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Properties

Label 14-9135e7-1.1-c1e7-0-2
Degree $14$
Conductor $5.308\times 10^{27}$
Sign $-1$
Analytic cond. $1.09875\times 10^{13}$
Root an. cond. $8.54068$
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  − 3·2-s + 4-s + 7·5-s − 7·7-s + 6·8-s − 21·10-s − 7·11-s + 5·13-s + 21·14-s − 8·16-s − 14·17-s − 9·19-s + 7·20-s + 21·22-s − 12·23-s + 28·25-s − 15·26-s − 7·28-s − 7·29-s − 7·31-s − 32-s + 42·34-s − 49·35-s + 23·37-s + 27·38-s + 42·40-s − 13·41-s + ⋯
L(s)  = 1  − 2.12·2-s + 1/2·4-s + 3.13·5-s − 2.64·7-s + 2.12·8-s − 6.64·10-s − 2.11·11-s + 1.38·13-s + 5.61·14-s − 2·16-s − 3.39·17-s − 2.06·19-s + 1.56·20-s + 4.47·22-s − 2.50·23-s + 28/5·25-s − 2.94·26-s − 1.32·28-s − 1.29·29-s − 1.25·31-s − 0.176·32-s + 7.20·34-s − 8.28·35-s + 3.78·37-s + 4.37·38-s + 6.64·40-s − 2.03·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{7} \cdot 7^{7} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{7} \cdot 7^{7} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(3^{14} \cdot 5^{7} \cdot 7^{7} \cdot 29^{7}\)
Sign: $-1$
Analytic conductor: \(1.09875\times 10^{13}\)
Root analytic conductor: \(8.54068\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(7\)
Selberg data: \((14,\ 3^{14} \cdot 5^{7} \cdot 7^{7} \cdot 29^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\)

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)$
bad3 \( 1 \)
5 \( ( 1 - T )^{7} \)
7 \( ( 1 + T )^{7} \)
29 \( ( 1 + T )^{7} \)
good2 \( 1 + 3 T + p^{3} T^{2} + 15 T^{3} + 27 T^{4} + 43 T^{5} + 9 p^{3} T^{6} + 101 T^{7} + 9 p^{4} T^{8} + 43 p^{2} T^{9} + 27 p^{3} T^{10} + 15 p^{4} T^{11} + p^{8} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 + 7 T + 69 T^{2} + 365 T^{3} + 2085 T^{4} + 8685 T^{5} + 3323 p T^{6} + 121406 T^{7} + 3323 p^{2} T^{8} + 8685 p^{2} T^{9} + 2085 p^{3} T^{10} + 365 p^{4} T^{11} + 69 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 - 5 T + 74 T^{2} - 297 T^{3} + 2492 T^{4} - 8127 T^{5} + 3833 p T^{6} - 132854 T^{7} + 3833 p^{2} T^{8} - 8127 p^{2} T^{9} + 2492 p^{3} T^{10} - 297 p^{4} T^{11} + 74 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 14 T + 162 T^{2} + 1299 T^{3} + 9092 T^{4} + 52110 T^{5} + 266041 T^{6} + 1158738 T^{7} + 266041 p T^{8} + 52110 p^{2} T^{9} + 9092 p^{3} T^{10} + 1299 p^{4} T^{11} + 162 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 9 T + 98 T^{2} + 493 T^{3} + 3070 T^{4} + 9611 T^{5} + 50837 T^{6} + 134746 T^{7} + 50837 p T^{8} + 9611 p^{2} T^{9} + 3070 p^{3} T^{10} + 493 p^{4} T^{11} + 98 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 + 12 T + 150 T^{2} + 1227 T^{3} + 404 p T^{4} + 58504 T^{5} + 332095 T^{6} + 1682274 T^{7} + 332095 p T^{8} + 58504 p^{2} T^{9} + 404 p^{4} T^{10} + 1227 p^{4} T^{11} + 150 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 + 7 T + 169 T^{2} + 819 T^{3} + 12347 T^{4} + 1515 p T^{5} + 563499 T^{6} + 1771342 T^{7} + 563499 p T^{8} + 1515 p^{3} T^{9} + 12347 p^{3} T^{10} + 819 p^{4} T^{11} + 169 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 - 23 T + 395 T^{2} - 4735 T^{3} + 48263 T^{4} - 404057 T^{5} + 3010765 T^{6} - 19284070 T^{7} + 3010765 p T^{8} - 404057 p^{2} T^{9} + 48263 p^{3} T^{10} - 4735 p^{4} T^{11} + 395 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 13 T + 229 T^{2} + 2145 T^{3} + 23501 T^{4} + 178127 T^{5} + 1475589 T^{6} + 9105314 T^{7} + 1475589 p T^{8} + 178127 p^{2} T^{9} + 23501 p^{3} T^{10} + 2145 p^{4} T^{11} + 229 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 - 31 T + 675 T^{2} - 10109 T^{3} + 2875 p T^{4} - 1214513 T^{5} + 10209823 T^{6} - 71925066 T^{7} + 10209823 p T^{8} - 1214513 p^{2} T^{9} + 2875 p^{4} T^{10} - 10109 p^{4} T^{11} + 675 p^{5} T^{12} - 31 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 6 T + 136 T^{2} - 587 T^{3} + 10526 T^{4} - 40110 T^{5} + 655703 T^{6} - 2340754 T^{7} + 655703 p T^{8} - 40110 p^{2} T^{9} + 10526 p^{3} T^{10} - 587 p^{4} T^{11} + 136 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 23 T + 458 T^{2} + 6051 T^{3} + 72236 T^{4} + 700573 T^{5} + 6237373 T^{6} + 47398162 T^{7} + 6237373 p T^{8} + 700573 p^{2} T^{9} + 72236 p^{3} T^{10} + 6051 p^{4} T^{11} + 458 p^{5} T^{12} + 23 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 + 4 T + 198 T^{2} + 1451 T^{3} + 22212 T^{4} + 192864 T^{5} + 1705771 T^{6} + 14933234 T^{7} + 1705771 p T^{8} + 192864 p^{2} T^{9} + 22212 p^{3} T^{10} + 1451 p^{4} T^{11} + 198 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 + 4 T + 387 T^{2} + 1312 T^{3} + 65993 T^{4} + 186676 T^{5} + 6465739 T^{6} + 14826384 T^{7} + 6465739 p T^{8} + 186676 p^{2} T^{9} + 65993 p^{3} T^{10} + 1312 p^{4} T^{11} + 387 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 - 13 T + 223 T^{2} - 2015 T^{3} + 17295 T^{4} - 83015 T^{5} + 375121 T^{6} - 990778 T^{7} + 375121 p T^{8} - 83015 p^{2} T^{9} + 17295 p^{3} T^{10} - 2015 p^{4} T^{11} + 223 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 11 T + 327 T^{2} - 3375 T^{3} + 54291 T^{4} - 497009 T^{5} + 5699805 T^{6} - 44501082 T^{7} + 5699805 p T^{8} - 497009 p^{2} T^{9} + 54291 p^{3} T^{10} - 3375 p^{4} T^{11} + 327 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 - 8 T + 182 T^{2} - 2299 T^{3} + 28556 T^{4} - 291860 T^{5} + 2759173 T^{6} - 27742306 T^{7} + 2759173 p T^{8} - 291860 p^{2} T^{9} + 28556 p^{3} T^{10} - 2299 p^{4} T^{11} + 182 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 + 456 T^{2} + 137 T^{3} + 95306 T^{4} + 37348 T^{5} + 11794163 T^{6} + 4230918 T^{7} + 11794163 p T^{8} + 37348 p^{2} T^{9} + 95306 p^{3} T^{10} + 137 p^{4} T^{11} + 456 p^{5} T^{12} + p^{7} T^{14} \)
83 \( 1 - 6 T + 146 T^{2} + 261 T^{3} + 12640 T^{4} + 53906 T^{5} + 1313699 T^{6} + 5426494 T^{7} + 1313699 p T^{8} + 53906 p^{2} T^{9} + 12640 p^{3} T^{10} + 261 p^{4} T^{11} + 146 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 - 5 T + 409 T^{2} - 2331 T^{3} + 84685 T^{4} - 462207 T^{5} + 11147049 T^{6} - 52749462 T^{7} + 11147049 p T^{8} - 462207 p^{2} T^{9} + 84685 p^{3} T^{10} - 2331 p^{4} T^{11} + 409 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 4 T + 558 T^{2} - 1801 T^{3} + 141872 T^{4} - 368000 T^{5} + 21392529 T^{6} - 44808054 T^{7} + 21392529 p T^{8} - 368000 p^{2} T^{9} + 141872 p^{3} T^{10} - 1801 p^{4} T^{11} + 558 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \)
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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

−3.88118829051268010903819914907, −3.83420905098753554672223010438, −3.80860411544407087743642595928, −3.48860419721454561169749671908, −3.20473271024669751275098186091, −3.16728044362583480521539509188, −3.15685693249483727099970102437, −3.12421545686086524777677399502, −2.72553201500830448979641281735, −2.58979167341482997002725796709, −2.56359693326170733991126422646, −2.42218748842780502577663863796, −2.39787918527702451599342846649, −2.20285914995522978562157061038, −2.13933200558277233479726750591, −2.13867777846407812572696792816, −2.09275832723084272701017553108, −2.01149635796106359258206004917, −1.72440809229337114670799463602, −1.35667995008983296728630868668, −1.14559935096147127069906466403, −1.12529962333764293457811499577, −1.02220490179533660238983092653, −0.985410314824654299158562540669, −0.965080095648316304643448357300, 0, 0, 0, 0, 0, 0, 0, 0.965080095648316304643448357300, 0.985410314824654299158562540669, 1.02220490179533660238983092653, 1.12529962333764293457811499577, 1.14559935096147127069906466403, 1.35667995008983296728630868668, 1.72440809229337114670799463602, 2.01149635796106359258206004917, 2.09275832723084272701017553108, 2.13867777846407812572696792816, 2.13933200558277233479726750591, 2.20285914995522978562157061038, 2.39787918527702451599342846649, 2.42218748842780502577663863796, 2.56359693326170733991126422646, 2.58979167341482997002725796709, 2.72553201500830448979641281735, 3.12421545686086524777677399502, 3.15685693249483727099970102437, 3.16728044362583480521539509188, 3.20473271024669751275098186091, 3.48860419721454561169749671908, 3.80860411544407087743642595928, 3.83420905098753554672223010438, 3.88118829051268010903819914907

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

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