Properties

Degree 14
Conductor $ 2^{7} \cdot 3^{7} \cdot 13^{7} \cdot 103^{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  − 7·2-s − 7·3-s + 28·4-s + 2·5-s + 49·6-s − 9·7-s − 84·8-s + 28·9-s − 14·10-s − 196·12-s + 7·13-s + 63·14-s − 14·15-s + 210·16-s + 3·17-s − 196·18-s − 16·19-s + 56·20-s + 63·21-s + 6·23-s + 588·24-s − 8·25-s − 49·26-s − 84·27-s − 252·28-s − 5·29-s + 98·30-s + ⋯
L(s)  = 1  − 4.94·2-s − 4.04·3-s + 14·4-s + 0.894·5-s + 20.0·6-s − 3.40·7-s − 29.6·8-s + 28/3·9-s − 4.42·10-s − 56.5·12-s + 1.94·13-s + 16.8·14-s − 3.61·15-s + 52.5·16-s + 0.727·17-s − 46.1·18-s − 3.67·19-s + 12.5·20-s + 13.7·21-s + 1.25·23-s + 120.·24-s − 8/5·25-s − 9.60·26-s − 16.1·27-s − 47.6·28-s − 0.928·29-s + 17.8·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 13^{7} \cdot 103^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 13^{7} \cdot 103^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(14\)
\( N \)  =  \(2^{7} \cdot 3^{7} \cdot 13^{7} \cdot 103^{7}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8034} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  7
Selberg data  =  $(14,\ 2^{7} \cdot 3^{7} \cdot 13^{7} \cdot 103^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;13,\;103\}$, \(F_p\) is a polynomial of degree 14. If $p \in \{2,\;3,\;13,\;103\}$, then $F_p$ is a polynomial of degree at most 13.
$p$$F_p$
bad2 \( ( 1 + T )^{7} \)
3 \( ( 1 + T )^{7} \)
13 \( ( 1 - T )^{7} \)
103 \( ( 1 - T )^{7} \)
good5 \( 1 - 2 T + 12 T^{2} - 19 T^{3} + 123 T^{4} - 209 T^{5} + 803 T^{6} - 983 T^{7} + 803 p T^{8} - 209 p^{2} T^{9} + 123 p^{3} T^{10} - 19 p^{4} T^{11} + 12 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
7 \( 1 + 9 T + 66 T^{2} + 327 T^{3} + 1446 T^{4} + 5155 T^{5} + 16867 T^{6} + 46481 T^{7} + 16867 p T^{8} + 5155 p^{2} T^{9} + 1446 p^{3} T^{10} + 327 p^{4} T^{11} + 66 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 + 50 T^{2} - 5 T^{3} + 111 p T^{4} - 112 T^{5} + 19364 T^{6} - 1257 T^{7} + 19364 p T^{8} - 112 p^{2} T^{9} + 111 p^{4} T^{10} - 5 p^{4} T^{11} + 50 p^{5} T^{12} + p^{7} T^{14} \)
17 \( 1 - 3 T + 48 T^{2} - 210 T^{3} + 1728 T^{4} - 6112 T^{5} + 41181 T^{6} - 130549 T^{7} + 41181 p T^{8} - 6112 p^{2} T^{9} + 1728 p^{3} T^{10} - 210 p^{4} T^{11} + 48 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 16 T + 225 T^{2} + 2028 T^{3} + 16324 T^{4} + 101598 T^{5} + 571970 T^{6} + 2616215 T^{7} + 571970 p T^{8} + 101598 p^{2} T^{9} + 16324 p^{3} T^{10} + 2028 p^{4} T^{11} + 225 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 6 T + 124 T^{2} - 579 T^{3} + 6515 T^{4} - 24568 T^{5} + 206736 T^{6} - 663561 T^{7} + 206736 p T^{8} - 24568 p^{2} T^{9} + 6515 p^{3} T^{10} - 579 p^{4} T^{11} + 124 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 5 T + 102 T^{2} + 612 T^{3} + 5141 T^{4} + 30066 T^{5} + 189821 T^{6} + 958991 T^{7} + 189821 p T^{8} + 30066 p^{2} T^{9} + 5141 p^{3} T^{10} + 612 p^{4} T^{11} + 102 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 + 16 T + 225 T^{2} + 2052 T^{3} + 592 p T^{4} + 128552 T^{5} + 886488 T^{6} + 4975541 T^{7} + 886488 p T^{8} + 128552 p^{2} T^{9} + 592 p^{4} T^{10} + 2052 p^{4} T^{11} + 225 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 - 17 T + 306 T^{2} - 3124 T^{3} + 32871 T^{4} - 245430 T^{5} + 1913427 T^{6} - 11334611 T^{7} + 1913427 p T^{8} - 245430 p^{2} T^{9} + 32871 p^{3} T^{10} - 3124 p^{4} T^{11} + 306 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 - 12 T + 4 p T^{2} - 964 T^{3} + 8573 T^{4} - 47076 T^{5} + 482781 T^{6} - 2693347 T^{7} + 482781 p T^{8} - 47076 p^{2} T^{9} + 8573 p^{3} T^{10} - 964 p^{4} T^{11} + 4 p^{6} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + 22 T + 325 T^{2} + 3220 T^{3} + 24121 T^{4} + 136680 T^{5} + 650694 T^{6} + 3418661 T^{7} + 650694 p T^{8} + 136680 p^{2} T^{9} + 24121 p^{3} T^{10} + 3220 p^{4} T^{11} + 325 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 + 131 T^{2} - 16 T^{3} + 11688 T^{4} + 6954 T^{5} + 704218 T^{6} + 308813 T^{7} + 704218 p T^{8} + 6954 p^{2} T^{9} + 11688 p^{3} T^{10} - 16 p^{4} T^{11} + 131 p^{5} T^{12} + p^{7} T^{14} \)
53 \( 1 - 2 T + 137 T^{2} + 393 T^{3} + 9678 T^{4} + 48589 T^{5} + 742820 T^{6} + 2496473 T^{7} + 742820 p T^{8} + 48589 p^{2} T^{9} + 9678 p^{3} T^{10} + 393 p^{4} T^{11} + 137 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 + 3 T + 210 T^{2} + 539 T^{3} + 25292 T^{4} + 53419 T^{5} + 2065249 T^{6} + 3896781 T^{7} + 2065249 p T^{8} + 53419 p^{2} T^{9} + 25292 p^{3} T^{10} + 539 p^{4} T^{11} + 210 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 + 6 T + 279 T^{2} + 1912 T^{3} + 39211 T^{4} + 258784 T^{5} + 3552046 T^{6} + 20060339 T^{7} + 3552046 p T^{8} + 258784 p^{2} T^{9} + 39211 p^{3} T^{10} + 1912 p^{4} T^{11} + 279 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 - T + 211 T^{2} + 2 p T^{3} + 21876 T^{4} + 41069 T^{5} + 1677249 T^{6} + 3917111 T^{7} + 1677249 p T^{8} + 41069 p^{2} T^{9} + 21876 p^{3} T^{10} + 2 p^{5} T^{11} + 211 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 15 T + 452 T^{2} - 4851 T^{3} + 85045 T^{4} - 716014 T^{5} + 9325472 T^{6} - 63589473 T^{7} + 9325472 p T^{8} - 716014 p^{2} T^{9} + 85045 p^{3} T^{10} - 4851 p^{4} T^{11} + 452 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 - 17 T + 530 T^{2} - 6819 T^{3} + 117094 T^{4} - 1175660 T^{5} + 14247390 T^{6} - 112197599 T^{7} + 14247390 p T^{8} - 1175660 p^{2} T^{9} + 117094 p^{3} T^{10} - 6819 p^{4} T^{11} + 530 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 + 27 T + 744 T^{2} + 12364 T^{3} + 197835 T^{4} + 2372854 T^{5} + 27173877 T^{6} + 247348701 T^{7} + 27173877 p T^{8} + 2372854 p^{2} T^{9} + 197835 p^{3} T^{10} + 12364 p^{4} T^{11} + 744 p^{5} T^{12} + 27 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 - 12 T + 446 T^{2} - 4743 T^{3} + 96239 T^{4} - 859806 T^{5} + 12423630 T^{6} - 91258057 T^{7} + 12423630 p T^{8} - 859806 p^{2} T^{9} + 96239 p^{3} T^{10} - 4743 p^{4} T^{11} + 446 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 + 9 T + 340 T^{2} + 1689 T^{3} + 41782 T^{4} - 2472 T^{5} + 2633172 T^{6} - 14512419 T^{7} + 2633172 p T^{8} - 2472 p^{2} T^{9} + 41782 p^{3} T^{10} + 1689 p^{4} T^{11} + 340 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 + 3 T + 307 T^{2} + 1782 T^{3} + 44866 T^{4} + 386761 T^{5} + 4801989 T^{6} + 46932733 T^{7} + 4801989 p T^{8} + 386761 p^{2} T^{9} + 44866 p^{3} T^{10} + 1782 p^{4} T^{11} + 307 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
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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

−3.84697083873084914402180276043, −3.80768723178610052054885029856, −3.78255092591082478791654089450, −3.69467298131860357973060529329, −3.38756499587384812261273501490, −3.20797307415258400429063837591, −3.05069653403504079397863069218, −2.95321217729174614152826210791, −2.91116779127382566635657858161, −2.85440007280551662543352117777, −2.73361930513164286575816266526, −2.14840037383694094793760962160, −2.13847159063878761120244822571, −2.13309196936108115146503501238, −2.13215641186555478645179106778, −2.03852573706414966735884581159, −1.88550776225120894918549782352, −1.81844847898486280794005411894, −1.32894695600967012567916634250, −1.22779395800480222303085605842, −1.21204324084739057014127770493, −1.12598799712367628695757540994, −1.05626427956969767782349388641, −0.866625426477179672904931116611, −0.78019607539215726766331184144, 0, 0, 0, 0, 0, 0, 0, 0.78019607539215726766331184144, 0.866625426477179672904931116611, 1.05626427956969767782349388641, 1.12598799712367628695757540994, 1.21204324084739057014127770493, 1.22779395800480222303085605842, 1.32894695600967012567916634250, 1.81844847898486280794005411894, 1.88550776225120894918549782352, 2.03852573706414966735884581159, 2.13215641186555478645179106778, 2.13309196936108115146503501238, 2.13847159063878761120244822571, 2.14840037383694094793760962160, 2.73361930513164286575816266526, 2.85440007280551662543352117777, 2.91116779127382566635657858161, 2.95321217729174614152826210791, 3.05069653403504079397863069218, 3.20797307415258400429063837591, 3.38756499587384812261273501490, 3.69467298131860357973060529329, 3.78255092591082478791654089450, 3.80768723178610052054885029856, 3.84697083873084914402180276043

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

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