Properties

Label 14-321e7-1.1-c1e7-0-0
Degree $14$
Conductor $3.512\times 10^{17}$
Sign $1$
Analytic cond. $726.900$
Root an. cond. $1.60099$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 7·3-s − 8·5-s + 6·7-s + 8-s + 28·9-s + 4·11-s + 6·13-s + 56·15-s + 16-s − 10·17-s + 8·19-s − 42·21-s + 6·23-s − 7·24-s + 23·25-s − 84·27-s + 16·31-s − 28·33-s − 48·35-s + 10·37-s − 42·39-s − 8·40-s − 2·41-s + 2·43-s − 224·45-s + 16·47-s − 7·48-s + ⋯
L(s)  = 1  − 4.04·3-s − 3.57·5-s + 2.26·7-s + 0.353·8-s + 28/3·9-s + 1.20·11-s + 1.66·13-s + 14.4·15-s + 1/4·16-s − 2.42·17-s + 1.83·19-s − 9.16·21-s + 1.25·23-s − 1.42·24-s + 23/5·25-s − 16.1·27-s + 2.87·31-s − 4.87·33-s − 8.11·35-s + 1.64·37-s − 6.72·39-s − 1.26·40-s − 0.312·41-s + 0.304·43-s − 33.3·45-s + 2.33·47-s − 1.01·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 107^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 107^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(3^{7} \cdot 107^{7}\)
Sign: $1$
Analytic conductor: \(726.900\)
Root analytic conductor: \(1.60099\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((14,\ 3^{7} \cdot 107^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6219628953\)
\(L(\frac12)\) \(\approx\) \(0.6219628953\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{7} \)
107 \( ( 1 - T )^{7} \)
good2 \( 1 - T^{3} - T^{4} + p^{2} T^{6} - 11 T^{7} + p^{3} T^{8} - p^{3} T^{10} - p^{4} T^{11} + p^{7} T^{14} \)
5 \( 1 + 8 T + 41 T^{2} + 164 T^{3} + 573 T^{4} + 344 p T^{5} + 914 p T^{6} + 86 p^{3} T^{7} + 914 p^{2} T^{8} + 344 p^{3} T^{9} + 573 p^{3} T^{10} + 164 p^{4} T^{11} + 41 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
7 \( 1 - 6 T + 34 T^{2} - 128 T^{3} + 537 T^{4} - 1726 T^{5} + 5536 T^{6} - 14312 T^{7} + 5536 p T^{8} - 1726 p^{2} T^{9} + 537 p^{3} T^{10} - 128 p^{4} T^{11} + 34 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 - 4 T + 4 p T^{2} - 152 T^{3} + 1003 T^{4} - 2942 T^{5} + 15240 T^{6} - 37612 T^{7} + 15240 p T^{8} - 2942 p^{2} T^{9} + 1003 p^{3} T^{10} - 152 p^{4} T^{11} + 4 p^{6} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 - 6 T + 71 T^{2} - 374 T^{3} + 2401 T^{4} - 10598 T^{5} + 288 p^{2} T^{6} - 175594 T^{7} + 288 p^{3} T^{8} - 10598 p^{2} T^{9} + 2401 p^{3} T^{10} - 374 p^{4} T^{11} + 71 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 10 T + 105 T^{2} + 600 T^{3} + 3845 T^{4} + 992 p T^{5} + 88598 T^{6} + 333598 T^{7} + 88598 p T^{8} + 992 p^{3} T^{9} + 3845 p^{3} T^{10} + 600 p^{4} T^{11} + 105 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 - 8 T + 60 T^{2} - 240 T^{3} + 1066 T^{4} - 3448 T^{5} + 20491 T^{6} - 73696 T^{7} + 20491 p T^{8} - 3448 p^{2} T^{9} + 1066 p^{3} T^{10} - 240 p^{4} T^{11} + 60 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 6 T + 88 T^{2} - 458 T^{3} + 185 p T^{4} - 18950 T^{5} + 6044 p T^{6} - 531476 T^{7} + 6044 p^{2} T^{8} - 18950 p^{2} T^{9} + 185 p^{4} T^{10} - 458 p^{4} T^{11} + 88 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 147 T^{2} + 8 T^{3} + 10421 T^{4} + 672 T^{5} + 456271 T^{6} + 27952 T^{7} + 456271 p T^{8} + 672 p^{2} T^{9} + 10421 p^{3} T^{10} + 8 p^{4} T^{11} + 147 p^{5} T^{12} + p^{7} T^{14} \)
31 \( 1 - 16 T + 234 T^{2} - 2266 T^{3} + 20813 T^{4} - 150880 T^{5} + 1039236 T^{6} - 5926108 T^{7} + 1039236 p T^{8} - 150880 p^{2} T^{9} + 20813 p^{3} T^{10} - 2266 p^{4} T^{11} + 234 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 - 10 T + 231 T^{2} - 1718 T^{3} + 23081 T^{4} - 136602 T^{5} + 1342912 T^{6} - 6404082 T^{7} + 1342912 p T^{8} - 136602 p^{2} T^{9} + 23081 p^{3} T^{10} - 1718 p^{4} T^{11} + 231 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 2 T + 135 T^{2} + 108 T^{3} + 7789 T^{4} + 638 T^{5} + 315547 T^{6} - 33048 T^{7} + 315547 p T^{8} + 638 p^{2} T^{9} + 7789 p^{3} T^{10} + 108 p^{4} T^{11} + 135 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 - 2 T + 230 T^{2} - 452 T^{3} + 25157 T^{4} - 43968 T^{5} + 38696 p T^{6} - 2443204 T^{7} + 38696 p^{2} T^{8} - 43968 p^{2} T^{9} + 25157 p^{3} T^{10} - 452 p^{4} T^{11} + 230 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 16 T + 264 T^{2} - 2400 T^{3} + 22703 T^{4} - 147438 T^{5} + 1130560 T^{6} - 6706116 T^{7} + 1130560 p T^{8} - 147438 p^{2} T^{9} + 22703 p^{3} T^{10} - 2400 p^{4} T^{11} + 264 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 16 T + 395 T^{2} + 4536 T^{3} + 64133 T^{4} + 561872 T^{5} + 5688439 T^{6} + 38839760 T^{7} + 5688439 p T^{8} + 561872 p^{2} T^{9} + 64133 p^{3} T^{10} + 4536 p^{4} T^{11} + 395 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 - 20 T + 341 T^{2} - 3976 T^{3} + 44261 T^{4} - 398860 T^{5} + 3524649 T^{6} - 26792816 T^{7} + 3524649 p T^{8} - 398860 p^{2} T^{9} + 44261 p^{3} T^{10} - 3976 p^{4} T^{11} + 341 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 - 2 T + 315 T^{2} - 330 T^{3} + 46593 T^{4} - 26774 T^{5} + 4267152 T^{6} - 1720786 T^{7} + 4267152 p T^{8} - 26774 p^{2} T^{9} + 46593 p^{3} T^{10} - 330 p^{4} T^{11} + 315 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 - 30 T + 525 T^{2} - 7348 T^{3} + 86565 T^{4} - 911826 T^{5} + 8641409 T^{6} - 73652376 T^{7} + 8641409 p T^{8} - 911826 p^{2} T^{9} + 86565 p^{3} T^{10} - 7348 p^{4} T^{11} + 525 p^{5} T^{12} - 30 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 32 T + 736 T^{2} - 12628 T^{3} + 180026 T^{4} - 2154944 T^{5} + 22403571 T^{6} - 201400600 T^{7} + 22403571 p T^{8} - 2154944 p^{2} T^{9} + 180026 p^{3} T^{10} - 12628 p^{4} T^{11} + 736 p^{5} T^{12} - 32 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 + 12 T + 399 T^{2} + 4480 T^{3} + 77349 T^{4} + 726932 T^{5} + 9008923 T^{6} + 67744320 T^{7} + 9008923 p T^{8} + 726932 p^{2} T^{9} + 77349 p^{3} T^{10} + 4480 p^{4} T^{11} + 399 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 - 36 T + 905 T^{2} - 16936 T^{3} + 257493 T^{4} - 3294620 T^{5} + 36204845 T^{6} - 344645424 T^{7} + 36204845 p T^{8} - 3294620 p^{2} T^{9} + 257493 p^{3} T^{10} - 16936 p^{4} T^{11} + 905 p^{5} T^{12} - 36 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 + 10 T + 300 T^{2} + 2598 T^{3} + 53483 T^{4} + 412906 T^{5} + 6275556 T^{6} + 40263276 T^{7} + 6275556 p T^{8} + 412906 p^{2} T^{9} + 53483 p^{3} T^{10} + 2598 p^{4} T^{11} + 300 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 + 4 T + 415 T^{2} + 768 T^{3} + 81141 T^{4} + 60700 T^{5} + 10292379 T^{6} + 4269504 T^{7} + 10292379 p T^{8} + 60700 p^{2} T^{9} + 81141 p^{3} T^{10} + 768 p^{4} T^{11} + 415 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 24 T + 615 T^{2} - 8664 T^{3} + 135829 T^{4} - 1473032 T^{5} + 18552067 T^{6} - 169784976 T^{7} + 18552067 p T^{8} - 1473032 p^{2} T^{9} + 135829 p^{3} T^{10} - 8664 p^{4} T^{11} + 615 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \)
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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

−5.60572447179940422997176061864, −5.47960885767898285153146103811, −5.38922670358218237836751368668, −5.20820955760158032049841198835, −4.87001044527657760231888000640, −4.80513066482585079616772401335, −4.70845690581262826390551161233, −4.67586042810795736106456054315, −4.37029257749621017336868197347, −4.36742637656010170110554653979, −4.22124692765771337285421135578, −3.92342669877929367880267278770, −3.68651211281698377826464692809, −3.63798760283125980665334556472, −3.52840072631811467101631439126, −3.43805266530106865015655675914, −2.86786795786206911069360908473, −2.45693363899077985813512820830, −2.12762861048756482316304041806, −2.05159483832002468720926552327, −1.56808130449252230541590689795, −1.13372128651302470361202529956, −0.895267680702344534611292650466, −0.844133441839036773437058151578, −0.56584731881642897615206869442, 0.56584731881642897615206869442, 0.844133441839036773437058151578, 0.895267680702344534611292650466, 1.13372128651302470361202529956, 1.56808130449252230541590689795, 2.05159483832002468720926552327, 2.12762861048756482316304041806, 2.45693363899077985813512820830, 2.86786795786206911069360908473, 3.43805266530106865015655675914, 3.52840072631811467101631439126, 3.63798760283125980665334556472, 3.68651211281698377826464692809, 3.92342669877929367880267278770, 4.22124692765771337285421135578, 4.36742637656010170110554653979, 4.37029257749621017336868197347, 4.67586042810795736106456054315, 4.70845690581262826390551161233, 4.80513066482585079616772401335, 4.87001044527657760231888000640, 5.20820955760158032049841198835, 5.38922670358218237836751368668, 5.47960885767898285153146103811, 5.60572447179940422997176061864

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

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