Properties

Degree 14
Conductor $ 2^{14} \cdot 3^{14} \cdot 251^{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  + 2·5-s − 6·7-s + 5·11-s − 13-s + 8·17-s − 15·19-s + 5·23-s − 20·25-s − 21·31-s − 12·35-s − 37-s + 10·41-s − 23·43-s + 10·47-s − 13·49-s + 53-s + 10·55-s + 4·59-s + 3·61-s − 2·65-s − 28·67-s + 18·71-s − 7·73-s − 30·77-s − 30·79-s − 13·83-s + 16·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 2.26·7-s + 1.50·11-s − 0.277·13-s + 1.94·17-s − 3.44·19-s + 1.04·23-s − 4·25-s − 3.77·31-s − 2.02·35-s − 0.164·37-s + 1.56·41-s − 3.50·43-s + 1.45·47-s − 1.85·49-s + 0.137·53-s + 1.34·55-s + 0.520·59-s + 0.384·61-s − 0.248·65-s − 3.42·67-s + 2.13·71-s − 0.819·73-s − 3.41·77-s − 3.37·79-s − 1.42·83-s + 1.73·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{14} \cdot 251^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{14} \cdot 251^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(14\)
\( N \)  =  \(2^{14} \cdot 3^{14} \cdot 251^{7}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{9036} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  7
Selberg data  =  $(14,\ 2^{14} \cdot 3^{14} \cdot 251^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;251\}$,\(F_p(T)\) is a polynomial of degree 14. If $p \in \{2,\;3,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 13.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
251 \( ( 1 + T )^{7} \)
good5 \( 1 - 2 T + 24 T^{2} - 7 p T^{3} + 276 T^{4} - 331 T^{5} + 407 p T^{6} - 2033 T^{7} + 407 p^{2} T^{8} - 331 p^{2} T^{9} + 276 p^{3} T^{10} - 7 p^{5} T^{11} + 24 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
7 \( 1 + 6 T + p^{2} T^{2} + 31 p T^{3} + 1012 T^{4} + 498 p T^{5} + 11665 T^{6} + 31629 T^{7} + 11665 p T^{8} + 498 p^{3} T^{9} + 1012 p^{3} T^{10} + 31 p^{5} T^{11} + p^{7} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 - 5 T + 57 T^{2} - 246 T^{3} + 1548 T^{4} - 511 p T^{5} + 25904 T^{6} - 77368 T^{7} + 25904 p T^{8} - 511 p^{3} T^{9} + 1548 p^{3} T^{10} - 246 p^{4} T^{11} + 57 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 + T + 49 T^{2} + 10 T^{3} + 1145 T^{4} - 534 T^{5} + 18318 T^{6} - 12855 T^{7} + 18318 p T^{8} - 534 p^{2} T^{9} + 1145 p^{3} T^{10} + 10 p^{4} T^{11} + 49 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 - 8 T + 112 T^{2} - 599 T^{3} + 4772 T^{4} - 19059 T^{5} + 115579 T^{6} - 380413 T^{7} + 115579 p T^{8} - 19059 p^{2} T^{9} + 4772 p^{3} T^{10} - 599 p^{4} T^{11} + 112 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 15 T + 198 T^{2} + 1705 T^{3} + 13153 T^{4} + 79977 T^{5} + 441300 T^{6} + 2015782 T^{7} + 441300 p T^{8} + 79977 p^{2} T^{9} + 13153 p^{3} T^{10} + 1705 p^{4} T^{11} + 198 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 5 T + 114 T^{2} - 562 T^{3} + 6547 T^{4} - 28211 T^{5} + 231026 T^{6} - 828297 T^{7} + 231026 p T^{8} - 28211 p^{2} T^{9} + 6547 p^{3} T^{10} - 562 p^{4} T^{11} + 114 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 132 T^{2} - 90 T^{3} + 8557 T^{4} - 8078 T^{5} + 357562 T^{6} - 322880 T^{7} + 357562 p T^{8} - 8078 p^{2} T^{9} + 8557 p^{3} T^{10} - 90 p^{4} T^{11} + 132 p^{5} T^{12} + p^{7} T^{14} \)
31 \( 1 + 21 T + 323 T^{2} + 3712 T^{3} + 34723 T^{4} + 275114 T^{5} + 1883364 T^{6} + 11173393 T^{7} + 1883364 p T^{8} + 275114 p^{2} T^{9} + 34723 p^{3} T^{10} + 3712 p^{4} T^{11} + 323 p^{5} T^{12} + 21 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 + T + 107 T^{2} - 80 T^{3} + 5714 T^{4} - 14045 T^{5} + 214630 T^{6} - 801168 T^{7} + 214630 p T^{8} - 14045 p^{2} T^{9} + 5714 p^{3} T^{10} - 80 p^{4} T^{11} + 107 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 - 10 T + 183 T^{2} - 1925 T^{3} + 18522 T^{4} - 156974 T^{5} + 1223611 T^{6} - 7778621 T^{7} + 1223611 p T^{8} - 156974 p^{2} T^{9} + 18522 p^{3} T^{10} - 1925 p^{4} T^{11} + 183 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + 23 T + 479 T^{2} + 6374 T^{3} + 76418 T^{4} + 709137 T^{5} + 5980176 T^{6} + 41069820 T^{7} + 5980176 p T^{8} + 709137 p^{2} T^{9} + 76418 p^{3} T^{10} + 6374 p^{4} T^{11} + 479 p^{5} T^{12} + 23 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 10 T + 175 T^{2} - 1405 T^{3} + 16332 T^{4} - 102894 T^{5} + 981162 T^{6} - 5586582 T^{7} + 981162 p T^{8} - 102894 p^{2} T^{9} + 16332 p^{3} T^{10} - 1405 p^{4} T^{11} + 175 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 - T + 341 T^{2} - 296 T^{3} + 51270 T^{4} - 37529 T^{5} + 4404456 T^{6} - 2612924 T^{7} + 4404456 p T^{8} - 37529 p^{2} T^{9} + 51270 p^{3} T^{10} - 296 p^{4} T^{11} + 341 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 - 4 T + 192 T^{2} - 766 T^{3} + 17341 T^{4} - 80260 T^{5} + 1108838 T^{6} - 5770956 T^{7} + 1108838 p T^{8} - 80260 p^{2} T^{9} + 17341 p^{3} T^{10} - 766 p^{4} T^{11} + 192 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 - 3 T + 162 T^{2} + p T^{3} + 11799 T^{4} + 53561 T^{5} + 692826 T^{6} + 4721098 T^{7} + 692826 p T^{8} + 53561 p^{2} T^{9} + 11799 p^{3} T^{10} + p^{5} T^{11} + 162 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 + 28 T + 648 T^{2} + 9931 T^{3} + 135284 T^{4} + 1472821 T^{5} + 14791793 T^{6} + 125329735 T^{7} + 14791793 p T^{8} + 1472821 p^{2} T^{9} + 135284 p^{3} T^{10} + 9931 p^{4} T^{11} + 648 p^{5} T^{12} + 28 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 18 T + 509 T^{2} - 6143 T^{3} + 99682 T^{4} - 906794 T^{5} + 10907742 T^{6} - 79739874 T^{7} + 10907742 p T^{8} - 906794 p^{2} T^{9} + 99682 p^{3} T^{10} - 6143 p^{4} T^{11} + 509 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 + 7 T + 277 T^{2} + 2034 T^{3} + 44471 T^{4} + 290456 T^{5} + 4664818 T^{6} + 26135103 T^{7} + 4664818 p T^{8} + 290456 p^{2} T^{9} + 44471 p^{3} T^{10} + 2034 p^{4} T^{11} + 277 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 + 30 T + 864 T^{2} + 15277 T^{3} + 251870 T^{4} + 3126597 T^{5} + 36178295 T^{6} + 332923983 T^{7} + 36178295 p T^{8} + 3126597 p^{2} T^{9} + 251870 p^{3} T^{10} + 15277 p^{4} T^{11} + 864 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 + 13 T + 485 T^{2} + 5360 T^{3} + 109330 T^{4} + 996823 T^{5} + 14416686 T^{6} + 106582888 T^{7} + 14416686 p T^{8} + 996823 p^{2} T^{9} + 109330 p^{3} T^{10} + 5360 p^{4} T^{11} + 485 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 + 109 T^{2} + 211 T^{3} + 18778 T^{4} - 41946 T^{5} + 1597555 T^{6} - 597701 T^{7} + 1597555 p T^{8} - 41946 p^{2} T^{9} + 18778 p^{3} T^{10} + 211 p^{4} T^{11} + 109 p^{5} T^{12} + p^{7} T^{14} \)
97 \( 1 + 2 T + 455 T^{2} - 86 T^{3} + 89144 T^{4} - 192034 T^{5} + 10939992 T^{6} - 31982204 T^{7} + 10939992 p T^{8} - 192034 p^{2} T^{9} + 89144 p^{3} T^{10} - 86 p^{4} T^{11} + 455 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \)
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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

−3.83865434738355801793398512679, −3.79649144760933204555594517155, −3.76152368637042720720801424454, −3.68332892679549857685004084236, −3.29684508944663809852145284181, −3.25360386900404939876376984563, −3.24773089520048471410481789320, −3.07443237000859342599757915885, −2.94713444689768723966074668416, −2.89813439238644333505680131488, −2.88784302235533235207541194747, −2.42207729277841433762271671245, −2.23340574366760145010237850814, −2.22147712990958123545574776161, −2.19309690604485020238790904467, −2.16668427083624228992803294578, −2.10335977680279020593948053417, −1.99344737750919091937341510044, −1.48934659998638266436912219604, −1.38368904415655663943239163753, −1.36945941745934456908033236700, −1.32181619930289266960357592570, −1.21095353457914458271724858244, −1.19850655153507633259169946259, −0.945364539623739597008874214982, 0, 0, 0, 0, 0, 0, 0, 0.945364539623739597008874214982, 1.19850655153507633259169946259, 1.21095353457914458271724858244, 1.32181619930289266960357592570, 1.36945941745934456908033236700, 1.38368904415655663943239163753, 1.48934659998638266436912219604, 1.99344737750919091937341510044, 2.10335977680279020593948053417, 2.16668427083624228992803294578, 2.19309690604485020238790904467, 2.22147712990958123545574776161, 2.23340574366760145010237850814, 2.42207729277841433762271671245, 2.88784302235533235207541194747, 2.89813439238644333505680131488, 2.94713444689768723966074668416, 3.07443237000859342599757915885, 3.24773089520048471410481789320, 3.25360386900404939876376984563, 3.29684508944663809852145284181, 3.68332892679549857685004084236, 3.76152368637042720720801424454, 3.79649144760933204555594517155, 3.83865434738355801793398512679

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

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