Properties

Label 14-4020e7-1.1-c1e7-0-0
Degree $14$
Conductor $1.697\times 10^{25}$
Sign $1$
Analytic cond. $3.51173\times 10^{10}$
Root an. cond. $5.66567$
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 − 7·5-s + 3·7-s + 28·9-s − 5·11-s + 5·13-s + 49·15-s + 3·17-s + 2·19-s − 21·21-s − 3·23-s + 28·25-s − 84·27-s − 8·29-s + 7·31-s + 35·33-s − 21·35-s − 5·37-s − 35·39-s + 7·41-s + 3·43-s − 196·45-s − 6·47-s − 12·49-s − 21·51-s − 9·53-s + 35·55-s + ⋯
L(s)  = 1  − 4.04·3-s − 3.13·5-s + 1.13·7-s + 28/3·9-s − 1.50·11-s + 1.38·13-s + 12.6·15-s + 0.727·17-s + 0.458·19-s − 4.58·21-s − 0.625·23-s + 28/5·25-s − 16.1·27-s − 1.48·29-s + 1.25·31-s + 6.09·33-s − 3.54·35-s − 0.821·37-s − 5.60·39-s + 1.09·41-s + 0.457·43-s − 29.2·45-s − 0.875·47-s − 1.71·49-s − 2.94·51-s − 1.23·53-s + 4.71·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(2^{14} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7}\)
Sign: $1$
Analytic conductor: \(3.51173\times 10^{10}\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((14,\ 2^{14} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8271683976\)
\(L(\frac12)\) \(\approx\) \(0.8271683976\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T )^{7} \)
5 \( ( 1 + T )^{7} \)
67 \( ( 1 + T )^{7} \)
good7 \( 1 - 3 T + 3 p T^{2} - 36 T^{3} + 192 T^{4} - 103 T^{5} + 1032 T^{6} + 188 T^{7} + 1032 p T^{8} - 103 p^{2} T^{9} + 192 p^{3} T^{10} - 36 p^{4} T^{11} + 3 p^{6} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 + 5 T + 21 T^{2} + 58 T^{3} + 292 T^{4} + 1049 T^{5} + 4080 T^{6} + 12192 T^{7} + 4080 p T^{8} + 1049 p^{2} T^{9} + 292 p^{3} T^{10} + 58 p^{4} T^{11} + 21 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 - 5 T + 34 T^{2} - 139 T^{3} + 635 T^{4} - 2607 T^{5} + 10740 T^{6} - 37974 T^{7} + 10740 p T^{8} - 2607 p^{2} T^{9} + 635 p^{3} T^{10} - 139 p^{4} T^{11} + 34 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 - 3 T + 31 T^{2} - 83 T^{3} + 651 T^{4} - 2330 T^{5} + 12209 T^{6} - 39446 T^{7} + 12209 p T^{8} - 2330 p^{2} T^{9} + 651 p^{3} T^{10} - 83 p^{4} T^{11} + 31 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 - 2 T + 25 T^{2} - 70 T^{3} + 977 T^{4} - 2063 T^{5} + 20081 T^{6} - 51322 T^{7} + 20081 p T^{8} - 2063 p^{2} T^{9} + 977 p^{3} T^{10} - 70 p^{4} T^{11} + 25 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 + 3 T + 87 T^{2} + 376 T^{3} + 3928 T^{4} + 19960 T^{5} + 119694 T^{6} + 598042 T^{7} + 119694 p T^{8} + 19960 p^{2} T^{9} + 3928 p^{3} T^{10} + 376 p^{4} T^{11} + 87 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 8 T + 195 T^{2} + 1174 T^{3} + 16017 T^{4} + 2627 p T^{5} + 746831 T^{6} + 2836022 T^{7} + 746831 p T^{8} + 2627 p^{3} T^{9} + 16017 p^{3} T^{10} + 1174 p^{4} T^{11} + 195 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 - 7 T + 103 T^{2} - 752 T^{3} + 6848 T^{4} - 42865 T^{5} + 308936 T^{6} - 1562360 T^{7} + 308936 p T^{8} - 42865 p^{2} T^{9} + 6848 p^{3} T^{10} - 752 p^{4} T^{11} + 103 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 + 5 T + 213 T^{2} + 1068 T^{3} + 20860 T^{4} + 95468 T^{5} + 1212318 T^{6} + 4647094 T^{7} + 1212318 p T^{8} + 95468 p^{2} T^{9} + 20860 p^{3} T^{10} + 1068 p^{4} T^{11} + 213 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 - 7 T + 201 T^{2} - 38 p T^{3} + 20510 T^{4} - 142227 T^{5} + 1320572 T^{6} - 7391784 T^{7} + 1320572 p T^{8} - 142227 p^{2} T^{9} + 20510 p^{3} T^{10} - 38 p^{5} T^{11} + 201 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 - 3 T + 211 T^{2} - 416 T^{3} + 19878 T^{4} - 22931 T^{5} + 1168364 T^{6} - 911236 T^{7} + 1168364 p T^{8} - 22931 p^{2} T^{9} + 19878 p^{3} T^{10} - 416 p^{4} T^{11} + 211 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 + 6 T + 281 T^{2} + 1420 T^{3} + 35583 T^{4} + 150355 T^{5} + 2641311 T^{6} + 9105486 T^{7} + 2641311 p T^{8} + 150355 p^{2} T^{9} + 35583 p^{3} T^{10} + 1420 p^{4} T^{11} + 281 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 9 T + 296 T^{2} + 2235 T^{3} + 39989 T^{4} + 254943 T^{5} + 3255898 T^{6} + 17152618 T^{7} + 3255898 p T^{8} + 254943 p^{2} T^{9} + 39989 p^{3} T^{10} + 2235 p^{4} T^{11} + 296 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 + 22 T + 499 T^{2} + 6902 T^{3} + 1561 p T^{4} + 937655 T^{5} + 9110405 T^{6} + 71721796 T^{7} + 9110405 p T^{8} + 937655 p^{2} T^{9} + 1561 p^{4} T^{10} + 6902 p^{4} T^{11} + 499 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 - 19 T + 472 T^{2} - 101 p T^{3} + 88213 T^{4} - 867977 T^{5} + 8973786 T^{6} - 68695318 T^{7} + 8973786 p T^{8} - 867977 p^{2} T^{9} + 88213 p^{3} T^{10} - 101 p^{5} T^{11} + 472 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 + 253 T^{2} - 419 T^{3} + 30766 T^{4} - 116730 T^{5} + 2499824 T^{6} - 12048814 T^{7} + 2499824 p T^{8} - 116730 p^{2} T^{9} + 30766 p^{3} T^{10} - 419 p^{4} T^{11} + 253 p^{5} T^{12} + p^{7} T^{14} \)
73 \( 1 - 23 T + 289 T^{2} - 1759 T^{3} + 6547 T^{4} - 96526 T^{5} + 2159079 T^{6} - 25271472 T^{7} + 2159079 p T^{8} - 96526 p^{2} T^{9} + 6547 p^{3} T^{10} - 1759 p^{4} T^{11} + 289 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 - 25 T + 651 T^{2} - 9530 T^{3} + 142430 T^{4} - 1499035 T^{5} + 16738302 T^{6} - 142494620 T^{7} + 16738302 p T^{8} - 1499035 p^{2} T^{9} + 142430 p^{3} T^{10} - 9530 p^{4} T^{11} + 651 p^{5} T^{12} - 25 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 + 20 T + 403 T^{2} + 4687 T^{3} + 61584 T^{4} + 626492 T^{5} + 7349918 T^{6} + 65413042 T^{7} + 7349918 p T^{8} + 626492 p^{2} T^{9} + 61584 p^{3} T^{10} + 4687 p^{4} T^{11} + 403 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 - T + 533 T^{2} - 347 T^{3} + 127527 T^{4} - 53386 T^{5} + 17869735 T^{6} - 5417092 T^{7} + 17869735 p T^{8} - 53386 p^{2} T^{9} + 127527 p^{3} T^{10} - 347 p^{4} T^{11} + 533 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 3 T + 454 T^{2} - 1093 T^{3} + 99495 T^{4} - 175093 T^{5} + 13812188 T^{6} - 19147754 T^{7} + 13812188 p T^{8} - 175093 p^{2} T^{9} + 99495 p^{3} T^{10} - 1093 p^{4} T^{11} + 454 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
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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

−3.85842850895220614573597254189, −3.84741834992210902815834000185, −3.78332671669224753064924167408, −3.77585172840126124846653963297, −3.17872858373228882626888283002, −3.15655268042277043597066463787, −3.07845985569576340882543261462, −3.06772142907504727712472574836, −3.06264264788907232673074317661, −3.04783235279062203807698693167, −2.66126185372537364006075635175, −2.12903543024031688994579914390, −2.06240197009087050248387806225, −1.93514291458683512039310063849, −1.90696024646604125346164910534, −1.69768034736087922304409793417, −1.65865349923133155144319986257, −1.58778804048236144925447393457, −1.00640211261018419478951482376, −0.862683541147386375001852126420, −0.857832763900832520751086018599, −0.57926817496702664226590570825, −0.55417827886987696585399934903, −0.51188023935418826187233209036, −0.20940619156881488401705517987, 0.20940619156881488401705517987, 0.51188023935418826187233209036, 0.55417827886987696585399934903, 0.57926817496702664226590570825, 0.857832763900832520751086018599, 0.862683541147386375001852126420, 1.00640211261018419478951482376, 1.58778804048236144925447393457, 1.65865349923133155144319986257, 1.69768034736087922304409793417, 1.90696024646604125346164910534, 1.93514291458683512039310063849, 2.06240197009087050248387806225, 2.12903543024031688994579914390, 2.66126185372537364006075635175, 3.04783235279062203807698693167, 3.06264264788907232673074317661, 3.06772142907504727712472574836, 3.07845985569576340882543261462, 3.15655268042277043597066463787, 3.17872858373228882626888283002, 3.77585172840126124846653963297, 3.78332671669224753064924167408, 3.84741834992210902815834000185, 3.85842850895220614573597254189

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

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