Properties

Label 18-6042e9-1.1-c1e9-0-3
Degree $18$
Conductor $1.073\times 10^{34}$
Sign $-1$
Analytic cond. $1.41618\times 10^{15}$
Root an. cond. $6.94590$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $9$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·2-s − 9·3-s + 45·4-s − 5·5-s − 81·6-s − 5·7-s + 165·8-s + 45·9-s − 45·10-s + 3·11-s − 405·12-s − 4·13-s − 45·14-s + 45·15-s + 495·16-s − 16·17-s + 405·18-s − 9·19-s − 225·20-s + 45·21-s + 27·22-s − 1.48e3·24-s − 6·25-s − 36·26-s − 165·27-s − 225·28-s − 2·29-s + ⋯
L(s)  = 1  + 6.36·2-s − 5.19·3-s + 45/2·4-s − 2.23·5-s − 33.0·6-s − 1.88·7-s + 58.3·8-s + 15·9-s − 14.2·10-s + 0.904·11-s − 116.·12-s − 1.10·13-s − 12.0·14-s + 11.6·15-s + 123.·16-s − 3.88·17-s + 95.4·18-s − 2.06·19-s − 50.3·20-s + 9.81·21-s + 5.75·22-s − 303.·24-s − 6/5·25-s − 7.06·26-s − 31.7·27-s − 42.5·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9}\)
Sign: $-1$
Analytic conductor: \(1.41618\times 10^{15}\)
Root analytic conductor: \(6.94590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(9\)
Selberg data: \((18,\ 2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{9} \)
3 \( ( 1 + T )^{9} \)
19 \( ( 1 + T )^{9} \)
53 \( ( 1 - T )^{9} \)
good5 \( 1 + p T + 31 T^{2} + 121 T^{3} + 481 T^{4} + 1519 T^{5} + 4687 T^{6} + 12508 T^{7} + 32108 T^{8} + 73198 T^{9} + 32108 p T^{10} + 12508 p^{2} T^{11} + 4687 p^{3} T^{12} + 1519 p^{4} T^{13} + 481 p^{5} T^{14} + 121 p^{6} T^{15} + 31 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} \)
7 \( 1 + 5 T + 51 T^{2} + 216 T^{3} + 1228 T^{4} + 626 p T^{5} + 18240 T^{6} + 54713 T^{7} + 183025 T^{8} + 460744 T^{9} + 183025 p T^{10} + 54713 p^{2} T^{11} + 18240 p^{3} T^{12} + 626 p^{5} T^{13} + 1228 p^{5} T^{14} + 216 p^{6} T^{15} + 51 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 - 3 T + 61 T^{2} - 161 T^{3} + 1854 T^{4} - 4418 T^{5} + 3386 p T^{6} - 80327 T^{7} + 49584 p T^{8} - 1039798 T^{9} + 49584 p^{2} T^{10} - 80327 p^{2} T^{11} + 3386 p^{4} T^{12} - 4418 p^{4} T^{13} + 1854 p^{5} T^{14} - 161 p^{6} T^{15} + 61 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 + 4 T + 53 T^{2} + 114 T^{3} + 1286 T^{4} + 1624 T^{5} + 23272 T^{6} + 20609 T^{7} + 371106 T^{8} + 309494 T^{9} + 371106 p T^{10} + 20609 p^{2} T^{11} + 23272 p^{3} T^{12} + 1624 p^{4} T^{13} + 1286 p^{5} T^{14} + 114 p^{6} T^{15} + 53 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 + 16 T + 169 T^{2} + 1248 T^{3} + 7778 T^{4} + 41056 T^{5} + 200696 T^{6} + 53253 p T^{7} + 3989280 T^{8} + 16648366 T^{9} + 3989280 p T^{10} + 53253 p^{3} T^{11} + 200696 p^{3} T^{12} + 41056 p^{4} T^{13} + 7778 p^{5} T^{14} + 1248 p^{6} T^{15} + 169 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 + 2 p T^{2} - 209 T^{3} + 1532 T^{4} - 7812 T^{5} + 67326 T^{6} - 234375 T^{7} + 1590143 T^{8} - 7529704 T^{9} + 1590143 p T^{10} - 234375 p^{2} T^{11} + 67326 p^{3} T^{12} - 7812 p^{4} T^{13} + 1532 p^{5} T^{14} - 209 p^{6} T^{15} + 2 p^{8} T^{16} + p^{9} T^{18} \)
29 \( 1 + 2 T + 148 T^{2} + 452 T^{3} + 11158 T^{4} + 40174 T^{5} + 579852 T^{6} + 2038992 T^{7} + 22500190 T^{8} + 69939618 T^{9} + 22500190 p T^{10} + 2038992 p^{2} T^{11} + 579852 p^{3} T^{12} + 40174 p^{4} T^{13} + 11158 p^{5} T^{14} + 452 p^{6} T^{15} + 148 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 + 10 T + 163 T^{2} + 39 p T^{3} + 10191 T^{4} + 53643 T^{5} + 300441 T^{6} + 924503 T^{7} + 4643316 T^{8} + 7780950 T^{9} + 4643316 p T^{10} + 924503 p^{2} T^{11} + 300441 p^{3} T^{12} + 53643 p^{4} T^{13} + 10191 p^{5} T^{14} + 39 p^{7} T^{15} + 163 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 + 14 T + 277 T^{2} + 2852 T^{3} + 32426 T^{4} + 260024 T^{5} + 2213082 T^{6} + 14674513 T^{7} + 105044438 T^{8} + 610422506 T^{9} + 105044438 p T^{10} + 14674513 p^{2} T^{11} + 2213082 p^{3} T^{12} + 260024 p^{4} T^{13} + 32426 p^{5} T^{14} + 2852 p^{6} T^{15} + 277 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 + 21 T + 365 T^{2} + 4359 T^{3} + 47586 T^{4} + 427874 T^{5} + 3668915 T^{6} + 27556397 T^{7} + 200665973 T^{8} + 1303747130 T^{9} + 200665973 p T^{10} + 27556397 p^{2} T^{11} + 3668915 p^{3} T^{12} + 427874 p^{4} T^{13} + 47586 p^{5} T^{14} + 4359 p^{6} T^{15} + 365 p^{7} T^{16} + 21 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 + 4 T + 178 T^{2} + 247 T^{3} + 15626 T^{4} + 5768 T^{5} + 1051956 T^{6} - 40543 T^{7} + 54205619 T^{8} - 17058136 T^{9} + 54205619 p T^{10} - 40543 p^{2} T^{11} + 1051956 p^{3} T^{12} + 5768 p^{4} T^{13} + 15626 p^{5} T^{14} + 247 p^{6} T^{15} + 178 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 8 T + 318 T^{2} + 1871 T^{3} + 44588 T^{4} + 196804 T^{5} + 3814163 T^{6} + 13087044 T^{7} + 232493014 T^{8} + 671128762 T^{9} + 232493014 p T^{10} + 13087044 p^{2} T^{11} + 3814163 p^{3} T^{12} + 196804 p^{4} T^{13} + 44588 p^{5} T^{14} + 1871 p^{6} T^{15} + 318 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 + 12 T + 446 T^{2} + 4422 T^{3} + 91338 T^{4} + 766258 T^{5} + 11414970 T^{6} + 81497346 T^{7} + 962393800 T^{8} + 5807367456 T^{9} + 962393800 p T^{10} + 81497346 p^{2} T^{11} + 11414970 p^{3} T^{12} + 766258 p^{4} T^{13} + 91338 p^{5} T^{14} + 4422 p^{6} T^{15} + 446 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 + 13 T + 259 T^{2} + 2270 T^{3} + 31025 T^{4} + 219131 T^{5} + 2605882 T^{6} + 15936701 T^{7} + 176122951 T^{8} + 981151638 T^{9} + 176122951 p T^{10} + 15936701 p^{2} T^{11} + 2605882 p^{3} T^{12} + 219131 p^{4} T^{13} + 31025 p^{5} T^{14} + 2270 p^{6} T^{15} + 259 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 - 22 T + 518 T^{2} - 7105 T^{3} + 101100 T^{4} - 1091212 T^{5} + 12308690 T^{6} - 115764263 T^{7} + 1113420091 T^{8} - 9097321308 T^{9} + 1113420091 p T^{10} - 115764263 p^{2} T^{11} + 12308690 p^{3} T^{12} - 1091212 p^{4} T^{13} + 101100 p^{5} T^{14} - 7105 p^{6} T^{15} + 518 p^{7} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 + 13 T + 526 T^{2} + 5338 T^{3} + 125478 T^{4} + 1057116 T^{5} + 18409668 T^{6} + 131338126 T^{7} + 1842242075 T^{8} + 11146727134 T^{9} + 1842242075 p T^{10} + 131338126 p^{2} T^{11} + 18409668 p^{3} T^{12} + 1057116 p^{4} T^{13} + 125478 p^{5} T^{14} + 5338 p^{6} T^{15} + 526 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 17 T + 497 T^{2} + 5433 T^{3} + 95518 T^{4} + 784314 T^{5} + 11404822 T^{6} + 80224631 T^{7} + 1057689490 T^{8} + 6617644778 T^{9} + 1057689490 p T^{10} + 80224631 p^{2} T^{11} + 11404822 p^{3} T^{12} + 784314 p^{4} T^{13} + 95518 p^{5} T^{14} + 5433 p^{6} T^{15} + 497 p^{7} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 + 205 T^{2} + 183 T^{3} + 24756 T^{4} + 12939 T^{5} + 2209335 T^{6} - 1734015 T^{7} + 166196755 T^{8} - 192830726 T^{9} + 166196755 p T^{10} - 1734015 p^{2} T^{11} + 2209335 p^{3} T^{12} + 12939 p^{4} T^{13} + 24756 p^{5} T^{14} + 183 p^{6} T^{15} + 205 p^{7} T^{16} + p^{9} T^{18} \)
83 \( 1 + 24 T + 707 T^{2} + 11464 T^{3} + 202314 T^{4} + 2564064 T^{5} + 34411744 T^{6} + 362681133 T^{7} + 4001007548 T^{8} + 35709483102 T^{9} + 4001007548 p T^{10} + 362681133 p^{2} T^{11} + 34411744 p^{3} T^{12} + 2564064 p^{4} T^{13} + 202314 p^{5} T^{14} + 11464 p^{6} T^{15} + 707 p^{7} T^{16} + 24 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 + 23 T + 406 T^{2} + 6208 T^{3} + 84232 T^{4} + 1114330 T^{5} + 13298214 T^{6} + 145868408 T^{7} + 1507642811 T^{8} + 14437010926 T^{9} + 1507642811 p T^{10} + 145868408 p^{2} T^{11} + 13298214 p^{3} T^{12} + 1114330 p^{4} T^{13} + 84232 p^{5} T^{14} + 6208 p^{6} T^{15} + 406 p^{7} T^{16} + 23 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 + 29 T + 661 T^{2} + 9715 T^{3} + 127350 T^{4} + 1251218 T^{5} + 11966062 T^{6} + 89327821 T^{7} + 799936422 T^{8} + 6332583586 T^{9} + 799936422 p T^{10} + 89327821 p^{2} T^{11} + 11966062 p^{3} T^{12} + 1251218 p^{4} T^{13} + 127350 p^{5} T^{14} + 9715 p^{6} T^{15} + 661 p^{7} T^{16} + 29 p^{8} T^{17} + p^{9} T^{18} \)
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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

−3.50927262989491616854300258922, −3.45153043860315239159494360650, −3.44868807898364939444911290158, −3.42411970317636698105628111503, −3.40597116183412493210231928665, −3.25311133706435193273410803397, −3.09358967077827886081647335824, −2.73146614422173436803347218048, −2.55205431550947851623001798224, −2.52277440492817133418734942649, −2.44227573511795468576887751662, −2.43843565284178894540927951901, −2.37199390326904127717543343782, −2.33038987468106981681908105876, −2.14722082416973317471123828010, −2.12384199318458520680332772520, −1.79363429059189686039444752035, −1.68237932739504729314317416924, −1.47241520020634380467919398517, −1.45768648609975416125077409102, −1.40753148701059788973705474667, −1.39358657579455314248074862350, −1.24195684447995353490767402546, −1.18802453202495941393689572359, −1.13513312629889467536712105779, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.13513312629889467536712105779, 1.18802453202495941393689572359, 1.24195684447995353490767402546, 1.39358657579455314248074862350, 1.40753148701059788973705474667, 1.45768648609975416125077409102, 1.47241520020634380467919398517, 1.68237932739504729314317416924, 1.79363429059189686039444752035, 2.12384199318458520680332772520, 2.14722082416973317471123828010, 2.33038987468106981681908105876, 2.37199390326904127717543343782, 2.43843565284178894540927951901, 2.44227573511795468576887751662, 2.52277440492817133418734942649, 2.55205431550947851623001798224, 2.73146614422173436803347218048, 3.09358967077827886081647335824, 3.25311133706435193273410803397, 3.40597116183412493210231928665, 3.42411970317636698105628111503, 3.44868807898364939444911290158, 3.45153043860315239159494360650, 3.50927262989491616854300258922

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

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