Properties

Label 26-6040e13-1.1-c1e13-0-0
Degree $26$
Conductor $1.424\times 10^{49}$
Sign $-1$
Analytic cond. $7.63981\times 10^{21}$
Root an. cond. $6.94475$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $13$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 13·5-s − 9·9-s − 14·11-s + 5·13-s + 52·15-s − 8·17-s + 16·19-s − 4·23-s + 91·25-s + 62·27-s − 6·29-s + 11·31-s + 56·33-s + 6·37-s − 20·39-s − 18·41-s + 7·43-s + 117·45-s − 22·47-s − 46·49-s + 32·51-s − 17·53-s + 182·55-s − 64·57-s − 6·59-s + 10·61-s + ⋯
L(s)  = 1  − 2.30·3-s − 5.81·5-s − 3·9-s − 4.22·11-s + 1.38·13-s + 13.4·15-s − 1.94·17-s + 3.67·19-s − 0.834·23-s + 91/5·25-s + 11.9·27-s − 1.11·29-s + 1.97·31-s + 9.74·33-s + 0.986·37-s − 3.20·39-s − 2.81·41-s + 1.06·43-s + 17.4·45-s − 3.20·47-s − 6.57·49-s + 4.48·51-s − 2.33·53-s + 24.5·55-s − 8.47·57-s − 0.781·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{39} \cdot 5^{13} \cdot 151^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{39} \cdot 5^{13} \cdot 151^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(26\)
Conductor: \(2^{39} \cdot 5^{13} \cdot 151^{13}\)
Sign: $-1$
Analytic conductor: \(7.63981\times 10^{21}\)
Root analytic conductor: \(6.94475\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(13\)
Selberg data: \((26,\ 2^{39} \cdot 5^{13} \cdot 151^{13} ,\ ( \ : [1/2]^{13} ),\ -1 )\)

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 \( 1 \)
5 \( ( 1 + T )^{13} \)
151 \( ( 1 - T )^{13} \)
good3 \( 1 + 4 T + 25 T^{2} + 74 T^{3} + 281 T^{4} + 679 T^{5} + 2008 T^{6} + 52 p^{4} T^{7} + 1186 p^{2} T^{8} + 6728 p T^{9} + 45823 T^{10} + 8822 p^{2} T^{11} + 164006 T^{12} + 260647 T^{13} + 164006 p T^{14} + 8822 p^{4} T^{15} + 45823 p^{3} T^{16} + 6728 p^{5} T^{17} + 1186 p^{7} T^{18} + 52 p^{10} T^{19} + 2008 p^{7} T^{20} + 679 p^{8} T^{21} + 281 p^{9} T^{22} + 74 p^{10} T^{23} + 25 p^{11} T^{24} + 4 p^{12} T^{25} + p^{13} T^{26} \)
7 \( 1 + 46 T^{2} + 3 T^{3} + 1095 T^{4} + 241 T^{5} + 17737 T^{6} + 7118 T^{7} + 217636 T^{8} + 121206 T^{9} + 2147799 T^{10} + 1391743 T^{11} + 17692018 T^{12} + 11431698 T^{13} + 17692018 p T^{14} + 1391743 p^{2} T^{15} + 2147799 p^{3} T^{16} + 121206 p^{4} T^{17} + 217636 p^{5} T^{18} + 7118 p^{6} T^{19} + 17737 p^{7} T^{20} + 241 p^{8} T^{21} + 1095 p^{9} T^{22} + 3 p^{10} T^{23} + 46 p^{11} T^{24} + p^{13} T^{26} \)
11 \( 1 + 14 T + 194 T^{2} + 1724 T^{3} + 14469 T^{4} + 8825 p T^{5} + 614158 T^{6} + 3344067 T^{7} + 17227512 T^{8} + 78940743 T^{9} + 343284525 T^{10} + 1349212866 T^{11} + 5042010643 T^{12} + 17144099998 T^{13} + 5042010643 p T^{14} + 1349212866 p^{2} T^{15} + 343284525 p^{3} T^{16} + 78940743 p^{4} T^{17} + 17227512 p^{5} T^{18} + 3344067 p^{6} T^{19} + 614158 p^{7} T^{20} + 8825 p^{9} T^{21} + 14469 p^{9} T^{22} + 1724 p^{10} T^{23} + 194 p^{11} T^{24} + 14 p^{12} T^{25} + p^{13} T^{26} \)
13 \( 1 - 5 T + 120 T^{2} - 556 T^{3} + 7162 T^{4} - 30241 T^{5} + 21356 p T^{6} - 1063383 T^{7} + 7751242 T^{8} - 26846327 T^{9} + 12625198 p T^{10} - 511602512 T^{11} + 2707150189 T^{12} - 7537299196 T^{13} + 2707150189 p T^{14} - 511602512 p^{2} T^{15} + 12625198 p^{4} T^{16} - 26846327 p^{4} T^{17} + 7751242 p^{5} T^{18} - 1063383 p^{6} T^{19} + 21356 p^{8} T^{20} - 30241 p^{8} T^{21} + 7162 p^{9} T^{22} - 556 p^{10} T^{23} + 120 p^{11} T^{24} - 5 p^{12} T^{25} + p^{13} T^{26} \)
17 \( 1 + 8 T + 180 T^{2} + 1129 T^{3} + 14594 T^{4} + 75971 T^{5} + 733874 T^{6} + 3285476 T^{7} + 26175553 T^{8} + 103059983 T^{9} + 709247720 T^{10} + 2486361391 T^{11} + 15114105422 T^{12} + 47380188116 T^{13} + 15114105422 p T^{14} + 2486361391 p^{2} T^{15} + 709247720 p^{3} T^{16} + 103059983 p^{4} T^{17} + 26175553 p^{5} T^{18} + 3285476 p^{6} T^{19} + 733874 p^{7} T^{20} + 75971 p^{8} T^{21} + 14594 p^{9} T^{22} + 1129 p^{10} T^{23} + 180 p^{11} T^{24} + 8 p^{12} T^{25} + p^{13} T^{26} \)
19 \( 1 - 16 T + 15 p T^{2} - 3085 T^{3} + 33317 T^{4} - 279515 T^{5} + 2286436 T^{6} - 15803987 T^{7} + 105870036 T^{8} - 622402185 T^{9} + 3540840453 T^{10} - 18006321491 T^{11} + 4662350992 p T^{12} - 392583388926 T^{13} + 4662350992 p^{2} T^{14} - 18006321491 p^{2} T^{15} + 3540840453 p^{3} T^{16} - 622402185 p^{4} T^{17} + 105870036 p^{5} T^{18} - 15803987 p^{6} T^{19} + 2286436 p^{7} T^{20} - 279515 p^{8} T^{21} + 33317 p^{9} T^{22} - 3085 p^{10} T^{23} + 15 p^{12} T^{24} - 16 p^{12} T^{25} + p^{13} T^{26} \)
23 \( 1 + 4 T + 208 T^{2} + 577 T^{3} + 19338 T^{4} + 28323 T^{5} + 1059011 T^{6} - 82927 T^{7} + 38203077 T^{8} - 77496151 T^{9} + 993251086 T^{10} - 4331036769 T^{11} + 21779891225 T^{12} - 128863329834 T^{13} + 21779891225 p T^{14} - 4331036769 p^{2} T^{15} + 993251086 p^{3} T^{16} - 77496151 p^{4} T^{17} + 38203077 p^{5} T^{18} - 82927 p^{6} T^{19} + 1059011 p^{7} T^{20} + 28323 p^{8} T^{21} + 19338 p^{9} T^{22} + 577 p^{10} T^{23} + 208 p^{11} T^{24} + 4 p^{12} T^{25} + p^{13} T^{26} \)
29 \( 1 + 6 T + 167 T^{2} + 1119 T^{3} + 14556 T^{4} + 95554 T^{5} + 867323 T^{6} + 5155775 T^{7} + 38673431 T^{8} + 204819528 T^{9} + 1375646200 T^{10} + 6728549019 T^{11} + 42400243934 T^{12} + 200835971850 T^{13} + 42400243934 p T^{14} + 6728549019 p^{2} T^{15} + 1375646200 p^{3} T^{16} + 204819528 p^{4} T^{17} + 38673431 p^{5} T^{18} + 5155775 p^{6} T^{19} + 867323 p^{7} T^{20} + 95554 p^{8} T^{21} + 14556 p^{9} T^{22} + 1119 p^{10} T^{23} + 167 p^{11} T^{24} + 6 p^{12} T^{25} + p^{13} T^{26} \)
31 \( 1 - 11 T + 238 T^{2} - 2146 T^{3} + 27423 T^{4} - 213083 T^{5} + 2107619 T^{6} - 14608062 T^{7} + 122405479 T^{8} - 768021726 T^{9} + 5651157323 T^{10} - 32220230074 T^{11} + 212385028049 T^{12} - 1101680720195 T^{13} + 212385028049 p T^{14} - 32220230074 p^{2} T^{15} + 5651157323 p^{3} T^{16} - 768021726 p^{4} T^{17} + 122405479 p^{5} T^{18} - 14608062 p^{6} T^{19} + 2107619 p^{7} T^{20} - 213083 p^{8} T^{21} + 27423 p^{9} T^{22} - 2146 p^{10} T^{23} + 238 p^{11} T^{24} - 11 p^{12} T^{25} + p^{13} T^{26} \)
37 \( 1 - 6 T + 288 T^{2} - 1580 T^{3} + 41221 T^{4} - 207761 T^{5} + 3882367 T^{6} - 18076309 T^{7} + 270389640 T^{8} - 1165896004 T^{9} + 14821805019 T^{10} - 59072018711 T^{11} + 662208710800 T^{12} - 2419941319722 T^{13} + 662208710800 p T^{14} - 59072018711 p^{2} T^{15} + 14821805019 p^{3} T^{16} - 1165896004 p^{4} T^{17} + 270389640 p^{5} T^{18} - 18076309 p^{6} T^{19} + 3882367 p^{7} T^{20} - 207761 p^{8} T^{21} + 41221 p^{9} T^{22} - 1580 p^{10} T^{23} + 288 p^{11} T^{24} - 6 p^{12} T^{25} + p^{13} T^{26} \)
41 \( 1 + 18 T + 402 T^{2} + 5278 T^{3} + 72254 T^{4} + 766107 T^{5} + 8137956 T^{6} + 73727947 T^{7} + 661285826 T^{8} + 5286002164 T^{9} + 41629266411 T^{10} + 298362914039 T^{11} + 2101215187046 T^{12} + 13579450259166 T^{13} + 2101215187046 p T^{14} + 298362914039 p^{2} T^{15} + 41629266411 p^{3} T^{16} + 5286002164 p^{4} T^{17} + 661285826 p^{5} T^{18} + 73727947 p^{6} T^{19} + 8137956 p^{7} T^{20} + 766107 p^{8} T^{21} + 72254 p^{9} T^{22} + 5278 p^{10} T^{23} + 402 p^{11} T^{24} + 18 p^{12} T^{25} + p^{13} T^{26} \)
43 \( 1 - 7 T + 332 T^{2} - 1690 T^{3} + 52911 T^{4} - 206356 T^{5} + 5602742 T^{6} - 17329187 T^{7} + 446353627 T^{8} - 1134442447 T^{9} + 28351878132 T^{10} - 61688069451 T^{11} + 1474448251091 T^{12} - 2861935589388 T^{13} + 1474448251091 p T^{14} - 61688069451 p^{2} T^{15} + 28351878132 p^{3} T^{16} - 1134442447 p^{4} T^{17} + 446353627 p^{5} T^{18} - 17329187 p^{6} T^{19} + 5602742 p^{7} T^{20} - 206356 p^{8} T^{21} + 52911 p^{9} T^{22} - 1690 p^{10} T^{23} + 332 p^{11} T^{24} - 7 p^{12} T^{25} + p^{13} T^{26} \)
47 \( 1 + 22 T + 597 T^{2} + 9204 T^{3} + 145292 T^{4} + 1749919 T^{5} + 20438727 T^{6} + 204362410 T^{7} + 1940934728 T^{8} + 16806052097 T^{9} + 137188101256 T^{10} + 1062343723182 T^{11} + 7745163593679 T^{12} + 54715625823788 T^{13} + 7745163593679 p T^{14} + 1062343723182 p^{2} T^{15} + 137188101256 p^{3} T^{16} + 16806052097 p^{4} T^{17} + 1940934728 p^{5} T^{18} + 204362410 p^{6} T^{19} + 20438727 p^{7} T^{20} + 1749919 p^{8} T^{21} + 145292 p^{9} T^{22} + 9204 p^{10} T^{23} + 597 p^{11} T^{24} + 22 p^{12} T^{25} + p^{13} T^{26} \)
53 \( 1 + 17 T + 517 T^{2} + 6573 T^{3} + 118122 T^{4} + 1226674 T^{5} + 16797017 T^{6} + 150152470 T^{7} + 1725931736 T^{8} + 13711431986 T^{9} + 138404681578 T^{10} + 992387726269 T^{11} + 8972206200805 T^{12} + 58256637673334 T^{13} + 8972206200805 p T^{14} + 992387726269 p^{2} T^{15} + 138404681578 p^{3} T^{16} + 13711431986 p^{4} T^{17} + 1725931736 p^{5} T^{18} + 150152470 p^{6} T^{19} + 16797017 p^{7} T^{20} + 1226674 p^{8} T^{21} + 118122 p^{9} T^{22} + 6573 p^{10} T^{23} + 517 p^{11} T^{24} + 17 p^{12} T^{25} + p^{13} T^{26} \)
59 \( 1 + 6 T + 9 p T^{2} + 3011 T^{3} + 137862 T^{4} + 739334 T^{5} + 23234635 T^{6} + 117265119 T^{7} + 2840117773 T^{8} + 13365969432 T^{9} + 266069574110 T^{10} + 1152143341551 T^{11} + 19658171978356 T^{12} + 76948369277422 T^{13} + 19658171978356 p T^{14} + 1152143341551 p^{2} T^{15} + 266069574110 p^{3} T^{16} + 13365969432 p^{4} T^{17} + 2840117773 p^{5} T^{18} + 117265119 p^{6} T^{19} + 23234635 p^{7} T^{20} + 739334 p^{8} T^{21} + 137862 p^{9} T^{22} + 3011 p^{10} T^{23} + 9 p^{12} T^{24} + 6 p^{12} T^{25} + p^{13} T^{26} \)
61 \( 1 - 10 T + 561 T^{2} - 5233 T^{3} + 154029 T^{4} - 1339188 T^{5} + 27415743 T^{6} - 221150773 T^{7} + 3527520185 T^{8} - 26196554633 T^{9} + 346113595797 T^{10} - 2341761498458 T^{11} + 26617840240960 T^{12} - 161858915327202 T^{13} + 26617840240960 p T^{14} - 2341761498458 p^{2} T^{15} + 346113595797 p^{3} T^{16} - 26196554633 p^{4} T^{17} + 3527520185 p^{5} T^{18} - 221150773 p^{6} T^{19} + 27415743 p^{7} T^{20} - 1339188 p^{8} T^{21} + 154029 p^{9} T^{22} - 5233 p^{10} T^{23} + 561 p^{11} T^{24} - 10 p^{12} T^{25} + p^{13} T^{26} \)
67 \( 1 - 12 T + 665 T^{2} - 7140 T^{3} + 208013 T^{4} - 2021955 T^{5} + 41130040 T^{6} - 364569506 T^{7} + 5807596282 T^{8} - 47005474760 T^{9} + 623398964827 T^{10} - 4584842194264 T^{11} + 52519403082702 T^{12} - 347271203945123 T^{13} + 52519403082702 p T^{14} - 4584842194264 p^{2} T^{15} + 623398964827 p^{3} T^{16} - 47005474760 p^{4} T^{17} + 5807596282 p^{5} T^{18} - 364569506 p^{6} T^{19} + 41130040 p^{7} T^{20} - 2021955 p^{8} T^{21} + 208013 p^{9} T^{22} - 7140 p^{10} T^{23} + 665 p^{11} T^{24} - 12 p^{12} T^{25} + p^{13} T^{26} \)
71 \( 1 + 16 T + 600 T^{2} + 9303 T^{3} + 191257 T^{4} + 2644019 T^{5} + 40586319 T^{6} + 491218314 T^{7} + 6227913258 T^{8} + 66330165858 T^{9} + 723295569155 T^{10} + 6828672615759 T^{11} + 65271622158806 T^{12} + 547640621279510 T^{13} + 65271622158806 p T^{14} + 6828672615759 p^{2} T^{15} + 723295569155 p^{3} T^{16} + 66330165858 p^{4} T^{17} + 6227913258 p^{5} T^{18} + 491218314 p^{6} T^{19} + 40586319 p^{7} T^{20} + 2644019 p^{8} T^{21} + 191257 p^{9} T^{22} + 9303 p^{10} T^{23} + 600 p^{11} T^{24} + 16 p^{12} T^{25} + p^{13} T^{26} \)
73 \( 1 + 24 T + 769 T^{2} + 11939 T^{3} + 226244 T^{4} + 2622242 T^{5} + 37785377 T^{6} + 347504349 T^{7} + 58239857 p T^{8} + 438592226 p T^{9} + 362496888988 T^{10} + 2337189970097 T^{11} + 26773382950040 T^{12} + 163671899929078 T^{13} + 26773382950040 p T^{14} + 2337189970097 p^{2} T^{15} + 362496888988 p^{3} T^{16} + 438592226 p^{5} T^{17} + 58239857 p^{6} T^{18} + 347504349 p^{6} T^{19} + 37785377 p^{7} T^{20} + 2622242 p^{8} T^{21} + 226244 p^{9} T^{22} + 11939 p^{10} T^{23} + 769 p^{11} T^{24} + 24 p^{12} T^{25} + p^{13} T^{26} \)
79 \( 1 - 36 T + 1053 T^{2} - 20715 T^{3} + 364974 T^{4} - 5262753 T^{5} + 71788931 T^{6} - 863654022 T^{7} + 10121381914 T^{8} - 107845195310 T^{9} + 1131093082394 T^{10} - 10924126266943 T^{11} + 104640315620113 T^{12} - 931578148956794 T^{13} + 104640315620113 p T^{14} - 10924126266943 p^{2} T^{15} + 1131093082394 p^{3} T^{16} - 107845195310 p^{4} T^{17} + 10121381914 p^{5} T^{18} - 863654022 p^{6} T^{19} + 71788931 p^{7} T^{20} - 5262753 p^{8} T^{21} + 364974 p^{9} T^{22} - 20715 p^{10} T^{23} + 1053 p^{11} T^{24} - 36 p^{12} T^{25} + p^{13} T^{26} \)
83 \( 1 - T + 303 T^{2} - 723 T^{3} + 71270 T^{4} - 154547 T^{5} + 11552741 T^{6} - 29244559 T^{7} + 1581532486 T^{8} - 3595884588 T^{9} + 177033915977 T^{10} - 414290554931 T^{11} + 17131502248919 T^{12} - 35242187307005 T^{13} + 17131502248919 p T^{14} - 414290554931 p^{2} T^{15} + 177033915977 p^{3} T^{16} - 3595884588 p^{4} T^{17} + 1581532486 p^{5} T^{18} - 29244559 p^{6} T^{19} + 11552741 p^{7} T^{20} - 154547 p^{8} T^{21} + 71270 p^{9} T^{22} - 723 p^{10} T^{23} + 303 p^{11} T^{24} - p^{12} T^{25} + p^{13} T^{26} \)
89 \( 1 + 53 T + 2002 T^{2} + 54398 T^{3} + 1233571 T^{4} + 23443428 T^{5} + 392806234 T^{6} + 5817829435 T^{7} + 78328510257 T^{8} + 961260034545 T^{9} + 10984234874496 T^{10} + 117218879048135 T^{11} + 1188942226035487 T^{12} + 11452514746162508 T^{13} + 1188942226035487 p T^{14} + 117218879048135 p^{2} T^{15} + 10984234874496 p^{3} T^{16} + 961260034545 p^{4} T^{17} + 78328510257 p^{5} T^{18} + 5817829435 p^{6} T^{19} + 392806234 p^{7} T^{20} + 23443428 p^{8} T^{21} + 1233571 p^{9} T^{22} + 54398 p^{10} T^{23} + 2002 p^{11} T^{24} + 53 p^{12} T^{25} + p^{13} T^{26} \)
97 \( 1 + 21 T + 651 T^{2} + 9907 T^{3} + 188399 T^{4} + 2321362 T^{5} + 32954330 T^{6} + 328855618 T^{7} + 3712298851 T^{8} + 29150841566 T^{9} + 273248536763 T^{10} + 1576167435891 T^{11} + 14933500499341 T^{12} + 82452835579798 T^{13} + 14933500499341 p T^{14} + 1576167435891 p^{2} T^{15} + 273248536763 p^{3} T^{16} + 29150841566 p^{4} T^{17} + 3712298851 p^{5} T^{18} + 328855618 p^{6} T^{19} + 32954330 p^{7} T^{20} + 2321362 p^{8} T^{21} + 188399 p^{9} T^{22} + 9907 p^{10} T^{23} + 651 p^{11} T^{24} + 21 p^{12} T^{25} + p^{13} T^{26} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.66337540215459799120401282089, −2.56493348377696765259462046412, −2.53009963194216423368767047985, −2.47010532132529311115717546422, −2.45801447744174781502181205451, −2.44303775666229769166339538495, −2.39632796486481338664602200516, −2.38209812245136938681045839669, −2.36564932272200576620701826541, −2.22474654958457470457260127327, −2.18956096737762319902770814678, −1.89281222951754739886703239990, −1.64480170115612434232611848103, −1.60825325352930740163463189723, −1.47170482354448976932708305296, −1.41400093420341802343025764761, −1.31468595247233094279402504946, −1.20630003925190479697497886245, −1.19261191875439507485474915702, −1.18153605182581714075511538337, −1.16523090747854808893134532838, −1.07780819104178926137905141863, −0.974750821415891282156599694153, −0.947060117721956217451121783845, −0.812212908564926605639902618068, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.812212908564926605639902618068, 0.947060117721956217451121783845, 0.974750821415891282156599694153, 1.07780819104178926137905141863, 1.16523090747854808893134532838, 1.18153605182581714075511538337, 1.19261191875439507485474915702, 1.20630003925190479697497886245, 1.31468595247233094279402504946, 1.41400093420341802343025764761, 1.47170482354448976932708305296, 1.60825325352930740163463189723, 1.64480170115612434232611848103, 1.89281222951754739886703239990, 2.18956096737762319902770814678, 2.22474654958457470457260127327, 2.36564932272200576620701826541, 2.38209812245136938681045839669, 2.39632796486481338664602200516, 2.44303775666229769166339538495, 2.45801447744174781502181205451, 2.47010532132529311115717546422, 2.53009963194216423368767047985, 2.56493348377696765259462046412, 2.66337540215459799120401282089

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

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