Properties

Label 14-4014e7-1.1-c1e7-0-1
Degree $14$
Conductor $1.679\times 10^{25}$
Sign $1$
Analytic cond. $3.47521\times 10^{10}$
Root an. cond. $5.66144$
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  − 7·2-s + 28·4-s − 6·5-s + 3·7-s − 84·8-s + 42·10-s + 11-s + 8·13-s − 21·14-s + 210·16-s − 16·17-s + 2·19-s − 168·20-s − 7·22-s − 8·23-s + 10·25-s − 56·26-s + 84·28-s − 4·29-s + 11·31-s − 462·32-s + 112·34-s − 18·35-s + 17·37-s − 14·38-s + 504·40-s − 18·41-s + ⋯
L(s)  = 1  − 4.94·2-s + 14·4-s − 2.68·5-s + 1.13·7-s − 29.6·8-s + 13.2·10-s + 0.301·11-s + 2.21·13-s − 5.61·14-s + 52.5·16-s − 3.88·17-s + 0.458·19-s − 37.5·20-s − 1.49·22-s − 1.66·23-s + 2·25-s − 10.9·26-s + 15.8·28-s − 0.742·29-s + 1.97·31-s − 81.6·32-s + 19.2·34-s − 3.04·35-s + 2.79·37-s − 2.27·38-s + 79.6·40-s − 2.81·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{14} \cdot 223^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{14} \cdot 223^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(2^{7} \cdot 3^{14} \cdot 223^{7}\)
Sign: $1$
Analytic conductor: \(3.47521\times 10^{10}\)
Root analytic conductor: \(5.66144\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((14,\ 2^{7} \cdot 3^{14} \cdot 223^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5546405301\)
\(L(\frac12)\) \(\approx\) \(0.5546405301\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{7} \)
3 \( 1 \)
223 \( ( 1 + T )^{7} \)
good5 \( 1 + 6 T + 26 T^{2} + 3 p^{2} T^{3} + 209 T^{4} + 466 T^{5} + 1064 T^{6} + 2146 T^{7} + 1064 p T^{8} + 466 p^{2} T^{9} + 209 p^{3} T^{10} + 3 p^{6} T^{11} + 26 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \)
7 \( 1 - 3 T + 17 T^{2} - 25 T^{3} + 19 p T^{4} - 113 T^{5} + 83 p T^{6} + 346 T^{7} + 83 p^{2} T^{8} - 113 p^{2} T^{9} + 19 p^{4} T^{10} - 25 p^{4} T^{11} + 17 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 - T + 29 T^{2} - 58 T^{3} + 525 T^{4} - 1158 T^{5} + 6713 T^{6} - 16614 T^{7} + 6713 p T^{8} - 1158 p^{2} T^{9} + 525 p^{3} T^{10} - 58 p^{4} T^{11} + 29 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 - 8 T + 87 T^{2} - 534 T^{3} + 3352 T^{4} - 15709 T^{5} + 72570 T^{6} - 263078 T^{7} + 72570 p T^{8} - 15709 p^{2} T^{9} + 3352 p^{3} T^{10} - 534 p^{4} T^{11} + 87 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 16 T + 182 T^{2} + 1459 T^{3} + 9888 T^{4} + 55744 T^{5} + 279481 T^{6} + 1215546 T^{7} + 279481 p T^{8} + 55744 p^{2} T^{9} + 9888 p^{3} T^{10} + 1459 p^{4} T^{11} + 182 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 - 2 T + 3 p T^{2} + 30 T^{3} + 1522 T^{4} + 1831 T^{5} + 43250 T^{6} + 14418 T^{7} + 43250 p T^{8} + 1831 p^{2} T^{9} + 1522 p^{3} T^{10} + 30 p^{4} T^{11} + 3 p^{6} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 + 8 T + 94 T^{2} + 389 T^{3} + 2978 T^{4} + 6480 T^{5} + 54941 T^{6} + 63078 T^{7} + 54941 p T^{8} + 6480 p^{2} T^{9} + 2978 p^{3} T^{10} + 389 p^{4} T^{11} + 94 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 4 T + 101 T^{2} + 216 T^{3} + 2780 T^{4} - 3931 T^{5} - 11736 T^{6} - 397206 T^{7} - 11736 p T^{8} - 3931 p^{2} T^{9} + 2780 p^{3} T^{10} + 216 p^{4} T^{11} + 101 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 - 11 T + 167 T^{2} - 1267 T^{3} + 11993 T^{4} - 74685 T^{5} + 552771 T^{6} - 2866186 T^{7} + 552771 p T^{8} - 74685 p^{2} T^{9} + 11993 p^{3} T^{10} - 1267 p^{4} T^{11} + 167 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 - 17 T + 280 T^{2} - 2814 T^{3} + 27841 T^{4} - 212197 T^{5} + 1603258 T^{6} - 9844176 T^{7} + 1603258 p T^{8} - 212197 p^{2} T^{9} + 27841 p^{3} T^{10} - 2814 p^{4} T^{11} + 280 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 18 T + 244 T^{2} + 2031 T^{3} + 15320 T^{4} + 87734 T^{5} + 576219 T^{6} + 3259314 T^{7} + 576219 p T^{8} + 87734 p^{2} T^{9} + 15320 p^{3} T^{10} + 2031 p^{4} T^{11} + 244 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + T + 19 T^{2} + 56 T^{3} + 1695 T^{4} - 6240 T^{5} - 6539 T^{6} - 47162 T^{7} - 6539 p T^{8} - 6240 p^{2} T^{9} + 1695 p^{3} T^{10} + 56 p^{4} T^{11} + 19 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 + 5 T + 61 T^{2} - 128 T^{3} + 2482 T^{4} - 14063 T^{5} + 41963 T^{6} - 1627436 T^{7} + 41963 p T^{8} - 14063 p^{2} T^{9} + 2482 p^{3} T^{10} - 128 p^{4} T^{11} + 61 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 6 T + 189 T^{2} + 1106 T^{3} + 17250 T^{4} + 106541 T^{5} + 1103088 T^{6} + 6768406 T^{7} + 1103088 p T^{8} + 106541 p^{2} T^{9} + 17250 p^{3} T^{10} + 1106 p^{4} T^{11} + 189 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 - 7 T + 221 T^{2} - 1540 T^{3} + 25061 T^{4} - 148516 T^{5} + 1965993 T^{6} - 9737770 T^{7} + 1965993 p T^{8} - 148516 p^{2} T^{9} + 25061 p^{3} T^{10} - 1540 p^{4} T^{11} + 221 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 - 24 T + 565 T^{2} - 8504 T^{3} + 115340 T^{4} - 1255803 T^{5} + 12166384 T^{6} - 100810822 T^{7} + 12166384 p T^{8} - 1255803 p^{2} T^{9} + 115340 p^{3} T^{10} - 8504 p^{4} T^{11} + 565 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 + 4 T + 334 T^{2} + 1393 T^{3} + 54365 T^{4} + 208604 T^{5} + 5511936 T^{6} + 17838814 T^{7} + 5511936 p T^{8} + 208604 p^{2} T^{9} + 54365 p^{3} T^{10} + 1393 p^{4} T^{11} + 334 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 7 T + 304 T^{2} - 2759 T^{3} + 42642 T^{4} - 454357 T^{5} + 3931755 T^{6} - 41713722 T^{7} + 3931755 p T^{8} - 454357 p^{2} T^{9} + 42642 p^{3} T^{10} - 2759 p^{4} T^{11} + 304 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 - 28 T + 674 T^{2} - 10822 T^{3} + 155858 T^{4} - 1792357 T^{5} + 18944622 T^{6} - 168136784 T^{7} + 18944622 p T^{8} - 1792357 p^{2} T^{9} + 155858 p^{3} T^{10} - 10822 p^{4} T^{11} + 674 p^{5} T^{12} - 28 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 + 278 T^{2} + 350 T^{3} + 42298 T^{4} + 68845 T^{5} + 4389542 T^{6} + 7723498 T^{7} + 4389542 p T^{8} + 68845 p^{2} T^{9} + 42298 p^{3} T^{10} + 350 p^{4} T^{11} + 278 p^{5} T^{12} + p^{7} T^{14} \)
83 \( 1 - 2 T + 124 T^{2} - 227 T^{3} + 9871 T^{4} - 46034 T^{5} + 748440 T^{6} - 4215522 T^{7} + 748440 p T^{8} - 46034 p^{2} T^{9} + 9871 p^{3} T^{10} - 227 p^{4} T^{11} + 124 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 + 5 T + 427 T^{2} + 1871 T^{3} + 90273 T^{4} + 340543 T^{5} + 11970235 T^{6} + 38144970 T^{7} + 11970235 p T^{8} + 340543 p^{2} T^{9} + 90273 p^{3} T^{10} + 1871 p^{4} T^{11} + 427 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 23 T + 477 T^{2} - 5871 T^{3} + 66981 T^{4} - 495073 T^{5} + 4100541 T^{6} - 25668914 T^{7} + 4100541 p T^{8} - 495073 p^{2} T^{9} + 66981 p^{3} T^{10} - 5871 p^{4} T^{11} + 477 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \)
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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

−3.77904802022638869271309710644, −3.76174406293905721617945093936, −3.67057132885269572661276924746, −3.62556165480323439137946115095, −3.09981768518117415271961038573, −3.07499336586676669807746869374, −3.01975942221540378859808263982, −2.72644876501465259074245770341, −2.71916242884844132419215537831, −2.65413020299433357294859817626, −2.51249333800936927383173619194, −2.07560332492089614934245704569, −1.98272686533307730611669891834, −1.86510385555534692843195022739, −1.81038240964575695895625880099, −1.80698609402567078853501296222, −1.77707038813988461673590633032, −1.56034687897262026486875557278, −1.13313378646169215926377379562, −0.923891473379773069137066170802, −0.866666945853861052927049627336, −0.52592987243098291179454690012, −0.44695084338139164166953540236, −0.43496941763941166322172235464, −0.37957003176339872126792791663, 0.37957003176339872126792791663, 0.43496941763941166322172235464, 0.44695084338139164166953540236, 0.52592987243098291179454690012, 0.866666945853861052927049627336, 0.923891473379773069137066170802, 1.13313378646169215926377379562, 1.56034687897262026486875557278, 1.77707038813988461673590633032, 1.80698609402567078853501296222, 1.81038240964575695895625880099, 1.86510385555534692843195022739, 1.98272686533307730611669891834, 2.07560332492089614934245704569, 2.51249333800936927383173619194, 2.65413020299433357294859817626, 2.71916242884844132419215537831, 2.72644876501465259074245770341, 3.01975942221540378859808263982, 3.07499336586676669807746869374, 3.09981768518117415271961038573, 3.62556165480323439137946115095, 3.67057132885269572661276924746, 3.76174406293905721617945093936, 3.77904802022638869271309710644

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

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