Properties

Degree $14$
Conductor $9.598\times 10^{23}$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $7$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 7·3-s − 3·4-s − 8·5-s − 14·6-s + 7·7-s + 7·8-s + 28·9-s + 16·10-s − 3·11-s − 21·12-s − 23·13-s − 14·14-s − 56·15-s + 5·16-s + 3·17-s − 56·18-s − 9·19-s + 24·20-s + 49·21-s + 6·22-s + 12·23-s + 49·24-s + 16·25-s + 46·26-s + 84·27-s − 21·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 4.04·3-s − 3/2·4-s − 3.57·5-s − 5.71·6-s + 2.64·7-s + 2.47·8-s + 28/3·9-s + 5.05·10-s − 0.904·11-s − 6.06·12-s − 6.37·13-s − 3.74·14-s − 14.4·15-s + 5/4·16-s + 0.727·17-s − 13.1·18-s − 2.06·19-s + 5.36·20-s + 10.6·21-s + 1.27·22-s + 2.50·23-s + 10.0·24-s + 16/5·25-s + 9.02·26-s + 16.1·27-s − 3.96·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 7^{7} \cdot 127^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 7^{7} \cdot 127^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(3^{7} \cdot 7^{7} \cdot 127^{7}\)
Sign: $-1$
Motivic weight: \(1\)
Character: induced by $\chi_{2667} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(7\)
Selberg data: \((14,\ 3^{7} \cdot 7^{7} \cdot 127^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\)

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 \( ( 1 - T )^{7} \)
7 \( ( 1 - T )^{7} \)
127 \( ( 1 + T )^{7} \)
good2 \( 1 + p T + 7 T^{2} + 13 T^{3} + 7 p^{2} T^{4} + 41 T^{5} + 73 T^{6} + 47 p T^{7} + 73 p T^{8} + 41 p^{2} T^{9} + 7 p^{5} T^{10} + 13 p^{4} T^{11} + 7 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \)
5 \( 1 + 8 T + 48 T^{2} + 198 T^{3} + 701 T^{4} + 2022 T^{5} + 5344 T^{6} + 2463 p T^{7} + 5344 p T^{8} + 2022 p^{2} T^{9} + 701 p^{3} T^{10} + 198 p^{4} T^{11} + 48 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 + 3 T + 31 T^{2} + 93 T^{3} + 656 T^{4} + 1715 T^{5} + 9199 T^{6} + 22080 T^{7} + 9199 p T^{8} + 1715 p^{2} T^{9} + 656 p^{3} T^{10} + 93 p^{4} T^{11} + 31 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 + 23 T + 289 T^{2} + 2532 T^{3} + 1316 p T^{4} + 93828 T^{5} + 430561 T^{6} + 1679255 T^{7} + 430561 p T^{8} + 93828 p^{2} T^{9} + 1316 p^{4} T^{10} + 2532 p^{4} T^{11} + 289 p^{5} T^{12} + 23 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 - 3 T + 50 T^{2} - 146 T^{3} + 1373 T^{4} - 4645 T^{5} + 30241 T^{6} - 99122 T^{7} + 30241 p T^{8} - 4645 p^{2} T^{9} + 1373 p^{3} T^{10} - 146 p^{4} T^{11} + 50 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 9 T + 127 T^{2} + 828 T^{3} + 6701 T^{4} + 34195 T^{5} + 201852 T^{6} + 825304 T^{7} + 201852 p T^{8} + 34195 p^{2} T^{9} + 6701 p^{3} T^{10} + 828 p^{4} T^{11} + 127 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 12 T + 150 T^{2} - 1067 T^{3} + 8328 T^{4} - 46995 T^{5} + 292164 T^{6} - 1356265 T^{7} + 292164 p T^{8} - 46995 p^{2} T^{9} + 8328 p^{3} T^{10} - 1067 p^{4} T^{11} + 150 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 9 T + 107 T^{2} + 652 T^{3} + 4764 T^{4} + 22810 T^{5} + 134487 T^{6} + 612145 T^{7} + 134487 p T^{8} + 22810 p^{2} T^{9} + 4764 p^{3} T^{10} + 652 p^{4} T^{11} + 107 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 + 33 T + 645 T^{2} + 8883 T^{3} + 95284 T^{4} + 826887 T^{5} + 5965621 T^{6} + 36144101 T^{7} + 5965621 p T^{8} + 826887 p^{2} T^{9} + 95284 p^{3} T^{10} + 8883 p^{4} T^{11} + 645 p^{5} T^{12} + 33 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 + 33 T + 653 T^{2} + 9315 T^{3} + 105122 T^{4} + 971473 T^{5} + 7548399 T^{6} + 49726081 T^{7} + 7548399 p T^{8} + 971473 p^{2} T^{9} + 105122 p^{3} T^{10} + 9315 p^{4} T^{11} + 653 p^{5} T^{12} + 33 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 3 T + 170 T^{2} + 50 T^{3} + 11626 T^{4} - 29278 T^{5} + 496448 T^{6} - 2150172 T^{7} + 496448 p T^{8} - 29278 p^{2} T^{9} + 11626 p^{3} T^{10} + 50 p^{4} T^{11} + 170 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + 9 T + 217 T^{2} + 1559 T^{3} + 22058 T^{4} + 131173 T^{5} + 1390533 T^{6} + 6894030 T^{7} + 1390533 p T^{8} + 131173 p^{2} T^{9} + 22058 p^{3} T^{10} + 1559 p^{4} T^{11} + 217 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 11 T + 145 T^{2} - 1339 T^{3} + 14705 T^{4} - 119758 T^{5} + 953008 T^{6} - 6346454 T^{7} + 953008 p T^{8} - 119758 p^{2} T^{9} + 14705 p^{3} T^{10} - 1339 p^{4} T^{11} + 145 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 - T + 216 T^{2} + 139 T^{3} + 22869 T^{4} + 30467 T^{5} + 1664224 T^{6} + 2175415 T^{7} + 1664224 p T^{8} + 30467 p^{2} T^{9} + 22869 p^{3} T^{10} + 139 p^{4} T^{11} + 216 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 + 30 T + 570 T^{2} + 7389 T^{3} + 75460 T^{4} + 617301 T^{5} + 4593588 T^{6} + 33613111 T^{7} + 4593588 p T^{8} + 617301 p^{2} T^{9} + 75460 p^{3} T^{10} + 7389 p^{4} T^{11} + 570 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 + 19 T + 7 p T^{2} + 5649 T^{3} + 75542 T^{4} + 765723 T^{5} + 7510227 T^{6} + 59966461 T^{7} + 7510227 p T^{8} + 765723 p^{2} T^{9} + 75542 p^{3} T^{10} + 5649 p^{4} T^{11} + 7 p^{6} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 + 30 T + 675 T^{2} + 11360 T^{3} + 157485 T^{4} + 1838995 T^{5} + 18608402 T^{6} + 162480644 T^{7} + 18608402 p T^{8} + 1838995 p^{2} T^{9} + 157485 p^{3} T^{10} + 11360 p^{4} T^{11} + 675 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 8 T + 246 T^{2} - 1787 T^{3} + 34468 T^{4} - 225343 T^{5} + 3268094 T^{6} - 18726838 T^{7} + 3268094 p T^{8} - 225343 p^{2} T^{9} + 34468 p^{3} T^{10} - 1787 p^{4} T^{11} + 246 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 + 20 T + 510 T^{2} + 6961 T^{3} + 102634 T^{4} + 1069123 T^{5} + 11655826 T^{6} + 97527765 T^{7} + 11655826 p T^{8} + 1069123 p^{2} T^{9} + 102634 p^{3} T^{10} + 6961 p^{4} T^{11} + 510 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 - 8 T + 273 T^{2} - 2858 T^{3} + 45901 T^{4} - 424499 T^{5} + 5436714 T^{6} - 39832374 T^{7} + 5436714 p T^{8} - 424499 p^{2} T^{9} + 45901 p^{3} T^{10} - 2858 p^{4} T^{11} + 273 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 + 34 T + 962 T^{2} + 18636 T^{3} + 304656 T^{4} + 4073018 T^{5} + 46715214 T^{6} + 458242049 T^{7} + 46715214 p T^{8} + 4073018 p^{2} T^{9} + 304656 p^{3} T^{10} + 18636 p^{4} T^{11} + 962 p^{5} T^{12} + 34 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 + 12 T + 151 T^{2} + 2659 T^{3} + 35240 T^{4} + 317069 T^{5} + 3805221 T^{6} + 40673821 T^{7} + 3805221 p T^{8} + 317069 p^{2} T^{9} + 35240 p^{3} T^{10} + 2659 p^{4} T^{11} + 151 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 7 T + 326 T^{2} - 2842 T^{3} + 55405 T^{4} - 456619 T^{5} + 7188211 T^{6} - 47880078 T^{7} + 7188211 p T^{8} - 456619 p^{2} T^{9} + 55405 p^{3} T^{10} - 2842 p^{4} T^{11} + 326 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
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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

−4.48019072528173150371199271021, −4.36664316523829153446406684357, −4.36235910045696442876869895615, −4.18486869708857855341031591002, −3.82083415866875313715198795380, −3.76081292997925957555670404027, −3.70940389272013307579683991354, −3.64286368313824415727375722646, −3.63321525868860762612857970041, −3.32042580854930806756995235928, −3.23898604671392589707376902050, −3.11133182178127022360390571296, −3.06076565460433692811602790337, −2.73082487966252966850486220613, −2.57028550379554700689755255949, −2.50431225623514826276977388219, −2.35067960140450992167580650730, −2.18132552610136066356746270128, −2.08995788913651934432165487272, −2.04887033637200668197075951327, −1.59943812692071389512082881175, −1.51945173405302223573002331817, −1.45554395550009807406699085580, −1.40355511455004245193478482159, −1.24025408756021894088158879940, 0, 0, 0, 0, 0, 0, 0, 1.24025408756021894088158879940, 1.40355511455004245193478482159, 1.45554395550009807406699085580, 1.51945173405302223573002331817, 1.59943812692071389512082881175, 2.04887033637200668197075951327, 2.08995788913651934432165487272, 2.18132552610136066356746270128, 2.35067960140450992167580650730, 2.50431225623514826276977388219, 2.57028550379554700689755255949, 2.73082487966252966850486220613, 3.06076565460433692811602790337, 3.11133182178127022360390571296, 3.23898604671392589707376902050, 3.32042580854930806756995235928, 3.63321525868860762612857970041, 3.64286368313824415727375722646, 3.70940389272013307579683991354, 3.76081292997925957555670404027, 3.82083415866875313715198795380, 4.18486869708857855341031591002, 4.36235910045696442876869895615, 4.36664316523829153446406684357, 4.48019072528173150371199271021

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

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