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Properties

Label 14-4026e7-1.1-c1e7-0-2
Degree $14$
Conductor $1.714\times 10^{25}$
Sign $-1$
Analytic cond. $3.54859\times 10^{10}$
Root an. cond. $5.66990$
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 − 7·3-s + 28·4-s − 2·5-s + 49·6-s + 7-s − 84·8-s + 28·9-s + 14·10-s − 7·11-s − 196·12-s − 7·14-s + 14·15-s + 210·16-s + 3·17-s − 196·18-s − 5·19-s − 56·20-s − 7·21-s + 49·22-s − 3·23-s + 588·24-s − 10·25-s − 84·27-s + 28·28-s − 14·29-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 + 0.377·7-s − 29.6·8-s + 28/3·9-s + 4.42·10-s − 2.11·11-s − 56.5·12-s − 1.87·14-s + 3.61·15-s + 52.5·16-s + 0.727·17-s − 46.1·18-s − 1.14·19-s − 12.5·20-s − 1.52·21-s + 10.4·22-s − 0.625·23-s + 120.·24-s − 2·25-s − 16.1·27-s + 5.29·28-s − 2.59·29-s − 17.8·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7}\)
Sign: $-1$
Analytic conductor: \(3.54859\times 10^{10}\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(7\)
Selberg data: \((14,\ 2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\)

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 \( ( 1 + T )^{7} \)
3 \( ( 1 + T )^{7} \)
11 \( ( 1 + T )^{7} \)
61 \( ( 1 + T )^{7} \)
good5 \( 1 + 2 T + 14 T^{2} + 21 T^{3} + 89 T^{4} + 14 p T^{5} + 364 T^{6} + 142 T^{7} + 364 p T^{8} + 14 p^{3} T^{9} + 89 p^{3} T^{10} + 21 p^{4} T^{11} + 14 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \)
7 \( 1 - T + 25 T^{2} - 32 T^{3} + 47 p T^{4} - 430 T^{5} + 2985 T^{6} - 526 p T^{7} + 2985 p T^{8} - 430 p^{2} T^{9} + 47 p^{4} T^{10} - 32 p^{4} T^{11} + 25 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 + 45 T^{2} + 28 T^{3} + 1098 T^{4} + 925 T^{5} + 19436 T^{6} + 14614 T^{7} + 19436 p T^{8} + 925 p^{2} T^{9} + 1098 p^{3} T^{10} + 28 p^{4} T^{11} + 45 p^{5} T^{12} + p^{7} T^{14} \)
17 \( 1 - 3 T + 52 T^{2} - 106 T^{3} + 5 p^{2} T^{4} - 2240 T^{5} + 30621 T^{6} - 44620 T^{7} + 30621 p T^{8} - 2240 p^{2} T^{9} + 5 p^{5} T^{10} - 106 p^{4} T^{11} + 52 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 5 T + 91 T^{2} + 450 T^{3} + 4229 T^{4} + 966 p T^{5} + 122757 T^{6} + 440582 T^{7} + 122757 p T^{8} + 966 p^{3} T^{9} + 4229 p^{3} T^{10} + 450 p^{4} T^{11} + 91 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 + 3 T + 84 T^{2} + 446 T^{3} + 3649 T^{4} + 23153 T^{5} + 117678 T^{6} + 28932 p T^{7} + 117678 p T^{8} + 23153 p^{2} T^{9} + 3649 p^{3} T^{10} + 446 p^{4} T^{11} + 84 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 14 T + 216 T^{2} + 2099 T^{3} + 18644 T^{4} + 138406 T^{5} + 888303 T^{6} + 5181922 T^{7} + 888303 p T^{8} + 138406 p^{2} T^{9} + 18644 p^{3} T^{10} + 2099 p^{4} T^{11} + 216 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 - 5 T + 161 T^{2} - 767 T^{3} + 12507 T^{4} - 52647 T^{5} + 591659 T^{6} - 2093338 T^{7} + 591659 p T^{8} - 52647 p^{2} T^{9} + 12507 p^{3} T^{10} - 767 p^{4} T^{11} + 161 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 - 14 T + 198 T^{2} - 2012 T^{3} + 17566 T^{4} - 132995 T^{5} + 939334 T^{6} - 5754020 T^{7} + 939334 p T^{8} - 132995 p^{2} T^{9} + 17566 p^{3} T^{10} - 2012 p^{4} T^{11} + 198 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 7 T + 224 T^{2} + 1439 T^{3} + 23360 T^{4} + 130625 T^{5} + 1470535 T^{6} + 6839346 T^{7} + 1470535 p T^{8} + 130625 p^{2} T^{9} + 23360 p^{3} T^{10} + 1439 p^{4} T^{11} + 224 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 - T + 144 T^{2} - 432 T^{3} + 8369 T^{4} - 58109 T^{5} + 288954 T^{6} - 3543844 T^{7} + 288954 p T^{8} - 58109 p^{2} T^{9} + 8369 p^{3} T^{10} - 432 p^{4} T^{11} + 144 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 + 143 T^{2} - 134 T^{3} + 218 p T^{4} - 3145 T^{5} + 606214 T^{6} + 298958 T^{7} + 606214 p T^{8} - 3145 p^{2} T^{9} + 218 p^{4} T^{10} - 134 p^{4} T^{11} + 143 p^{5} T^{12} + p^{7} T^{14} \)
53 \( 1 + 3 T + 193 T^{2} + 46 T^{3} + 17551 T^{4} - 24658 T^{5} + 1181021 T^{6} - 1977370 T^{7} + 1181021 p T^{8} - 24658 p^{2} T^{9} + 17551 p^{3} T^{10} + 46 p^{4} T^{11} + 193 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 + 14 T + 7 p T^{2} + 4220 T^{3} + 70038 T^{4} + 557747 T^{5} + 6665088 T^{6} + 42205670 T^{7} + 6665088 p T^{8} + 557747 p^{2} T^{9} + 70038 p^{3} T^{10} + 4220 p^{4} T^{11} + 7 p^{6} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 + 249 T^{2} - 612 T^{3} + 32316 T^{4} - 112395 T^{5} + 2965226 T^{6} - 9819458 T^{7} + 2965226 p T^{8} - 112395 p^{2} T^{9} + 32316 p^{3} T^{10} - 612 p^{4} T^{11} + 249 p^{5} T^{12} + p^{7} T^{14} \)
71 \( 1 + 22 T + 644 T^{2} + 9527 T^{3} + 156593 T^{4} + 1704816 T^{5} + 276490 p T^{6} + 161780134 T^{7} + 276490 p^{2} T^{8} + 1704816 p^{2} T^{9} + 156593 p^{3} T^{10} + 9527 p^{4} T^{11} + 644 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 - T + 364 T^{2} - 296 T^{3} + 61993 T^{4} - 30297 T^{5} + 6604954 T^{6} - 2048028 T^{7} + 6604954 p T^{8} - 30297 p^{2} T^{9} + 61993 p^{3} T^{10} - 296 p^{4} T^{11} + 364 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 - 10 T + 363 T^{2} - 3814 T^{3} + 62168 T^{4} - 656547 T^{5} + 6793226 T^{6} - 66031034 T^{7} + 6793226 p T^{8} - 656547 p^{2} T^{9} + 62168 p^{3} T^{10} - 3814 p^{4} T^{11} + 363 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 + 17 T + 542 T^{2} + 6484 T^{3} + 119053 T^{4} + 1098807 T^{5} + 15033196 T^{6} + 112538360 T^{7} + 15033196 p T^{8} + 1098807 p^{2} T^{9} + 119053 p^{3} T^{10} + 6484 p^{4} T^{11} + 542 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 + 18 T + 554 T^{2} + 7305 T^{3} + 124975 T^{4} + 1304968 T^{5} + 16362238 T^{6} + 141939378 T^{7} + 16362238 p T^{8} + 1304968 p^{2} T^{9} + 124975 p^{3} T^{10} + 7305 p^{4} T^{11} + 554 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 25 T + 569 T^{2} - 7000 T^{3} + 76902 T^{4} - 419559 T^{5} + 2045781 T^{6} + 4237706 T^{7} + 2045781 p T^{8} - 419559 p^{2} T^{9} + 76902 p^{3} T^{10} - 7000 p^{4} T^{11} + 569 p^{5} T^{12} - 25 p^{6} T^{13} + p^{7} T^{14} \)
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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

−4.27020571522567580038337393486, −4.25837770274494831191350596390, −4.14501826066679910962726861920, −4.13157366212424016756038735885, −3.54903432989011896460413062356, −3.47691897808482479205335829544, −3.44629403855785069758159091908, −3.32476063274637045568928169555, −3.27738484614866948972820811187, −3.22631765709254904147185327229, −2.99956356772756128733110902454, −2.53359398700913601427177158751, −2.41356269069207527998618223225, −2.38344072506593267356552540089, −2.24745684835060206277537017804, −2.16951206457221715105576083753, −2.09867001406717774771307664965, −2.02006765030001935916242400194, −1.44911504245292241066059230885, −1.44353482053148605145789349781, −1.41325197757227575605254124140, −1.19467183400164060324866050318, −1.13462689851539855184130214023, −1.05231468273781062762517470249, −0.893372534040728935528300116652, 0, 0, 0, 0, 0, 0, 0, 0.893372534040728935528300116652, 1.05231468273781062762517470249, 1.13462689851539855184130214023, 1.19467183400164060324866050318, 1.41325197757227575605254124140, 1.44353482053148605145789349781, 1.44911504245292241066059230885, 2.02006765030001935916242400194, 2.09867001406717774771307664965, 2.16951206457221715105576083753, 2.24745684835060206277537017804, 2.38344072506593267356552540089, 2.41356269069207527998618223225, 2.53359398700913601427177158751, 2.99956356772756128733110902454, 3.22631765709254904147185327229, 3.27738484614866948972820811187, 3.32476063274637045568928169555, 3.44629403855785069758159091908, 3.47691897808482479205335829544, 3.54903432989011896460413062356, 4.13157366212424016756038735885, 4.14501826066679910962726861920, 4.25837770274494831191350596390, 4.27020571522567580038337393486

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

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