Properties

Degree 18
Conductor $ 2^{27} \cdot 5^{9} \cdot 151^{9} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 9

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·5-s − 2·7-s − 15·9-s − 6·11-s − 9·13-s − 2·17-s − 10·19-s − 6·23-s + 45·25-s + 4·27-s − 6·29-s + 9·31-s − 18·35-s − 12·37-s − 20·41-s + 43-s − 135·45-s + 22·47-s − 44·49-s − 35·53-s − 54·55-s + 14·59-s − 22·61-s + 30·63-s − 81·65-s + 4·67-s − 22·71-s + ⋯
L(s)  = 1  + 4.02·5-s − 0.755·7-s − 5·9-s − 1.80·11-s − 2.49·13-s − 0.485·17-s − 2.29·19-s − 1.25·23-s + 9·25-s + 0.769·27-s − 1.11·29-s + 1.61·31-s − 3.04·35-s − 1.97·37-s − 3.12·41-s + 0.152·43-s − 20.1·45-s + 3.20·47-s − 6.28·49-s − 4.80·53-s − 7.28·55-s + 1.82·59-s − 2.81·61-s + 3.77·63-s − 10.0·65-s + 0.488·67-s − 2.61·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{27} \cdot 5^{9} \cdot 151^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{27} \cdot 5^{9} \cdot 151^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(18\)
\( N \)  =  \(2^{27} \cdot 5^{9} \cdot 151^{9}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6040} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  9
Selberg data  =  $(18,\ 2^{27} \cdot 5^{9} \cdot 151^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;151\}$, \(F_p\) is a polynomial of degree 18. If $p \in \{2,\;5,\;151\}$, then $F_p$ is a polynomial of degree at most 17.
$p$$F_p$
bad2 \( 1 \)
5 \( ( 1 - T )^{9} \)
151 \( ( 1 + T )^{9} \)
good3 \( 1 + 5 p T^{2} - 4 T^{3} + 118 T^{4} - 47 T^{5} + 635 T^{6} - 266 T^{7} + 2527 T^{8} - 967 T^{9} + 2527 p T^{10} - 266 p^{2} T^{11} + 635 p^{3} T^{12} - 47 p^{4} T^{13} + 118 p^{5} T^{14} - 4 p^{6} T^{15} + 5 p^{8} T^{16} + p^{9} T^{18} \)
7 \( 1 + 2 T + 48 T^{2} + 87 T^{3} + 1101 T^{4} + 1789 T^{5} + 15787 T^{6} + 22602 T^{7} + 155425 T^{8} + 191188 T^{9} + 155425 p T^{10} + 22602 p^{2} T^{11} + 15787 p^{3} T^{12} + 1789 p^{4} T^{13} + 1101 p^{5} T^{14} + 87 p^{6} T^{15} + 48 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 + 6 T + 79 T^{2} + 412 T^{3} + 2914 T^{4} + 1191 p T^{5} + 66735 T^{6} + 255668 T^{7} + 1043569 T^{8} + 3375305 T^{9} + 1043569 p T^{10} + 255668 p^{2} T^{11} + 66735 p^{3} T^{12} + 1191 p^{5} T^{13} + 2914 p^{5} T^{14} + 412 p^{6} T^{15} + 79 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 + 9 T + 99 T^{2} + 567 T^{3} + 265 p T^{4} + 13766 T^{5} + 57760 T^{6} + 170408 T^{7} + 605954 T^{8} + 1754767 T^{9} + 605954 p T^{10} + 170408 p^{2} T^{11} + 57760 p^{3} T^{12} + 13766 p^{4} T^{13} + 265 p^{6} T^{14} + 567 p^{6} T^{15} + 99 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 + 2 T + 111 T^{2} + 263 T^{3} + 5989 T^{4} + 14562 T^{5} + 205296 T^{6} + 468556 T^{7} + 4877847 T^{8} + 9765154 T^{9} + 4877847 p T^{10} + 468556 p^{2} T^{11} + 205296 p^{3} T^{12} + 14562 p^{4} T^{13} + 5989 p^{5} T^{14} + 263 p^{6} T^{15} + 111 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 + 10 T + 178 T^{2} + 1313 T^{3} + 13385 T^{4} + 78486 T^{5} + 582240 T^{6} + 2797447 T^{7} + 16391825 T^{8} + 65033423 T^{9} + 16391825 p T^{10} + 2797447 p^{2} T^{11} + 582240 p^{3} T^{12} + 78486 p^{4} T^{13} + 13385 p^{5} T^{14} + 1313 p^{6} T^{15} + 178 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 + 6 T + 105 T^{2} + 601 T^{3} + 11 p^{2} T^{4} + 31388 T^{5} + 229739 T^{6} + 1112785 T^{7} + 13124 p^{2} T^{8} + 29389157 T^{9} + 13124 p^{3} T^{10} + 1112785 p^{2} T^{11} + 229739 p^{3} T^{12} + 31388 p^{4} T^{13} + 11 p^{7} T^{14} + 601 p^{6} T^{15} + 105 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 + 6 T + 196 T^{2} + 769 T^{3} + 15986 T^{4} + 36539 T^{5} + 754774 T^{6} + 762460 T^{7} + 25518646 T^{8} + 11770109 T^{9} + 25518646 p T^{10} + 762460 p^{2} T^{11} + 754774 p^{3} T^{12} + 36539 p^{4} T^{13} + 15986 p^{5} T^{14} + 769 p^{6} T^{15} + 196 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 - 9 T + 234 T^{2} - 1732 T^{3} + 25480 T^{4} - 158678 T^{5} + 1691399 T^{6} - 8921024 T^{7} + 75364190 T^{8} - 334756311 T^{9} + 75364190 p T^{10} - 8921024 p^{2} T^{11} + 1691399 p^{3} T^{12} - 158678 p^{4} T^{13} + 25480 p^{5} T^{14} - 1732 p^{6} T^{15} + 234 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 + 12 T + 276 T^{2} + 2684 T^{3} + 35909 T^{4} + 288167 T^{5} + 2851203 T^{6} + 19150173 T^{7} + 151750959 T^{8} + 855268604 T^{9} + 151750959 p T^{10} + 19150173 p^{2} T^{11} + 2851203 p^{3} T^{12} + 288167 p^{4} T^{13} + 35909 p^{5} T^{14} + 2684 p^{6} T^{15} + 276 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 + 20 T + 408 T^{2} + 5382 T^{3} + 66752 T^{4} + 662951 T^{5} + 6182298 T^{6} + 49271005 T^{7} + 370828921 T^{8} + 2440211572 T^{9} + 370828921 p T^{10} + 49271005 p^{2} T^{11} + 6182298 p^{3} T^{12} + 662951 p^{4} T^{13} + 66752 p^{5} T^{14} + 5382 p^{6} T^{15} + 408 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 - T + 142 T^{2} - 10 T^{3} + 12667 T^{4} + 5978 T^{5} + 866152 T^{6} + 495413 T^{7} + 46190614 T^{8} + 27052868 T^{9} + 46190614 p T^{10} + 495413 p^{2} T^{11} + 866152 p^{3} T^{12} + 5978 p^{4} T^{13} + 12667 p^{5} T^{14} - 10 p^{6} T^{15} + 142 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 - 22 T + 8 p T^{2} - 4404 T^{3} + 42260 T^{4} - 310347 T^{5} + 1850858 T^{6} - 7701577 T^{7} + 22600745 T^{8} - 51499192 T^{9} + 22600745 p T^{10} - 7701577 p^{2} T^{11} + 1850858 p^{3} T^{12} - 310347 p^{4} T^{13} + 42260 p^{5} T^{14} - 4404 p^{6} T^{15} + 8 p^{8} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 + 35 T + 850 T^{2} + 14180 T^{3} + 194080 T^{4} + 2153766 T^{5} + 21034560 T^{6} + 179691819 T^{7} + 1435774373 T^{8} + 10623299708 T^{9} + 1435774373 p T^{10} + 179691819 p^{2} T^{11} + 21034560 p^{3} T^{12} + 2153766 p^{4} T^{13} + 194080 p^{5} T^{14} + 14180 p^{6} T^{15} + 850 p^{7} T^{16} + 35 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 - 14 T + 498 T^{2} - 5447 T^{3} + 108700 T^{4} - 970433 T^{5} + 14022556 T^{6} - 104378350 T^{7} + 1196918896 T^{8} - 7458610785 T^{9} + 1196918896 p T^{10} - 104378350 p^{2} T^{11} + 14022556 p^{3} T^{12} - 970433 p^{4} T^{13} + 108700 p^{5} T^{14} - 5447 p^{6} T^{15} + 498 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 + 22 T + 620 T^{2} + 9359 T^{3} + 152725 T^{4} + 1760123 T^{5} + 21079971 T^{6} + 195959042 T^{7} + 1876678719 T^{8} + 14447579032 T^{9} + 1876678719 p T^{10} + 195959042 p^{2} T^{11} + 21079971 p^{3} T^{12} + 1760123 p^{4} T^{13} + 152725 p^{5} T^{14} + 9359 p^{6} T^{15} + 620 p^{7} T^{16} + 22 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 - 4 T + 423 T^{2} - 1756 T^{3} + 85338 T^{4} - 363041 T^{5} + 10913597 T^{6} - 45397750 T^{7} + 987881219 T^{8} - 3719759597 T^{9} + 987881219 p T^{10} - 45397750 p^{2} T^{11} + 10913597 p^{3} T^{12} - 363041 p^{4} T^{13} + 85338 p^{5} T^{14} - 1756 p^{6} T^{15} + 423 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 + 22 T + 614 T^{2} + 9413 T^{3} + 156739 T^{4} + 1912189 T^{5} + 23879047 T^{6} + 241533430 T^{7} + 2432107105 T^{8} + 20614628884 T^{9} + 2432107105 p T^{10} + 241533430 p^{2} T^{11} + 23879047 p^{3} T^{12} + 1912189 p^{4} T^{13} + 156739 p^{5} T^{14} + 9413 p^{6} T^{15} + 614 p^{7} T^{16} + 22 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 34 T + 964 T^{2} + 18065 T^{3} + 297862 T^{4} + 3921795 T^{5} + 647466 p T^{6} + 488688038 T^{7} + 4783584224 T^{8} + 41674504091 T^{9} + 4783584224 p T^{10} + 488688038 p^{2} T^{11} + 647466 p^{4} T^{12} + 3921795 p^{4} T^{13} + 297862 p^{5} T^{14} + 18065 p^{6} T^{15} + 964 p^{7} T^{16} + 34 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 - 8 T + 361 T^{2} - 3121 T^{3} + 65108 T^{4} - 603177 T^{5} + 7790243 T^{6} - 75572458 T^{7} + 721910187 T^{8} - 6894365900 T^{9} + 721910187 p T^{10} - 75572458 p^{2} T^{11} + 7790243 p^{3} T^{12} - 603177 p^{4} T^{13} + 65108 p^{5} T^{14} - 3121 p^{6} T^{15} + 361 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 + 3 T + 511 T^{2} + 1653 T^{3} + 125955 T^{4} + 425658 T^{5} + 19783154 T^{6} + 65916258 T^{7} + 2208335548 T^{8} + 6686212487 T^{9} + 2208335548 p T^{10} + 65916258 p^{2} T^{11} + 19783154 p^{3} T^{12} + 425658 p^{4} T^{13} + 125955 p^{5} T^{14} + 1653 p^{6} T^{15} + 511 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 + 5 T + 494 T^{2} + 1934 T^{3} + 115913 T^{4} + 331352 T^{5} + 17489158 T^{6} + 35044073 T^{7} + 1953083326 T^{8} + 3088810680 T^{9} + 1953083326 p T^{10} + 35044073 p^{2} T^{11} + 17489158 p^{3} T^{12} + 331352 p^{4} T^{13} + 115913 p^{5} T^{14} + 1934 p^{6} T^{15} + 494 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 + 33 T + 1084 T^{2} + 21618 T^{3} + 420709 T^{4} + 6238212 T^{5} + 91075788 T^{6} + 1091216741 T^{7} + 12895551382 T^{8} + 127980390584 T^{9} + 12895551382 p T^{10} + 1091216741 p^{2} T^{11} + 91075788 p^{3} T^{12} + 6238212 p^{4} T^{13} + 420709 p^{5} T^{14} + 21618 p^{6} T^{15} + 1084 p^{7} T^{16} + 33 p^{8} T^{17} + p^{9} T^{18} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.20214529425128953804875903106, −3.17270788425848429169067250007, −3.10168230642286775285003098760, −3.08319142522861840056536955649, −2.99331352791671227635514808124, −2.97520130322760833077331801771, −2.94140182627710275229707339751, −2.56706738042353185078031330294, −2.52081227440498712392612148022, −2.43536233040450065063624322559, −2.40034872528875233770648228034, −2.36037426181602344169641644563, −2.31609462998700022563528253407, −2.31549546203952146857035629788, −2.30612612059409720093110126665, −2.07063487157153659723220764878, −1.80291184482599512488421391609, −1.70102033360078356689653026280, −1.45791412774876054039557994235, −1.42413722760490484930917916240, −1.41276329652155994689393060864, −1.39130717123968754416963849660, −1.15099172947200930871463421380, −1.14494982582733615719429592562, −1.08895334580445830326054490229, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.08895334580445830326054490229, 1.14494982582733615719429592562, 1.15099172947200930871463421380, 1.39130717123968754416963849660, 1.41276329652155994689393060864, 1.42413722760490484930917916240, 1.45791412774876054039557994235, 1.70102033360078356689653026280, 1.80291184482599512488421391609, 2.07063487157153659723220764878, 2.30612612059409720093110126665, 2.31549546203952146857035629788, 2.31609462998700022563528253407, 2.36037426181602344169641644563, 2.40034872528875233770648228034, 2.43536233040450065063624322559, 2.52081227440498712392612148022, 2.56706738042353185078031330294, 2.94140182627710275229707339751, 2.97520130322760833077331801771, 2.99331352791671227635514808124, 3.08319142522861840056536955649, 3.10168230642286775285003098760, 3.17270788425848429169067250007, 3.20214529425128953804875903106

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

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