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Properties

Label 14-8030e7-1.1-c1e7-0-1
Degree $14$
Conductor $2.153\times 10^{27}$
Sign $-1$
Analytic cond. $4.45603\times 10^{12}$
Root an. cond. $8.00748$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $7$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 7·2-s + 2·3-s + 28·4-s + 7·5-s − 14·6-s − 7·7-s − 84·8-s − 5·9-s − 49·10-s − 7·11-s + 56·12-s − 11·13-s + 49·14-s + 14·15-s + 210·16-s − 7·17-s + 35·18-s + 196·20-s − 14·21-s + 49·22-s − 4·23-s − 168·24-s + 28·25-s + 77·26-s − 6·27-s − 196·28-s + 4·29-s + ⋯
L(s)  = 1  − 4.94·2-s + 1.15·3-s + 14·4-s + 3.13·5-s − 5.71·6-s − 2.64·7-s − 29.6·8-s − 5/3·9-s − 15.4·10-s − 2.11·11-s + 16.1·12-s − 3.05·13-s + 13.0·14-s + 3.61·15-s + 52.5·16-s − 1.69·17-s + 8.24·18-s + 43.8·20-s − 3.05·21-s + 10.4·22-s − 0.834·23-s − 34.2·24-s + 28/5·25-s + 15.1·26-s − 1.15·27-s − 37.0·28-s + 0.742·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 5^{7} \cdot 11^{7} \cdot 73^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 5^{7} \cdot 11^{7} \cdot 73^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(2^{7} \cdot 5^{7} \cdot 11^{7} \cdot 73^{7}\)
Sign: $-1$
Analytic conductor: \(4.45603\times 10^{12}\)
Root analytic conductor: \(8.00748\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(7\)
Selberg data: \((14,\ 2^{7} \cdot 5^{7} \cdot 11^{7} \cdot 73^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{7} \)
5 \( ( 1 - T )^{7} \)
11 \( ( 1 + T )^{7} \)
73 \( ( 1 - T )^{7} \)
good3 \( 1 - 2 T + p^{2} T^{2} - 22 T^{3} + 7 p^{2} T^{4} - 115 T^{5} + 269 T^{6} - 436 T^{7} + 269 p T^{8} - 115 p^{2} T^{9} + 7 p^{5} T^{10} - 22 p^{4} T^{11} + p^{7} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
7 \( 1 + p T + 39 T^{2} + 166 T^{3} + 652 T^{4} + 2176 T^{5} + 978 p T^{6} + 18744 T^{7} + 978 p^{2} T^{8} + 2176 p^{2} T^{9} + 652 p^{3} T^{10} + 166 p^{4} T^{11} + 39 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \)
13 \( 1 + 11 T + 93 T^{2} + 568 T^{3} + 3072 T^{4} + 14293 T^{5} + 61112 T^{6} + 230772 T^{7} + 61112 p T^{8} + 14293 p^{2} T^{9} + 3072 p^{3} T^{10} + 568 p^{4} T^{11} + 93 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 7 T + 94 T^{2} + 490 T^{3} + 3972 T^{4} + 16851 T^{5} + 103025 T^{6} + 358903 T^{7} + 103025 p T^{8} + 16851 p^{2} T^{9} + 3972 p^{3} T^{10} + 490 p^{4} T^{11} + 94 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 4 p T^{2} + 110 T^{3} + 2518 T^{4} + 7775 T^{5} + 53783 T^{6} + 216241 T^{7} + 53783 p T^{8} + 7775 p^{2} T^{9} + 2518 p^{3} T^{10} + 110 p^{4} T^{11} + 4 p^{6} T^{12} + p^{7} T^{14} \)
23 \( 1 + 4 T + 136 T^{2} + 510 T^{3} + 8393 T^{4} + 27884 T^{5} + 304527 T^{6} + 840418 T^{7} + 304527 p T^{8} + 27884 p^{2} T^{9} + 8393 p^{3} T^{10} + 510 p^{4} T^{11} + 136 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 - 4 T + 149 T^{2} - 544 T^{3} + 10665 T^{4} - 34093 T^{5} + 468282 T^{6} - 1258015 T^{7} + 468282 p T^{8} - 34093 p^{2} T^{9} + 10665 p^{3} T^{10} - 544 p^{4} T^{11} + 149 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 - 3 T + 68 T^{2} - 129 T^{3} + 1896 T^{4} - 2585 T^{5} + 47805 T^{6} - 89814 T^{7} + 47805 p T^{8} - 2585 p^{2} T^{9} + 1896 p^{3} T^{10} - 129 p^{4} T^{11} + 68 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 + 22 T + 10 p T^{2} + 4316 T^{3} + 44172 T^{4} + 368389 T^{5} + 2770751 T^{6} + 17670967 T^{7} + 2770751 p T^{8} + 368389 p^{2} T^{9} + 44172 p^{3} T^{10} + 4316 p^{4} T^{11} + 10 p^{6} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 8 T + 208 T^{2} + 1686 T^{3} + 21129 T^{4} + 156272 T^{5} + 1321753 T^{6} + 8256054 T^{7} + 1321753 p T^{8} + 156272 p^{2} T^{9} + 21129 p^{3} T^{10} + 1686 p^{4} T^{11} + 208 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + 19 T + 374 T^{2} + 4641 T^{3} + 52833 T^{4} + 481849 T^{5} + 3922753 T^{6} + 27307106 T^{7} + 3922753 p T^{8} + 481849 p^{2} T^{9} + 52833 p^{3} T^{10} + 4641 p^{4} T^{11} + 374 p^{5} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 3 T + 139 T^{2} - 538 T^{3} + 10398 T^{4} - 37815 T^{5} + 550904 T^{6} - 1984392 T^{7} + 550904 p T^{8} - 37815 p^{2} T^{9} + 10398 p^{3} T^{10} - 538 p^{4} T^{11} + 139 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 126 T^{2} - 188 T^{3} + 10881 T^{4} - 32605 T^{5} + 699732 T^{6} - 1920568 T^{7} + 699732 p T^{8} - 32605 p^{2} T^{9} + 10881 p^{3} T^{10} - 188 p^{4} T^{11} + 126 p^{5} T^{12} + p^{7} T^{14} \)
59 \( 1 - 23 T + 374 T^{2} - 3601 T^{3} + 19801 T^{4} + 27693 T^{5} - 1821044 T^{6} + 19026766 T^{7} - 1821044 p T^{8} + 27693 p^{2} T^{9} + 19801 p^{3} T^{10} - 3601 p^{4} T^{11} + 374 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 + 25 T + 513 T^{2} + 7285 T^{3} + 93267 T^{4} + 967572 T^{5} + 9246463 T^{6} + 74924432 T^{7} + 9246463 p T^{8} + 967572 p^{2} T^{9} + 93267 p^{3} T^{10} + 7285 p^{4} T^{11} + 513 p^{5} T^{12} + 25 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 - 27 T + 545 T^{2} - 7212 T^{3} + 87120 T^{4} - 870707 T^{5} + 8567014 T^{6} - 71525228 T^{7} + 8567014 p T^{8} - 870707 p^{2} T^{9} + 87120 p^{3} T^{10} - 7212 p^{4} T^{11} + 545 p^{5} T^{12} - 27 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 11 T + 295 T^{2} - 2327 T^{3} + 39159 T^{4} - 229466 T^{5} + 3345361 T^{6} - 16799656 T^{7} + 3345361 p T^{8} - 229466 p^{2} T^{9} + 39159 p^{3} T^{10} - 2327 p^{4} T^{11} + 295 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 - 34 T + 696 T^{2} - 8936 T^{3} + 76739 T^{4} - 331626 T^{5} - 933038 T^{6} + 24628668 T^{7} - 933038 p T^{8} - 331626 p^{2} T^{9} + 76739 p^{3} T^{10} - 8936 p^{4} T^{11} + 696 p^{5} T^{12} - 34 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 + 27 T + 666 T^{2} + 10295 T^{3} + 155589 T^{4} + 1808263 T^{5} + 20746600 T^{6} + 190851998 T^{7} + 20746600 p T^{8} + 1808263 p^{2} T^{9} + 155589 p^{3} T^{10} + 10295 p^{4} T^{11} + 666 p^{5} T^{12} + 27 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 + 16 T + 461 T^{2} + 4552 T^{3} + 75197 T^{4} + 473441 T^{5} + 6807496 T^{6} + 34707885 T^{7} + 6807496 p T^{8} + 473441 p^{2} T^{9} + 75197 p^{3} T^{10} + 4552 p^{4} T^{11} + 461 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 + 25 T + 412 T^{2} + 3637 T^{3} + 25311 T^{4} - 71105 T^{5} - 3211384 T^{6} - 51762810 T^{7} - 3211384 p T^{8} - 71105 p^{2} T^{9} + 25311 p^{3} T^{10} + 3637 p^{4} T^{11} + 412 p^{5} T^{12} + 25 p^{6} T^{13} + p^{7} T^{14} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.82435792070440542780292260272, −3.61136007768310813785306143171, −3.57099226972241876280421819041, −3.47050387330472834345357186470, −2.99241980679608035242186044812, −2.97404490377105922830724840303, −2.94839424537467798328272200013, −2.88161399326979718562678482528, −2.86020410544702757137396031043, −2.73480278729292401393293976526, −2.57626756906230973422251532018, −2.56750448271013545249926447152, −2.38809974024747091455408859630, −2.37072899152346346691351798085, −2.22994815711227358204495719101, −2.08679002452303440646412663170, −2.06355606768277861549319198340, −1.80451691794659179073381964565, −1.72363640274673344031485621292, −1.49336294265847662128090630224, −1.43153014026244687216725012174, −1.26289612491887302458793885052, −1.02501916762221506733718092335, −0.955030410839004977006747901776, −0.796027071384122645432522716871, 0, 0, 0, 0, 0, 0, 0, 0.796027071384122645432522716871, 0.955030410839004977006747901776, 1.02501916762221506733718092335, 1.26289612491887302458793885052, 1.43153014026244687216725012174, 1.49336294265847662128090630224, 1.72363640274673344031485621292, 1.80451691794659179073381964565, 2.06355606768277861549319198340, 2.08679002452303440646412663170, 2.22994815711227358204495719101, 2.37072899152346346691351798085, 2.38809974024747091455408859630, 2.56750448271013545249926447152, 2.57626756906230973422251532018, 2.73480278729292401393293976526, 2.86020410544702757137396031043, 2.88161399326979718562678482528, 2.94839424537467798328272200013, 2.97404490377105922830724840303, 2.99241980679608035242186044812, 3.47050387330472834345357186470, 3.57099226972241876280421819041, 3.61136007768310813785306143171, 3.82435792070440542780292260272

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

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