582913171122691, 2.34180953249202073045309440597, 2.65909357733922534396551561506, 2.92085090668140378732424238600, 2.92257086407155040199688626178, 2.97063612682992012344910720465, 3.46378699876149319479581571165, 3.62667725569529418950905313400, 3.64712847963526473298885731700, 3.70796332719033368946537794199, 3.87257680585912608416081449467, 3.92604854862271023476549349182, 4.13600241439185324358004936455, 4.35750975276117889559370098573, 4.54583406848384331012347932305, 5.12184302437284254906344434441, 5.12656048832169010749327539910, 5.24371186392165324650429315453, 5.26299004028852380059841902230, 5.28395059909668945775915579126, 5.40145521112950869943737404335

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

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