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Properties

Label 14-8030e7-1.1-c1e7-0-0
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 − 3·7-s − 84·8-s − 6·9-s − 49·10-s − 7·11-s − 56·12-s − 3·13-s + 21·14-s − 14·15-s + 210·16-s − 11·17-s + 42·18-s − 8·19-s + 196·20-s + 6·21-s + 49·22-s + 8·23-s + 168·24-s + 28·25-s + 21·26-s + 13·27-s − 84·28-s + ⋯
L(s)  = 1  − 4.94·2-s − 1.15·3-s + 14·4-s + 3.13·5-s + 5.71·6-s − 1.13·7-s − 29.6·8-s − 2·9-s − 15.4·10-s − 2.11·11-s − 16.1·12-s − 0.832·13-s + 5.61·14-s − 3.61·15-s + 52.5·16-s − 2.66·17-s + 9.89·18-s − 1.83·19-s + 43.8·20-s + 1.30·21-s + 10.4·22-s + 1.66·23-s + 34.2·24-s + 28/5·25-s + 4.11·26-s + 2.50·27-s − 15.8·28-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 \( 1 + 2 T + 10 T^{2} + 19 T^{3} + 49 T^{4} + 25 p T^{5} + 166 T^{6} + 73 p T^{7} + 166 p T^{8} + 25 p^{3} T^{9} + 49 p^{3} T^{10} + 19 p^{4} T^{11} + 10 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \)
7 \( 1 + 3 T + 36 T^{2} + 99 T^{3} + 627 T^{4} + 1494 T^{5} + 6714 T^{6} + 13276 T^{7} + 6714 p T^{8} + 1494 p^{2} T^{9} + 627 p^{3} T^{10} + 99 p^{4} T^{11} + 36 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 + 3 T + 51 T^{2} + 213 T^{3} + 1467 T^{4} + 5628 T^{5} + 29676 T^{6} + 6758 p T^{7} + 29676 p T^{8} + 5628 p^{2} T^{9} + 1467 p^{3} T^{10} + 213 p^{4} T^{11} + 51 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 11 T + 118 T^{2} + 933 T^{3} + 6131 T^{4} + 35038 T^{5} + 176338 T^{6} + 760152 T^{7} + 176338 p T^{8} + 35038 p^{2} T^{9} + 6131 p^{3} T^{10} + 933 p^{4} T^{11} + 118 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 8 T + 81 T^{2} + 417 T^{3} + 2629 T^{4} + 11788 T^{5} + 3407 p T^{6} + 263970 T^{7} + 3407 p^{2} T^{8} + 11788 p^{2} T^{9} + 2629 p^{3} T^{10} + 417 p^{4} T^{11} + 81 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 8 T + 32 T^{2} - 168 T^{3} + 1683 T^{4} - 10398 T^{5} + 42460 T^{6} - 173612 T^{7} + 42460 p T^{8} - 10398 p^{2} T^{9} + 1683 p^{3} T^{10} - 168 p^{4} T^{11} + 32 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 9 T + 156 T^{2} + 947 T^{3} + 9101 T^{4} + 1384 p T^{5} + 306396 T^{6} + 1174192 T^{7} + 306396 p T^{8} + 1384 p^{3} T^{9} + 9101 p^{3} T^{10} + 947 p^{4} T^{11} + 156 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 + 9 T + 153 T^{2} + 1104 T^{3} + 11517 T^{4} + 68331 T^{5} + 537021 T^{6} + 85304 p T^{7} + 537021 p T^{8} + 68331 p^{2} T^{9} + 11517 p^{3} T^{10} + 1104 p^{4} T^{11} + 153 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 - 21 T + 286 T^{2} - 3047 T^{3} + 27561 T^{4} - 218540 T^{5} + 1562772 T^{6} - 10003200 T^{7} + 1562772 p T^{8} - 218540 p^{2} T^{9} + 27561 p^{3} T^{10} - 3047 p^{4} T^{11} + 286 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 10 T + 125 T^{2} + 1320 T^{3} + 9455 T^{4} + 74846 T^{5} + 537587 T^{6} + 3074976 T^{7} + 537587 p T^{8} + 74846 p^{2} T^{9} + 9455 p^{3} T^{10} + 1320 p^{4} T^{11} + 125 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + 5 T + 140 T^{2} + 606 T^{3} + 10010 T^{4} + 32755 T^{5} + 502547 T^{6} + 1331780 T^{7} + 502547 p T^{8} + 32755 p^{2} T^{9} + 10010 p^{3} T^{10} + 606 p^{4} T^{11} + 140 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 19 T + 344 T^{2} - 3976 T^{3} + 44346 T^{4} - 384357 T^{5} + 3243187 T^{6} - 22473760 T^{7} + 3243187 p T^{8} - 384357 p^{2} T^{9} + 44346 p^{3} T^{10} - 3976 p^{4} T^{11} + 344 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 13 T + 291 T^{2} + 3246 T^{3} + 39721 T^{4} + 377495 T^{5} + 3259579 T^{6} + 25546588 T^{7} + 3259579 p T^{8} + 377495 p^{2} T^{9} + 39721 p^{3} T^{10} + 3246 p^{4} T^{11} + 291 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 - 5 T + 216 T^{2} - 762 T^{3} + 21288 T^{4} - 45067 T^{5} + 1404017 T^{6} - 1955428 T^{7} + 1404017 p T^{8} - 45067 p^{2} T^{9} + 21288 p^{3} T^{10} - 762 p^{4} T^{11} + 216 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 + 12 T + 439 T^{2} + 4147 T^{3} + 81185 T^{4} + 610212 T^{5} + 8278983 T^{6} + 49031670 T^{7} + 8278983 p T^{8} + 610212 p^{2} T^{9} + 81185 p^{3} T^{10} + 4147 p^{4} T^{11} + 439 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 + 23 T + 641 T^{2} + 9711 T^{3} + 151955 T^{4} + 1671970 T^{5} + 18259586 T^{6} + 150693512 T^{7} + 18259586 p T^{8} + 1671970 p^{2} T^{9} + 151955 p^{3} T^{10} + 9711 p^{4} T^{11} + 641 p^{5} T^{12} + 23 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 22 T + 384 T^{2} - 5255 T^{3} + 66887 T^{4} - 702007 T^{5} + 7071852 T^{6} - 61995037 T^{7} + 7071852 p T^{8} - 702007 p^{2} T^{9} + 66887 p^{3} T^{10} - 5255 p^{4} T^{11} + 384 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 + 32 T + 889 T^{2} + 200 p T^{3} + 252693 T^{4} + 3128704 T^{5} + 35529917 T^{6} + 328848336 T^{7} + 35529917 p T^{8} + 3128704 p^{2} T^{9} + 252693 p^{3} T^{10} + 200 p^{5} T^{11} + 889 p^{5} T^{12} + 32 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 - T + 327 T^{2} + 441 T^{3} + 52315 T^{4} + 116476 T^{5} + 5958442 T^{6} + 11988110 T^{7} + 5958442 p T^{8} + 116476 p^{2} T^{9} + 52315 p^{3} T^{10} + 441 p^{4} T^{11} + 327 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 - 20 T + 418 T^{2} - 6459 T^{3} + 95339 T^{4} - 1113667 T^{5} + 12736072 T^{6} - 124045027 T^{7} + 12736072 p T^{8} - 1113667 p^{2} T^{9} + 95339 p^{3} T^{10} - 6459 p^{4} T^{11} + 418 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 + 2 T + 288 T^{2} - 1206 T^{3} + 34063 T^{4} - 476966 T^{5} + 2341844 T^{6} - 67629788 T^{7} + 2341844 p T^{8} - 476966 p^{2} T^{9} + 34063 p^{3} T^{10} - 1206 p^{4} T^{11} + 288 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \)
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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

−3.68676184134116822520205572118, −3.61263675165942453348652111842, −3.50228971476183962734292657726, −3.46758551253509801381466547919, −3.16087372670995341664637335131, −3.00719404703916640093969032845, −2.95763485025048147613454798466, −2.92627728661700255386501682717, −2.76802923144351855595159906711, −2.76155205308752094831576722118, −2.74813547576369683585492038264, −2.35316270191351374435935614608, −2.21251161401874244472290456146, −2.13450764916461636238488152394, −2.12796233299484660245309574629, −2.07268994327572177848109156036, −2.06723838568243500925784825217, −2.01401595009666316470154500929, −1.69385371585986478085208140449, −1.31242738740780204691487145447, −1.31100055865281995257273272078, −1.15076027066879326773032718148, −1.07706981266973021538952561315, −0.996223619222238923874939006055, −0.74300326349841667724711114908, 0, 0, 0, 0, 0, 0, 0, 0.74300326349841667724711114908, 0.996223619222238923874939006055, 1.07706981266973021538952561315, 1.15076027066879326773032718148, 1.31100055865281995257273272078, 1.31242738740780204691487145447, 1.69385371585986478085208140449, 2.01401595009666316470154500929, 2.06723838568243500925784825217, 2.07268994327572177848109156036, 2.12796233299484660245309574629, 2.13450764916461636238488152394, 2.21251161401874244472290456146, 2.35316270191351374435935614608, 2.74813547576369683585492038264, 2.76155205308752094831576722118, 2.76802923144351855595159906711, 2.92627728661700255386501682717, 2.95763485025048147613454798466, 3.00719404703916640093969032845, 3.16087372670995341664637335131, 3.46758551253509801381466547919, 3.50228971476183962734292657726, 3.61263675165942453348652111842, 3.68676184134116822520205572118

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

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