Properties

Degree 18
Conductor $ 2^{9} \cdot 3^{9} \cdot 11^{9} \cdot 61^{9} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·2-s + 9·3-s + 45·4-s + 8·5-s + 81·6-s + 9·7-s + 165·8-s + 45·9-s + 72·10-s + 9·11-s + 405·12-s + 8·13-s + 81·14-s + 72·15-s + 495·16-s + 17-s + 405·18-s + 5·19-s + 360·20-s + 81·21-s + 81·22-s − 23-s + 1.48e3·24-s + 21·25-s + 72·26-s + 165·27-s + 405·28-s + ⋯
L(s)  = 1  + 6.36·2-s + 5.19·3-s + 45/2·4-s + 3.57·5-s + 33.0·6-s + 3.40·7-s + 58.3·8-s + 15·9-s + 22.7·10-s + 2.71·11-s + 116.·12-s + 2.21·13-s + 21.6·14-s + 18.5·15-s + 123.·16-s + 0.242·17-s + 95.4·18-s + 1.14·19-s + 80.4·20-s + 17.6·21-s + 17.2·22-s − 0.208·23-s + 303.·24-s + 21/5·25-s + 14.1·26-s + 31.7·27-s + 76.5·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 11^{9} \cdot 61^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 11^{9} \cdot 61^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(18\)
\( N \)  =  \(2^{9} \cdot 3^{9} \cdot 11^{9} \cdot 61^{9}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4026} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(18,\ 2^{9} \cdot 3^{9} \cdot 11^{9} \cdot 61^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )$
$L(1)$  $\approx$  $780657.8137$
$L(\frac12)$  $\approx$  $780657.8137$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11,\;61\}$, \(F_p\) is a polynomial of degree 18. If $p \in \{2,\;3,\;11,\;61\}$, then $F_p$ is a polynomial of degree at most 17.
$p$$F_p$
bad2 \( ( 1 - T )^{9} \)
3 \( ( 1 - T )^{9} \)
11 \( ( 1 - T )^{9} \)
61 \( ( 1 + T )^{9} \)
good5 \( 1 - 8 T + 43 T^{2} - 173 T^{3} + 596 T^{4} - 1821 T^{5} + 5181 T^{6} - 13837 T^{7} + 34611 T^{8} - 16082 p T^{9} + 34611 p T^{10} - 13837 p^{2} T^{11} + 5181 p^{3} T^{12} - 1821 p^{4} T^{13} + 596 p^{5} T^{14} - 173 p^{6} T^{15} + 43 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \)
7 \( 1 - 9 T + 57 T^{2} - 254 T^{3} + 1016 T^{4} - 3511 T^{5} + 11672 T^{6} - 34914 T^{7} + 14480 p T^{8} - 270112 T^{9} + 14480 p^{2} T^{10} - 34914 p^{2} T^{11} + 11672 p^{3} T^{12} - 3511 p^{4} T^{13} + 1016 p^{5} T^{14} - 254 p^{6} T^{15} + 57 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 - 8 T + 6 p T^{2} - 432 T^{3} + 2434 T^{4} - 10355 T^{5} + 265 p^{2} T^{6} - 159867 T^{7} + 628598 T^{8} - 2099508 T^{9} + 628598 p T^{10} - 159867 p^{2} T^{11} + 265 p^{5} T^{12} - 10355 p^{4} T^{13} + 2434 p^{5} T^{14} - 432 p^{6} T^{15} + 6 p^{8} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 - T + 90 T^{2} + 36 T^{3} + 3566 T^{4} + 7439 T^{5} + 84611 T^{6} + 338722 T^{7} + 1507336 T^{8} + 7734304 T^{9} + 1507336 p T^{10} + 338722 p^{2} T^{11} + 84611 p^{3} T^{12} + 7439 p^{4} T^{13} + 3566 p^{5} T^{14} + 36 p^{6} T^{15} + 90 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 - 5 T + 71 T^{2} - 346 T^{3} + 2884 T^{4} - 789 p T^{5} + 90776 T^{6} - 444626 T^{7} + 2160454 T^{8} - 9570800 T^{9} + 2160454 p T^{10} - 444626 p^{2} T^{11} + 90776 p^{3} T^{12} - 789 p^{5} T^{13} + 2884 p^{5} T^{14} - 346 p^{6} T^{15} + 71 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 + T + 76 T^{2} + 222 T^{3} + 3176 T^{4} + 592 p T^{5} + 111494 T^{6} + 19838 p T^{7} + 3308121 T^{8} + 11436478 T^{9} + 3308121 p T^{10} + 19838 p^{3} T^{11} + 111494 p^{3} T^{12} + 592 p^{5} T^{13} + 3176 p^{5} T^{14} + 222 p^{6} T^{15} + 76 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 + 14 T + 245 T^{2} + 2481 T^{3} + 903 p T^{4} + 207825 T^{5} + 1655733 T^{6} + 10754403 T^{7} + 69370466 T^{8} + 12946274 p T^{9} + 69370466 p T^{10} + 10754403 p^{2} T^{11} + 1655733 p^{3} T^{12} + 207825 p^{4} T^{13} + 903 p^{6} T^{14} + 2481 p^{6} T^{15} + 245 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 - 25 T + 462 T^{2} - 6008 T^{3} + 65765 T^{4} - 19275 p T^{5} + 4812095 T^{6} - 34069348 T^{7} + 219678951 T^{8} - 1275714300 T^{9} + 219678951 p T^{10} - 34069348 p^{2} T^{11} + 4812095 p^{3} T^{12} - 19275 p^{5} T^{13} + 65765 p^{5} T^{14} - 6008 p^{6} T^{15} + 462 p^{7} T^{16} - 25 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 - 16 T + 276 T^{2} - 2638 T^{3} + 23833 T^{4} - 133961 T^{5} + 638446 T^{6} - 297518 T^{7} - 11273464 T^{8} + 137497450 T^{9} - 11273464 p T^{10} - 297518 p^{2} T^{11} + 638446 p^{3} T^{12} - 133961 p^{4} T^{13} + 23833 p^{5} T^{14} - 2638 p^{6} T^{15} + 276 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 - 5 T + 139 T^{2} - 704 T^{3} + 12993 T^{4} - 63019 T^{5} + 880519 T^{6} - 3866896 T^{7} + 45589660 T^{8} - 183888032 T^{9} + 45589660 p T^{10} - 3866896 p^{2} T^{11} + 880519 p^{3} T^{12} - 63019 p^{4} T^{13} + 12993 p^{5} T^{14} - 704 p^{6} T^{15} + 139 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 - 5 T + 34 T^{2} - 310 T^{3} + 4140 T^{4} - 22044 T^{5} + 167878 T^{6} - 1100770 T^{7} + 9159643 T^{8} - 42641774 T^{9} + 9159643 p T^{10} - 1100770 p^{2} T^{11} + 167878 p^{3} T^{12} - 22044 p^{4} T^{13} + 4140 p^{5} T^{14} - 310 p^{6} T^{15} + 34 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 - 8 T + 239 T^{2} - 1310 T^{3} + 24469 T^{4} - 102001 T^{5} + 1708823 T^{6} - 6577734 T^{7} + 100330640 T^{8} - 361481814 T^{9} + 100330640 p T^{10} - 6577734 p^{2} T^{11} + 1708823 p^{3} T^{12} - 102001 p^{4} T^{13} + 24469 p^{5} T^{14} - 1310 p^{6} T^{15} + 239 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 - T + 201 T^{2} - 76 T^{3} + 21676 T^{4} - 4509 T^{5} + 1738900 T^{6} + 33540 T^{7} + 111004234 T^{8} + 21886428 T^{9} + 111004234 p T^{10} + 33540 p^{2} T^{11} + 1738900 p^{3} T^{12} - 4509 p^{4} T^{13} + 21676 p^{5} T^{14} - 76 p^{6} T^{15} + 201 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 - 4 T + 326 T^{2} - 690 T^{3} + 48930 T^{4} - 14953 T^{5} + 4598993 T^{6} + 5391123 T^{7} + 322789418 T^{8} + 553826680 T^{9} + 322789418 p T^{10} + 5391123 p^{2} T^{11} + 4598993 p^{3} T^{12} - 14953 p^{4} T^{13} + 48930 p^{5} T^{14} - 690 p^{6} T^{15} + 326 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 + 4 T + 441 T^{2} + 2236 T^{3} + 90751 T^{4} + 530455 T^{5} + 11694625 T^{6} + 71082416 T^{7} + 1060815410 T^{8} + 5946422690 T^{9} + 1060815410 p T^{10} + 71082416 p^{2} T^{11} + 11694625 p^{3} T^{12} + 530455 p^{4} T^{13} + 90751 p^{5} T^{14} + 2236 p^{6} T^{15} + 441 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 - 20 T + 504 T^{2} - 7263 T^{3} + 113232 T^{4} - 1332428 T^{5} + 15920938 T^{6} - 158399153 T^{7} + 1557834753 T^{8} - 13249712128 T^{9} + 1557834753 p T^{10} - 158399153 p^{2} T^{11} + 15920938 p^{3} T^{12} - 1332428 p^{4} T^{13} + 113232 p^{5} T^{14} - 7263 p^{6} T^{15} + 504 p^{7} T^{16} - 20 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 - 15 T + 458 T^{2} - 5686 T^{3} + 104740 T^{4} - 1099794 T^{5} + 15228338 T^{6} - 137002922 T^{7} + 1543536703 T^{8} - 11861010366 T^{9} + 1543536703 p T^{10} - 137002922 p^{2} T^{11} + 15228338 p^{3} T^{12} - 1099794 p^{4} T^{13} + 104740 p^{5} T^{14} - 5686 p^{6} T^{15} + 458 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 + 2 T + 285 T^{2} + 432 T^{3} + 38669 T^{4} + 20555 T^{5} + 3899611 T^{6} + 509508 T^{7} + 349149658 T^{8} + 94271678 T^{9} + 349149658 p T^{10} + 509508 p^{2} T^{11} + 3899611 p^{3} T^{12} + 20555 p^{4} T^{13} + 38669 p^{5} T^{14} + 432 p^{6} T^{15} + 285 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 - 21 T + 536 T^{2} - 8472 T^{3} + 135858 T^{4} - 1728950 T^{5} + 21746156 T^{6} - 232136504 T^{7} + 2452072257 T^{8} - 22440930506 T^{9} + 2452072257 p T^{10} - 232136504 p^{2} T^{11} + 21746156 p^{3} T^{12} - 1728950 p^{4} T^{13} + 135858 p^{5} T^{14} - 8472 p^{6} T^{15} + 536 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 - 10 T + 593 T^{2} - 5039 T^{3} + 167476 T^{4} - 1218985 T^{5} + 332951 p T^{6} - 185487881 T^{7} + 3642987927 T^{8} - 19566775250 T^{9} + 3642987927 p T^{10} - 185487881 p^{2} T^{11} + 332951 p^{4} T^{12} - 1218985 p^{4} T^{13} + 167476 p^{5} T^{14} - 5039 p^{6} T^{15} + 593 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 - 3 T + 242 T^{2} - 419 T^{3} + 45376 T^{4} - 11678 T^{5} + 5944026 T^{6} + 1244419 T^{7} + 663596572 T^{8} + 666908176 T^{9} + 663596572 p T^{10} + 1244419 p^{2} T^{11} + 5944026 p^{3} T^{12} - 11678 p^{4} T^{13} + 45376 p^{5} T^{14} - 419 p^{6} T^{15} + 242 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \)
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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

−3.23661165563173644875372829011, −2.99323337233510096108083679797, −2.98942932835518122744083189083, −2.92208674387423020236062806379, −2.90249919544708770058440216347, −2.78390027195843155630066161382, −2.62887761785164225856460799792, −2.35341892986409573582347032842, −2.32044253992205971673530168289, −2.26894028276143987109713952680, −2.13745591278548257553611793246, −2.12162360611696354053190322571, −2.04687185457531416279492577786, −2.04495510877173696622475967223, −2.02561692451421263865853388123, −1.92683784393446946269315239326, −1.57614262881653932199022393951, −1.35201793970015482301976063819, −1.32736676760791531120739594514, −1.30295381433341018942114956649, −1.26139052765123108472750141313, −1.11249619646599480697799202311, −1.07464930759986874719253602787, −1.03890674944333899940515016160, −0.795915076819202164711666993086, 0.795915076819202164711666993086, 1.03890674944333899940515016160, 1.07464930759986874719253602787, 1.11249619646599480697799202311, 1.26139052765123108472750141313, 1.30295381433341018942114956649, 1.32736676760791531120739594514, 1.35201793970015482301976063819, 1.57614262881653932199022393951, 1.92683784393446946269315239326, 2.02561692451421263865853388123, 2.04495510877173696622475967223, 2.04687185457531416279492577786, 2.12162360611696354053190322571, 2.13745591278548257553611793246, 2.26894028276143987109713952680, 2.32044253992205971673530168289, 2.35341892986409573582347032842, 2.62887761785164225856460799792, 2.78390027195843155630066161382, 2.90249919544708770058440216347, 2.92208674387423020236062806379, 2.98942932835518122744083189083, 2.99323337233510096108083679797, 3.23661165563173644875372829011

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

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