Properties

Degree 14
Conductor $ 2^{28} \cdot 251^{7} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2·5-s + 6·7-s − 6·9-s + 5·11-s − 13-s − 6·15-s − 8·17-s + 15·19-s + 18·21-s + 5·23-s − 20·25-s − 27·27-s + 21·31-s + 15·33-s − 12·35-s − 37-s − 3·39-s − 10·41-s + 23·43-s + 12·45-s + 10·47-s − 13·49-s − 24·51-s − 53-s − 10·55-s + 45·57-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.894·5-s + 2.26·7-s − 2·9-s + 1.50·11-s − 0.277·13-s − 1.54·15-s − 1.94·17-s + 3.44·19-s + 3.92·21-s + 1.04·23-s − 4·25-s − 5.19·27-s + 3.77·31-s + 2.61·33-s − 2.02·35-s − 0.164·37-s − 0.480·39-s − 1.56·41-s + 3.50·43-s + 1.78·45-s + 1.45·47-s − 1.85·49-s − 3.36·51-s − 0.137·53-s − 1.34·55-s + 5.96·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 251^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 251^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(14\)
\( N \)  =  \(2^{28} \cdot 251^{7}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4016} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(14,\ 2^{28} \cdot 251^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )$
$L(1)$  $\approx$  $31.82471732$
$L(\frac12)$  $\approx$  $31.82471732$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;251\}$,\(F_p(T)\) is a polynomial of degree 14. If $p \in \{2,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 13.
$p$$F_p(T)$
bad2 \( 1 \)
251 \( ( 1 + T )^{7} \)
good3 \( 1 - p T + 5 p T^{2} - 4 p^{2} T^{3} + 107 T^{4} - 206 T^{5} + 52 p^{2} T^{6} - 751 T^{7} + 52 p^{3} T^{8} - 206 p^{2} T^{9} + 107 p^{3} T^{10} - 4 p^{6} T^{11} + 5 p^{6} T^{12} - p^{7} T^{13} + p^{7} T^{14} \)
5 \( 1 + 2 T + 24 T^{2} + 7 p T^{3} + 276 T^{4} + 331 T^{5} + 407 p T^{6} + 2033 T^{7} + 407 p^{2} T^{8} + 331 p^{2} T^{9} + 276 p^{3} T^{10} + 7 p^{5} T^{11} + 24 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \)
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

−3.79174000209206429113757111520, −3.74154557439109419882114565654, −3.67509115375160407341723550049, −3.53342751799167033511759307183, −3.17743047874291837502981265778, −3.09607967857738804527961741729, −3.00179615210389152173555874428, −2.99004362511325250376539888481, −2.88724248669968195854544902518, −2.84942987297462058278158839652, −2.71919105990402096771452854513, −2.26210333199226590064341514490, −2.21812808458645933143628542712, −2.12902293862787225918694722189, −1.94154115018158437946204349055, −1.93515393976980912760580319735, −1.92817789147770722494401165372, −1.73244615216372368272700576123, −1.33481479269873841022474058530, −1.05409168097639532888011641531, −1.01493568557011409454043413181, −0.852472314786856807289469878206, −0.55647462105700252076438897002, −0.47402549824066669448991644431, −0.39442739359651450890787571390, 0.39442739359651450890787571390, 0.47402549824066669448991644431, 0.55647462105700252076438897002, 0.852472314786856807289469878206, 1.01493568557011409454043413181, 1.05409168097639532888011641531, 1.33481479269873841022474058530, 1.73244615216372368272700576123, 1.92817789147770722494401165372, 1.93515393976980912760580319735, 1.94154115018158437946204349055, 2.12902293862787225918694722189, 2.21812808458645933143628542712, 2.26210333199226590064341514490, 2.71919105990402096771452854513, 2.84942987297462058278158839652, 2.88724248669968195854544902518, 2.99004362511325250376539888481, 3.00179615210389152173555874428, 3.09607967857738804527961741729, 3.17743047874291837502981265778, 3.53342751799167033511759307183, 3.67509115375160407341723550049, 3.74154557439109419882114565654, 3.79174000209206429113757111520

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

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