Properties

Degree 18
Conductor $ 2^{9} \cdot 3^{27} \cdot 149^{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  + 9·2-s + 45·4-s − 4·5-s − 4·7-s + 165·8-s − 36·10-s − 4·11-s − 8·13-s − 36·14-s + 495·16-s − 17-s − 10·19-s − 180·20-s − 36·22-s − 8·23-s − 16·25-s − 72·26-s − 180·28-s − 4·29-s − 17·31-s + 1.28e3·32-s − 9·34-s + 16·35-s − 11·37-s − 90·38-s − 660·40-s − 16·43-s + ⋯
L(s)  = 1  + 6.36·2-s + 45/2·4-s − 1.78·5-s − 1.51·7-s + 58.3·8-s − 11.3·10-s − 1.20·11-s − 2.21·13-s − 9.62·14-s + 123.·16-s − 0.242·17-s − 2.29·19-s − 40.2·20-s − 7.67·22-s − 1.66·23-s − 3.19·25-s − 14.1·26-s − 34.0·28-s − 0.742·29-s − 3.05·31-s + 227.·32-s − 1.54·34-s + 2.70·35-s − 1.80·37-s − 14.5·38-s − 104.·40-s − 2.43·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{27} \cdot 149^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{27} \cdot 149^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(18\)
\( N \)  =  \(2^{9} \cdot 3^{27} \cdot 149^{9}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8046} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  9
Selberg data  =  $(18,\ 2^{9} \cdot 3^{27} \cdot 149^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;149\}$, \(F_p(T)\) is a polynomial of degree 18. If $p \in \{2,\;3,\;149\}$, then $F_p(T)$ is a polynomial of degree at most 17.
$p$$F_p(T)$
bad2 \( ( 1 - T )^{9} \)
3 \( 1 \)
149 \( ( 1 + T )^{9} \)
good5 \( 1 + 4 T + 32 T^{2} + 104 T^{3} + 488 T^{4} + 272 p T^{5} + 4759 T^{6} + 2294 p T^{7} + 32617 T^{8} + 67718 T^{9} + 32617 p T^{10} + 2294 p^{3} T^{11} + 4759 p^{3} T^{12} + 272 p^{5} T^{13} + 488 p^{5} T^{14} + 104 p^{6} T^{15} + 32 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
7 \( 1 + 4 T + 6 p T^{2} + 141 T^{3} + 794 T^{4} + 2265 T^{5} + 9183 T^{6} + 3301 p T^{7} + 77577 T^{8} + 179045 T^{9} + 77577 p T^{10} + 3301 p^{3} T^{11} + 9183 p^{3} T^{12} + 2265 p^{4} T^{13} + 794 p^{5} T^{14} + 141 p^{6} T^{15} + 6 p^{8} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
11 \( 1 + 4 T + 68 T^{2} + 284 T^{3} + 2307 T^{4} + 9201 T^{5} + 50752 T^{6} + 180614 T^{7} + 781453 T^{8} + 2385860 T^{9} + 781453 p T^{10} + 180614 p^{2} T^{11} + 50752 p^{3} T^{12} + 9201 p^{4} T^{13} + 2307 p^{5} T^{14} + 284 p^{6} T^{15} + 68 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
13 \( 1 + 8 T + 113 T^{2} + 684 T^{3} + 5484 T^{4} + 26690 T^{5} + 155755 T^{6} + 628402 T^{7} + 2920764 T^{8} + 9865738 T^{9} + 2920764 p T^{10} + 628402 p^{2} T^{11} + 155755 p^{3} T^{12} + 26690 p^{4} T^{13} + 5484 p^{5} T^{14} + 684 p^{6} T^{15} + 113 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 + T + 93 T^{2} + 19 T^{3} + 4351 T^{4} - 1013 T^{5} + 135092 T^{6} - 59947 T^{7} + 3044452 T^{8} - 1420770 T^{9} + 3044452 p T^{10} - 59947 p^{2} T^{11} + 135092 p^{3} T^{12} - 1013 p^{4} T^{13} + 4351 p^{5} T^{14} + 19 p^{6} T^{15} + 93 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \)
19 \( 1 + 10 T + 154 T^{2} + 1087 T^{3} + 9648 T^{4} + 54358 T^{5} + 361242 T^{6} + 1729463 T^{7} + 9424332 T^{8} + 38786652 T^{9} + 9424332 p T^{10} + 1729463 p^{2} T^{11} + 361242 p^{3} T^{12} + 54358 p^{4} T^{13} + 9648 p^{5} T^{14} + 1087 p^{6} T^{15} + 154 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 + 8 T + 177 T^{2} + 1015 T^{3} + 12770 T^{4} + 54339 T^{5} + 524816 T^{6} + 1728021 T^{7} + 14993524 T^{8} + 42219669 T^{9} + 14993524 p T^{10} + 1728021 p^{2} T^{11} + 524816 p^{3} T^{12} + 54339 p^{4} T^{13} + 12770 p^{5} T^{14} + 1015 p^{6} T^{15} + 177 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 + 4 T + 140 T^{2} + 228 T^{3} + 8814 T^{4} + 1232 T^{5} + 406708 T^{6} - 76890 T^{7} + 15758047 T^{8} - 990911 T^{9} + 15758047 p T^{10} - 76890 p^{2} T^{11} + 406708 p^{3} T^{12} + 1232 p^{4} T^{13} + 8814 p^{5} T^{14} + 228 p^{6} T^{15} + 140 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 + 17 T + 294 T^{2} + 3288 T^{3} + 35058 T^{4} + 299254 T^{5} + 2415185 T^{6} + 538327 p T^{7} + 109028017 T^{8} + 624140757 T^{9} + 109028017 p T^{10} + 538327 p^{3} T^{11} + 2415185 p^{3} T^{12} + 299254 p^{4} T^{13} + 35058 p^{5} T^{14} + 3288 p^{6} T^{15} + 294 p^{7} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 + 11 T + 232 T^{2} + 1750 T^{3} + 21288 T^{4} + 115727 T^{5} + 1094681 T^{6} + 4383700 T^{7} + 40610911 T^{8} + 143060408 T^{9} + 40610911 p T^{10} + 4383700 p^{2} T^{11} + 1094681 p^{3} T^{12} + 115727 p^{4} T^{13} + 21288 p^{5} T^{14} + 1750 p^{6} T^{15} + 232 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 + 219 T^{2} - 187 T^{3} + 24091 T^{4} - 33174 T^{5} + 1783023 T^{6} - 2683062 T^{7} + 2370443 p T^{8} - 134191801 T^{9} + 2370443 p^{2} T^{10} - 2683062 p^{2} T^{11} + 1783023 p^{3} T^{12} - 33174 p^{4} T^{13} + 24091 p^{5} T^{14} - 187 p^{6} T^{15} + 219 p^{7} T^{16} + p^{9} T^{18} \)
43 \( 1 + 16 T + 299 T^{2} + 2907 T^{3} + 31033 T^{4} + 203017 T^{5} + 1564829 T^{6} + 6998481 T^{7} + 50274985 T^{8} + 203934498 T^{9} + 50274985 p T^{10} + 6998481 p^{2} T^{11} + 1564829 p^{3} T^{12} + 203017 p^{4} T^{13} + 31033 p^{5} T^{14} + 2907 p^{6} T^{15} + 299 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 7 T + 233 T^{2} + 1150 T^{3} + 24491 T^{4} + 90119 T^{5} + 1667498 T^{6} + 4659039 T^{7} + 87538662 T^{8} + 208700788 T^{9} + 87538662 p T^{10} + 4659039 p^{2} T^{11} + 1667498 p^{3} T^{12} + 90119 p^{4} T^{13} + 24491 p^{5} T^{14} + 1150 p^{6} T^{15} + 233 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 + 12 T + 251 T^{2} + 2123 T^{3} + 28110 T^{4} + 185955 T^{5} + 38810 p T^{6} + 11875405 T^{7} + 123270464 T^{8} + 660472201 T^{9} + 123270464 p T^{10} + 11875405 p^{2} T^{11} + 38810 p^{4} T^{12} + 185955 p^{4} T^{13} + 28110 p^{5} T^{14} + 2123 p^{6} T^{15} + 251 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 + 6 T + 286 T^{2} + 1739 T^{3} + 43076 T^{4} + 249230 T^{5} + 4459908 T^{6} + 23858923 T^{7} + 343771964 T^{8} + 1647933976 T^{9} + 343771964 p T^{10} + 23858923 p^{2} T^{11} + 4459908 p^{3} T^{12} + 249230 p^{4} T^{13} + 43076 p^{5} T^{14} + 1739 p^{6} T^{15} + 286 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 + 13 T + 484 T^{2} + 5224 T^{3} + 107984 T^{4} + 976962 T^{5} + 14476899 T^{6} + 110318977 T^{7} + 1285534845 T^{8} + 8199304700 T^{9} + 1285534845 p T^{10} + 110318977 p^{2} T^{11} + 14476899 p^{3} T^{12} + 976962 p^{4} T^{13} + 107984 p^{5} T^{14} + 5224 p^{6} T^{15} + 484 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 + 14 T + 404 T^{2} + 4765 T^{3} + 1140 p T^{4} + 793798 T^{5} + 9308190 T^{6} + 86206581 T^{7} + 824351120 T^{8} + 6730075334 T^{9} + 824351120 p T^{10} + 86206581 p^{2} T^{11} + 9308190 p^{3} T^{12} + 793798 p^{4} T^{13} + 1140 p^{6} T^{14} + 4765 p^{6} T^{15} + 404 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 + 30 T + 690 T^{2} + 12048 T^{3} + 178293 T^{4} + 2311374 T^{5} + 26830140 T^{6} + 282338634 T^{7} + 2711601060 T^{8} + 23874389457 T^{9} + 2711601060 p T^{10} + 282338634 p^{2} T^{11} + 26830140 p^{3} T^{12} + 2311374 p^{4} T^{13} + 178293 p^{5} T^{14} + 12048 p^{6} T^{15} + 690 p^{7} T^{16} + 30 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 12 T + 377 T^{2} + 4929 T^{3} + 78527 T^{4} + 927103 T^{5} + 11214517 T^{6} + 111105985 T^{7} + 1134999931 T^{8} + 9491974800 T^{9} + 1134999931 p T^{10} + 111105985 p^{2} T^{11} + 11214517 p^{3} T^{12} + 927103 p^{4} T^{13} + 78527 p^{5} T^{14} + 4929 p^{6} T^{15} + 377 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 + 35 T + 974 T^{2} + 19318 T^{3} + 329779 T^{4} + 4739591 T^{5} + 60921526 T^{6} + 691590981 T^{7} + 7153951615 T^{8} + 66417803924 T^{9} + 7153951615 p T^{10} + 691590981 p^{2} T^{11} + 60921526 p^{3} T^{12} + 4739591 p^{4} T^{13} + 329779 p^{5} T^{14} + 19318 p^{6} T^{15} + 974 p^{7} T^{16} + 35 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 + 5 T + 419 T^{2} + 2365 T^{3} + 86890 T^{4} + 462426 T^{5} + 11913060 T^{6} + 56786404 T^{7} + 1216257923 T^{8} + 5289477148 T^{9} + 1216257923 p T^{10} + 56786404 p^{2} T^{11} + 11913060 p^{3} T^{12} + 462426 p^{4} T^{13} + 86890 p^{5} T^{14} + 2365 p^{6} T^{15} + 419 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \)
89 \( 1 + 23 T + 858 T^{2} + 15157 T^{3} + 314746 T^{4} + 4452798 T^{5} + 66088684 T^{6} + 764102797 T^{7} + 8849517519 T^{8} + 83953359151 T^{9} + 8849517519 p T^{10} + 764102797 p^{2} T^{11} + 66088684 p^{3} T^{12} + 4452798 p^{4} T^{13} + 314746 p^{5} T^{14} + 15157 p^{6} T^{15} + 858 p^{7} T^{16} + 23 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 + 21 T + 846 T^{2} + 14729 T^{3} + 324320 T^{4} + 4641521 T^{5} + 73149346 T^{6} + 8870019 p T^{7} + 10636874310 T^{8} + 102739172088 T^{9} + 10636874310 p T^{10} + 8870019 p^{3} T^{11} + 73149346 p^{3} T^{12} + 4641521 p^{4} T^{13} + 324320 p^{5} T^{14} + 14729 p^{6} T^{15} + 846 p^{7} T^{16} + 21 p^{8} T^{17} + p^{9} T^{18} \)
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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

−3.26782205360561070293066218534, −3.24472516454757243947694563681, −3.22689379766815340820490517518, −3.18047784287463671012766992736, −3.17022276823231056170974240188, −3.05816765419673316044942202459, −2.90933134584048003152153981378, −2.61441870260798722420396290280, −2.60272799542593071084289332055, −2.47877497715247089093561173255, −2.41960644768587308284598281818, −2.41680377049469375870152238854, −2.33872161378786591706111269798, −2.19290692189994986296235755555, −2.16123111618924675425393508721, −2.15006443792993029971209988832, −1.81814527069704662786472965717, −1.59372808277006851229008052603, −1.58352882615974936117732675492, −1.50336970869346827549236182273, −1.47890566537837333370781901050, −1.41582879122772396150512583547, −1.41062944589022578218294908970, −1.21678975731420977251942521132, −1.21275825187232792931119436913, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.21275825187232792931119436913, 1.21678975731420977251942521132, 1.41062944589022578218294908970, 1.41582879122772396150512583547, 1.47890566537837333370781901050, 1.50336970869346827549236182273, 1.58352882615974936117732675492, 1.59372808277006851229008052603, 1.81814527069704662786472965717, 2.15006443792993029971209988832, 2.16123111618924675425393508721, 2.19290692189994986296235755555, 2.33872161378786591706111269798, 2.41680377049469375870152238854, 2.41960644768587308284598281818, 2.47877497715247089093561173255, 2.60272799542593071084289332055, 2.61441870260798722420396290280, 2.90933134584048003152153981378, 3.05816765419673316044942202459, 3.17022276823231056170974240188, 3.18047784287463671012766992736, 3.22689379766815340820490517518, 3.24472516454757243947694563681, 3.26782205360561070293066218534

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

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