Properties

Label 18-6018e9-1.1-c1e9-0-1
Degree $18$
Conductor $1.035\times 10^{34}$
Sign $-1$
Analytic cond. $1.36635\times 10^{15}$
Root an. cond. $6.93209$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $9$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·2-s − 9·3-s + 45·4-s + 2·5-s + 81·6-s − 5·7-s − 165·8-s + 45·9-s − 18·10-s − 11-s − 405·12-s − 4·13-s + 45·14-s − 18·15-s + 495·16-s + 9·17-s − 405·18-s − 7·19-s + 90·20-s + 45·21-s + 9·22-s − 8·23-s + 1.48e3·24-s − 18·25-s + 36·26-s − 165·27-s − 225·28-s + ⋯
L(s)  = 1  − 6.36·2-s − 5.19·3-s + 45/2·4-s + 0.894·5-s + 33.0·6-s − 1.88·7-s − 58.3·8-s + 15·9-s − 5.69·10-s − 0.301·11-s − 116.·12-s − 1.10·13-s + 12.0·14-s − 4.64·15-s + 123.·16-s + 2.18·17-s − 95.4·18-s − 1.60·19-s + 20.1·20-s + 9.81·21-s + 1.91·22-s − 1.66·23-s + 303.·24-s − 3.59·25-s + 7.06·26-s − 31.7·27-s − 42.5·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9}\)
Sign: $-1$
Analytic conductor: \(1.36635\times 10^{15}\)
Root analytic conductor: \(6.93209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(9\)
Selberg data: \((18,\ 2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\)

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 \( ( 1 + T )^{9} \)
17 \( ( 1 - T )^{9} \)
59 \( ( 1 - T )^{9} \)
good5 \( 1 - 2 T + 22 T^{2} - 47 T^{3} + 287 T^{4} - 547 T^{5} + 501 p T^{6} - 4362 T^{7} + 3271 p T^{8} - 24936 T^{9} + 3271 p^{2} T^{10} - 4362 p^{2} T^{11} + 501 p^{4} T^{12} - 547 p^{4} T^{13} + 287 p^{5} T^{14} - 47 p^{6} T^{15} + 22 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \)
7 \( 1 + 5 T + 5 p T^{2} + 145 T^{3} + 642 T^{4} + 2271 T^{5} + 7935 T^{6} + 23809 T^{7} + 72650 T^{8} + 189556 T^{9} + 72650 p T^{10} + 23809 p^{2} T^{11} + 7935 p^{3} T^{12} + 2271 p^{4} T^{13} + 642 p^{5} T^{14} + 145 p^{6} T^{15} + 5 p^{8} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 + T + 45 T^{2} + 18 T^{3} + 998 T^{4} + 169 T^{5} + 15749 T^{6} + 2895 T^{7} + 202333 T^{8} + 42538 T^{9} + 202333 p T^{10} + 2895 p^{2} T^{11} + 15749 p^{3} T^{12} + 169 p^{4} T^{13} + 998 p^{5} T^{14} + 18 p^{6} T^{15} + 45 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 + 4 T + 6 p T^{2} + 284 T^{3} + 2884 T^{4} + 9600 T^{5} + 68334 T^{6} + 205772 T^{7} + 1169167 T^{8} + 3124392 T^{9} + 1169167 p T^{10} + 205772 p^{2} T^{11} + 68334 p^{3} T^{12} + 9600 p^{4} T^{13} + 2884 p^{5} T^{14} + 284 p^{6} T^{15} + 6 p^{8} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 + 7 T + 115 T^{2} + 634 T^{3} + 6157 T^{4} + 27955 T^{5} + 208097 T^{6} + 807510 T^{7} + 5097894 T^{8} + 17385180 T^{9} + 5097894 p T^{10} + 807510 p^{2} T^{11} + 208097 p^{3} T^{12} + 27955 p^{4} T^{13} + 6157 p^{5} T^{14} + 634 p^{6} T^{15} + 115 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 + 8 T + 144 T^{2} + 962 T^{3} + 9987 T^{4} + 57911 T^{5} + 441863 T^{6} + 2236974 T^{7} + 600804 p T^{8} + 60708394 T^{9} + 600804 p^{2} T^{10} + 2236974 p^{2} T^{11} + 441863 p^{3} T^{12} + 57911 p^{4} T^{13} + 9987 p^{5} T^{14} + 962 p^{6} T^{15} + 144 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 - 6 T + 162 T^{2} - 885 T^{3} + 13643 T^{4} - 66193 T^{5} + 749731 T^{6} - 3220346 T^{7} + 29472239 T^{8} - 110106212 T^{9} + 29472239 p T^{10} - 3220346 p^{2} T^{11} + 749731 p^{3} T^{12} - 66193 p^{4} T^{13} + 13643 p^{5} T^{14} - 885 p^{6} T^{15} + 162 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 + 17 T + 227 T^{2} + 2266 T^{3} + 20673 T^{4} + 159466 T^{5} + 1165811 T^{6} + 7644830 T^{7} + 47861988 T^{8} + 8761510 p T^{9} + 47861988 p T^{10} + 7644830 p^{2} T^{11} + 1165811 p^{3} T^{12} + 159466 p^{4} T^{13} + 20673 p^{5} T^{14} + 2266 p^{6} T^{15} + 227 p^{7} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 - 2 T + 48 T^{2} - 27 T^{3} + 3551 T^{4} + 2193 T^{5} + 151893 T^{6} - 142272 T^{7} + 7156071 T^{8} + 4961232 T^{9} + 7156071 p T^{10} - 142272 p^{2} T^{11} + 151893 p^{3} T^{12} + 2193 p^{4} T^{13} + 3551 p^{5} T^{14} - 27 p^{6} T^{15} + 48 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 - 14 T + 264 T^{2} - 2862 T^{3} + 30777 T^{4} - 271721 T^{5} + 2235801 T^{6} - 16697792 T^{7} + 118205270 T^{8} - 771056792 T^{9} + 118205270 p T^{10} - 16697792 p^{2} T^{11} + 2235801 p^{3} T^{12} - 271721 p^{4} T^{13} + 30777 p^{5} T^{14} - 2862 p^{6} T^{15} + 264 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 + 27 T + 623 T^{2} + 9796 T^{3} + 134988 T^{4} + 1519071 T^{5} + 15288387 T^{6} + 132641947 T^{7} + 1040179099 T^{8} + 7162763078 T^{9} + 1040179099 p T^{10} + 132641947 p^{2} T^{11} + 15288387 p^{3} T^{12} + 1519071 p^{4} T^{13} + 134988 p^{5} T^{14} + 9796 p^{6} T^{15} + 623 p^{7} T^{16} + 27 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 18 T + 507 T^{2} + 6567 T^{3} + 2231 p T^{4} + 1053980 T^{5} + 12087715 T^{6} + 97507985 T^{7} + 870758740 T^{8} + 5697200500 T^{9} + 870758740 p T^{10} + 97507985 p^{2} T^{11} + 12087715 p^{3} T^{12} + 1053980 p^{4} T^{13} + 2231 p^{6} T^{14} + 6567 p^{6} T^{15} + 507 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 - 4 T + 385 T^{2} - 1356 T^{3} + 69288 T^{4} - 213625 T^{5} + 7667926 T^{6} - 20492876 T^{7} + 576286884 T^{8} - 1312745238 T^{9} + 576286884 p T^{10} - 20492876 p^{2} T^{11} + 7667926 p^{3} T^{12} - 213625 p^{4} T^{13} + 69288 p^{5} T^{14} - 1356 p^{6} T^{15} + 385 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 - 5 T + 267 T^{2} - 1488 T^{3} + 34062 T^{4} - 237001 T^{5} + 2959303 T^{6} - 25206437 T^{7} + 207842667 T^{8} - 1848437882 T^{9} + 207842667 p T^{10} - 25206437 p^{2} T^{11} + 2959303 p^{3} T^{12} - 237001 p^{4} T^{13} + 34062 p^{5} T^{14} - 1488 p^{6} T^{15} + 267 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 + 22 T + 569 T^{2} + 8923 T^{3} + 134769 T^{4} + 1652654 T^{5} + 18588196 T^{6} + 188369670 T^{7} + 1727722827 T^{8} + 14884634158 T^{9} + 1727722827 p T^{10} + 188369670 p^{2} T^{11} + 18588196 p^{3} T^{12} + 1652654 p^{4} T^{13} + 134769 p^{5} T^{14} + 8923 p^{6} T^{15} + 569 p^{7} T^{16} + 22 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 - 16 T + 655 T^{2} - 8487 T^{3} + 188239 T^{4} - 2007554 T^{5} + 31343121 T^{6} - 276706657 T^{7} + 3343827312 T^{8} - 24314212844 T^{9} + 3343827312 p T^{10} - 276706657 p^{2} T^{11} + 31343121 p^{3} T^{12} - 2007554 p^{4} T^{13} + 188239 p^{5} T^{14} - 8487 p^{6} T^{15} + 655 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 12 T + 362 T^{2} + 4748 T^{3} + 64151 T^{4} + 800069 T^{5} + 7986895 T^{6} + 80640506 T^{7} + 770032900 T^{8} + 6258925320 T^{9} + 770032900 p T^{10} + 80640506 p^{2} T^{11} + 7986895 p^{3} T^{12} + 800069 p^{4} T^{13} + 64151 p^{5} T^{14} + 4748 p^{6} T^{15} + 362 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 + 9 T + 475 T^{2} + 3387 T^{3} + 102791 T^{4} + 559471 T^{5} + 13701201 T^{6} + 57000880 T^{7} + 1329287220 T^{8} + 4662605594 T^{9} + 1329287220 p T^{10} + 57000880 p^{2} T^{11} + 13701201 p^{3} T^{12} + 559471 p^{4} T^{13} + 102791 p^{5} T^{14} + 3387 p^{6} T^{15} + 475 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 - 10 T + 376 T^{2} - 40 p T^{3} + 72694 T^{4} - 617937 T^{5} + 9918590 T^{6} - 79421708 T^{7} + 1028885207 T^{8} - 7529751666 T^{9} + 1028885207 p T^{10} - 79421708 p^{2} T^{11} + 9918590 p^{3} T^{12} - 617937 p^{4} T^{13} + 72694 p^{5} T^{14} - 40 p^{7} T^{15} + 376 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 - 15 T + 696 T^{2} - 8180 T^{3} + 216540 T^{4} - 2103941 T^{5} + 40815863 T^{6} - 334484591 T^{7} + 5185712987 T^{8} - 35850759452 T^{9} + 5185712987 p T^{10} - 334484591 p^{2} T^{11} + 40815863 p^{3} T^{12} - 2103941 p^{4} T^{13} + 216540 p^{5} T^{14} - 8180 p^{6} T^{15} + 696 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 + 33 T + 1196 T^{2} + 25561 T^{3} + 540434 T^{4} + 8574681 T^{5} + 131840820 T^{6} + 1644606739 T^{7} + 19785644749 T^{8} + 198510330292 T^{9} + 19785644749 p T^{10} + 1644606739 p^{2} T^{11} + 131840820 p^{3} T^{12} + 8574681 p^{4} T^{13} + 540434 p^{5} T^{14} + 25561 p^{6} T^{15} + 1196 p^{7} T^{16} + 33 p^{8} T^{17} + p^{9} T^{18} \)
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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

−3.28580462940783529476102063836, −3.18183741275496625553284574003, −3.08815077028944885375549270647, −3.00291279602380240876149393304, −2.99194905380910420985342845625, −2.99184513728796674846617657926, −2.60877288594549146136676184640, −2.52068027827879124579359858399, −2.28371167703609131777909506713, −2.22944094666046863515652284800, −2.20774787372446168443460327767, −2.19106006073692698076346359195, −2.03997354777611283213500793295, −2.01026386590786927901323001393, −1.86563163062318753829452358823, −1.78741377236415281815228907201, −1.53152370859834241047870482765, −1.41917824682969714585807009462, −1.26715072178102289804394692514, −1.20735633286687027459327613819, −1.19353565007293576667196710842, −1.18134903739465789389831651384, −1.11175096201656082744417351548, −0.841211616722436998060354851548, −0.75647768396970652079939128567, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.75647768396970652079939128567, 0.841211616722436998060354851548, 1.11175096201656082744417351548, 1.18134903739465789389831651384, 1.19353565007293576667196710842, 1.20735633286687027459327613819, 1.26715072178102289804394692514, 1.41917824682969714585807009462, 1.53152370859834241047870482765, 1.78741377236415281815228907201, 1.86563163062318753829452358823, 2.01026386590786927901323001393, 2.03997354777611283213500793295, 2.19106006073692698076346359195, 2.20774787372446168443460327767, 2.22944094666046863515652284800, 2.28371167703609131777909506713, 2.52068027827879124579359858399, 2.60877288594549146136676184640, 2.99184513728796674846617657926, 2.99194905380910420985342845625, 3.00291279602380240876149393304, 3.08815077028944885375549270647, 3.18183741275496625553284574003, 3.28580462940783529476102063836

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

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