Properties

Degree 14
Conductor $ 3^{14} \cdot 7^{7} \cdot 127^{7} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 7

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·4-s + 8·5-s + 7·7-s − 7·8-s + 16·10-s + 3·11-s − 23·13-s + 14·14-s + 5·16-s − 3·17-s − 9·19-s − 24·20-s + 6·22-s − 12·23-s + 16·25-s − 46·26-s − 21·28-s + 9·29-s − 33·31-s + 5·32-s − 6·34-s + 56·35-s − 33·37-s − 18·38-s − 56·40-s + 3·41-s + ⋯
L(s)  = 1  + 1.41·2-s − 3/2·4-s + 3.57·5-s + 2.64·7-s − 2.47·8-s + 5.05·10-s + 0.904·11-s − 6.37·13-s + 3.74·14-s + 5/4·16-s − 0.727·17-s − 2.06·19-s − 5.36·20-s + 1.27·22-s − 2.50·23-s + 16/5·25-s − 9.02·26-s − 3.96·28-s + 1.67·29-s − 5.92·31-s + 0.883·32-s − 1.02·34-s + 9.46·35-s − 5.42·37-s − 2.91·38-s − 8.85·40-s + 0.468·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 7^{7} \cdot 127^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 7^{7} \cdot 127^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(14\)
\( N \)  =  \(3^{14} \cdot 7^{7} \cdot 127^{7}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8001} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(7\)
Selberg data  =  \((14,\ 3^{14} \cdot 7^{7} \cdot 127^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;127\}$,\(F_p(T)\) is a polynomial of degree 14. If $p \in \{3,\;7,\;127\}$, then $F_p(T)$ is a polynomial of degree at most 13.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( ( 1 - T )^{7} \)
127 \( ( 1 + T )^{7} \)
good2 \( 1 - p T + 7 T^{2} - 13 T^{3} + 7 p^{2} T^{4} - 41 T^{5} + 73 T^{6} - 47 p T^{7} + 73 p T^{8} - 41 p^{2} T^{9} + 7 p^{5} T^{10} - 13 p^{4} T^{11} + 7 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \)
5 \( 1 - 8 T + 48 T^{2} - 198 T^{3} + 701 T^{4} - 2022 T^{5} + 5344 T^{6} - 2463 p T^{7} + 5344 p T^{8} - 2022 p^{2} T^{9} + 701 p^{3} T^{10} - 198 p^{4} T^{11} + 48 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 - 3 T + 31 T^{2} - 93 T^{3} + 656 T^{4} - 1715 T^{5} + 9199 T^{6} - 22080 T^{7} + 9199 p T^{8} - 1715 p^{2} T^{9} + 656 p^{3} T^{10} - 93 p^{4} T^{11} + 31 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 + 23 T + 289 T^{2} + 2532 T^{3} + 1316 p T^{4} + 93828 T^{5} + 430561 T^{6} + 1679255 T^{7} + 430561 p T^{8} + 93828 p^{2} T^{9} + 1316 p^{4} T^{10} + 2532 p^{4} T^{11} + 289 p^{5} T^{12} + 23 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 3 T + 50 T^{2} + 146 T^{3} + 1373 T^{4} + 4645 T^{5} + 30241 T^{6} + 99122 T^{7} + 30241 p T^{8} + 4645 p^{2} T^{9} + 1373 p^{3} T^{10} + 146 p^{4} T^{11} + 50 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 9 T + 127 T^{2} + 828 T^{3} + 6701 T^{4} + 34195 T^{5} + 201852 T^{6} + 825304 T^{7} + 201852 p T^{8} + 34195 p^{2} T^{9} + 6701 p^{3} T^{10} + 828 p^{4} T^{11} + 127 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 + 12 T + 150 T^{2} + 1067 T^{3} + 8328 T^{4} + 46995 T^{5} + 292164 T^{6} + 1356265 T^{7} + 292164 p T^{8} + 46995 p^{2} T^{9} + 8328 p^{3} T^{10} + 1067 p^{4} T^{11} + 150 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 - 9 T + 107 T^{2} - 652 T^{3} + 4764 T^{4} - 22810 T^{5} + 134487 T^{6} - 612145 T^{7} + 134487 p T^{8} - 22810 p^{2} T^{9} + 4764 p^{3} T^{10} - 652 p^{4} T^{11} + 107 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 + 33 T + 645 T^{2} + 8883 T^{3} + 95284 T^{4} + 826887 T^{5} + 5965621 T^{6} + 36144101 T^{7} + 5965621 p T^{8} + 826887 p^{2} T^{9} + 95284 p^{3} T^{10} + 8883 p^{4} T^{11} + 645 p^{5} T^{12} + 33 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 + 33 T + 653 T^{2} + 9315 T^{3} + 105122 T^{4} + 971473 T^{5} + 7548399 T^{6} + 49726081 T^{7} + 7548399 p T^{8} + 971473 p^{2} T^{9} + 105122 p^{3} T^{10} + 9315 p^{4} T^{11} + 653 p^{5} T^{12} + 33 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 - 3 T + 170 T^{2} - 50 T^{3} + 11626 T^{4} + 29278 T^{5} + 496448 T^{6} + 2150172 T^{7} + 496448 p T^{8} + 29278 p^{2} T^{9} + 11626 p^{3} T^{10} - 50 p^{4} T^{11} + 170 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + 9 T + 217 T^{2} + 1559 T^{3} + 22058 T^{4} + 131173 T^{5} + 1390533 T^{6} + 6894030 T^{7} + 1390533 p T^{8} + 131173 p^{2} T^{9} + 22058 p^{3} T^{10} + 1559 p^{4} T^{11} + 217 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 + 11 T + 145 T^{2} + 1339 T^{3} + 14705 T^{4} + 119758 T^{5} + 953008 T^{6} + 6346454 T^{7} + 953008 p T^{8} + 119758 p^{2} T^{9} + 14705 p^{3} T^{10} + 1339 p^{4} T^{11} + 145 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + T + 216 T^{2} - 139 T^{3} + 22869 T^{4} - 30467 T^{5} + 1664224 T^{6} - 2175415 T^{7} + 1664224 p T^{8} - 30467 p^{2} T^{9} + 22869 p^{3} T^{10} - 139 p^{4} T^{11} + 216 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 - 30 T + 570 T^{2} - 7389 T^{3} + 75460 T^{4} - 617301 T^{5} + 4593588 T^{6} - 33613111 T^{7} + 4593588 p T^{8} - 617301 p^{2} T^{9} + 75460 p^{3} T^{10} - 7389 p^{4} T^{11} + 570 p^{5} T^{12} - 30 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 + 19 T + 7 p T^{2} + 5649 T^{3} + 75542 T^{4} + 765723 T^{5} + 7510227 T^{6} + 59966461 T^{7} + 7510227 p T^{8} + 765723 p^{2} T^{9} + 75542 p^{3} T^{10} + 5649 p^{4} T^{11} + 7 p^{6} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 + 30 T + 675 T^{2} + 11360 T^{3} + 157485 T^{4} + 1838995 T^{5} + 18608402 T^{6} + 162480644 T^{7} + 18608402 p T^{8} + 1838995 p^{2} T^{9} + 157485 p^{3} T^{10} + 11360 p^{4} T^{11} + 675 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 + 8 T + 246 T^{2} + 1787 T^{3} + 34468 T^{4} + 225343 T^{5} + 3268094 T^{6} + 18726838 T^{7} + 3268094 p T^{8} + 225343 p^{2} T^{9} + 34468 p^{3} T^{10} + 1787 p^{4} T^{11} + 246 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 + 20 T + 510 T^{2} + 6961 T^{3} + 102634 T^{4} + 1069123 T^{5} + 11655826 T^{6} + 97527765 T^{7} + 11655826 p T^{8} + 1069123 p^{2} T^{9} + 102634 p^{3} T^{10} + 6961 p^{4} T^{11} + 510 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 - 8 T + 273 T^{2} - 2858 T^{3} + 45901 T^{4} - 424499 T^{5} + 5436714 T^{6} - 39832374 T^{7} + 5436714 p T^{8} - 424499 p^{2} T^{9} + 45901 p^{3} T^{10} - 2858 p^{4} T^{11} + 273 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 - 34 T + 962 T^{2} - 18636 T^{3} + 304656 T^{4} - 4073018 T^{5} + 46715214 T^{6} - 458242049 T^{7} + 46715214 p T^{8} - 4073018 p^{2} T^{9} + 304656 p^{3} T^{10} - 18636 p^{4} T^{11} + 962 p^{5} T^{12} - 34 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 - 12 T + 151 T^{2} - 2659 T^{3} + 35240 T^{4} - 317069 T^{5} + 3805221 T^{6} - 40673821 T^{7} + 3805221 p T^{8} - 317069 p^{2} T^{9} + 35240 p^{3} T^{10} - 2659 p^{4} T^{11} + 151 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 7 T + 326 T^{2} - 2842 T^{3} + 55405 T^{4} - 456619 T^{5} + 7188211 T^{6} - 47880078 T^{7} + 7188211 p T^{8} - 456619 p^{2} T^{9} + 55405 p^{3} T^{10} - 2842 p^{4} T^{11} + 326 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.00135965857059444806225029315, −3.96713110418069451655333286871, −3.93995324189305407984113147747, −3.76954604134175659866502237157, −3.50662786607944416646579748805, −3.50162832179824173452788866040, −3.09937233980481292456569367148, −3.08120065647249204041561772042, −3.03817039343586453247763620269, −3.00822341110264516281721200623, −2.48843884456712230385783681781, −2.38091953429631983289000196751, −2.36971574895449476526623964965, −2.28353828024938539006382076250, −2.15306947128086954361354234104, −2.00482167194477589797158999486, −1.94105546729738287151163311347, −1.92792248334769426393719457636, −1.91285762042880458483414421913, −1.89957056949398439560296119748, −1.55690183422270188191248524935, −1.38055266914188754748785379500, −1.35544145584558757428642471806, −1.07475042600521424257386195034, −1.03589665077963974341060406936, 0, 0, 0, 0, 0, 0, 0, 1.03589665077963974341060406936, 1.07475042600521424257386195034, 1.35544145584558757428642471806, 1.38055266914188754748785379500, 1.55690183422270188191248524935, 1.89957056949398439560296119748, 1.91285762042880458483414421913, 1.92792248334769426393719457636, 1.94105546729738287151163311347, 2.00482167194477589797158999486, 2.15306947128086954361354234104, 2.28353828024938539006382076250, 2.36971574895449476526623964965, 2.38091953429631983289000196751, 2.48843884456712230385783681781, 3.00822341110264516281721200623, 3.03817039343586453247763620269, 3.08120065647249204041561772042, 3.09937233980481292456569367148, 3.50162832179824173452788866040, 3.50662786607944416646579748805, 3.76954604134175659866502237157, 3.93995324189305407984113147747, 3.96713110418069451655333286871, 4.00135965857059444806225029315

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

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