Properties

Degree 14
Conductor $ 2^{28} \cdot 3^{7} \cdot 167^{7} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 7

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·3-s − 3·5-s − 8·7-s + 28·9-s − 11-s − 2·13-s − 21·15-s + 11·17-s − 2·19-s − 56·21-s − 17·23-s − 11·25-s + 84·27-s − 7·29-s − 10·31-s − 7·33-s + 24·35-s − 21·37-s − 14·39-s + 8·41-s + 12·43-s − 84·45-s − 25·47-s + 4·49-s + 77·51-s − 7·53-s + 3·55-s + ⋯
L(s)  = 1  + 4.04·3-s − 1.34·5-s − 3.02·7-s + 28/3·9-s − 0.301·11-s − 0.554·13-s − 5.42·15-s + 2.66·17-s − 0.458·19-s − 12.2·21-s − 3.54·23-s − 2.19·25-s + 16.1·27-s − 1.29·29-s − 1.79·31-s − 1.21·33-s + 4.05·35-s − 3.45·37-s − 2.24·39-s + 1.24·41-s + 1.82·43-s − 12.5·45-s − 3.64·47-s + 4/7·49-s + 10.7·51-s − 0.961·53-s + 0.404·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{7} \cdot 167^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{7} \cdot 167^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(14\)
\( N \)  =  \(2^{28} \cdot 3^{7} \cdot 167^{7}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8016} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(7\)
Selberg data  =  \((14,\ 2^{28} \cdot 3^{7} \cdot 167^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\(F_p(T)\) is a polynomial of degree 14. If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 13.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - T )^{7} \)
167 \( ( 1 - T )^{7} \)
good5 \( 1 + 3 T + 4 p T^{2} + 8 p T^{3} + 173 T^{4} + 258 T^{5} + 1026 T^{6} + 1326 T^{7} + 1026 p T^{8} + 258 p^{2} T^{9} + 173 p^{3} T^{10} + 8 p^{5} T^{11} + 4 p^{6} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
7 \( 1 + 8 T + 60 T^{2} + 281 T^{3} + 181 p T^{4} + 4377 T^{5} + 2092 p T^{6} + 39420 T^{7} + 2092 p^{2} T^{8} + 4377 p^{2} T^{9} + 181 p^{4} T^{10} + 281 p^{4} T^{11} + 60 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 + T + 46 T^{2} + 28 T^{3} + 1061 T^{4} + 414 T^{5} + 16252 T^{6} + 5010 T^{7} + 16252 p T^{8} + 414 p^{2} T^{9} + 1061 p^{3} T^{10} + 28 p^{4} T^{11} + 46 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 + 2 T + 48 T^{2} + 20 T^{3} + 854 T^{4} - 1667 T^{5} + 8225 T^{6} - 41310 T^{7} + 8225 p T^{8} - 1667 p^{2} T^{9} + 854 p^{3} T^{10} + 20 p^{4} T^{11} + 48 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 - 11 T + 125 T^{2} - 904 T^{3} + 6292 T^{4} - 34384 T^{5} + 176026 T^{6} - 750594 T^{7} + 176026 p T^{8} - 34384 p^{2} T^{9} + 6292 p^{3} T^{10} - 904 p^{4} T^{11} + 125 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 2 T + 68 T^{2} + 280 T^{3} + 2522 T^{4} + 11611 T^{5} + 70571 T^{6} + 268206 T^{7} + 70571 p T^{8} + 11611 p^{2} T^{9} + 2522 p^{3} T^{10} + 280 p^{4} T^{11} + 68 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 + 17 T + 264 T^{2} + 2580 T^{3} + 22869 T^{4} + 155172 T^{5} + 962930 T^{6} + 4821022 T^{7} + 962930 p T^{8} + 155172 p^{2} T^{9} + 22869 p^{3} T^{10} + 2580 p^{4} T^{11} + 264 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 7 T + 146 T^{2} + 790 T^{3} + 9969 T^{4} + 44070 T^{5} + 422776 T^{6} + 1553690 T^{7} + 422776 p T^{8} + 44070 p^{2} T^{9} + 9969 p^{3} T^{10} + 790 p^{4} T^{11} + 146 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 + 10 T + 211 T^{2} + 1575 T^{3} + 18783 T^{4} + 110477 T^{5} + 946125 T^{6} + 4420900 T^{7} + 946125 p T^{8} + 110477 p^{2} T^{9} + 18783 p^{3} T^{10} + 1575 p^{4} T^{11} + 211 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 + 21 T + 360 T^{2} + 4288 T^{3} + 43987 T^{4} + 370784 T^{5} + 2772232 T^{6} + 17825150 T^{7} + 2772232 p T^{8} + 370784 p^{2} T^{9} + 43987 p^{3} T^{10} + 4288 p^{4} T^{11} + 360 p^{5} T^{12} + 21 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 - 8 T + 5 p T^{2} - 1417 T^{3} + 20339 T^{4} - 118045 T^{5} + 1250235 T^{6} - 6008956 T^{7} + 1250235 p T^{8} - 118045 p^{2} T^{9} + 20339 p^{3} T^{10} - 1417 p^{4} T^{11} + 5 p^{6} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 - 12 T + 145 T^{2} - 23 p T^{3} + 7307 T^{4} - 31281 T^{5} + 187515 T^{6} - 541084 T^{7} + 187515 p T^{8} - 31281 p^{2} T^{9} + 7307 p^{3} T^{10} - 23 p^{5} T^{11} + 145 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 + 25 T + 440 T^{2} + 5547 T^{3} + 60316 T^{4} + 549620 T^{5} + 4505677 T^{6} + 32294448 T^{7} + 4505677 p T^{8} + 549620 p^{2} T^{9} + 60316 p^{3} T^{10} + 5547 p^{4} T^{11} + 440 p^{5} T^{12} + 25 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 7 T + 166 T^{2} + 1295 T^{3} + 19306 T^{4} + 115366 T^{5} + 1391231 T^{6} + 7778352 T^{7} + 1391231 p T^{8} + 115366 p^{2} T^{9} + 19306 p^{3} T^{10} + 1295 p^{4} T^{11} + 166 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 + 3 T + 362 T^{2} + 997 T^{3} + 58918 T^{4} + 141144 T^{5} + 5562009 T^{6} + 10948080 T^{7} + 5562009 p T^{8} + 141144 p^{2} T^{9} + 58918 p^{3} T^{10} + 997 p^{4} T^{11} + 362 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 + 14 T + 304 T^{2} + 3390 T^{3} + 45370 T^{4} + 399265 T^{5} + 4077845 T^{6} + 29981926 T^{7} + 4077845 p T^{8} + 399265 p^{2} T^{9} + 45370 p^{3} T^{10} + 3390 p^{4} T^{11} + 304 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 + 4 T + 382 T^{2} + 1324 T^{3} + 66982 T^{4} + 198497 T^{5} + 6988981 T^{6} + 17114094 T^{7} + 6988981 p T^{8} + 198497 p^{2} T^{9} + 66982 p^{3} T^{10} + 1324 p^{4} T^{11} + 382 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 + 27 T + 731 T^{2} + 12149 T^{3} + 188468 T^{4} + 2220242 T^{5} + 24216158 T^{6} + 212111532 T^{7} + 24216158 p T^{8} + 2220242 p^{2} T^{9} + 188468 p^{3} T^{10} + 12149 p^{4} T^{11} + 731 p^{5} T^{12} + 27 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 + 12 T + 286 T^{2} + 1814 T^{3} + 32184 T^{4} + 167073 T^{5} + 3162525 T^{6} + 15750450 T^{7} + 3162525 p T^{8} + 167073 p^{2} T^{9} + 32184 p^{3} T^{10} + 1814 p^{4} T^{11} + 286 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 + 8 T + 463 T^{2} + 2721 T^{3} + 93163 T^{4} + 415133 T^{5} + 11094341 T^{6} + 39502788 T^{7} + 11094341 p T^{8} + 415133 p^{2} T^{9} + 93163 p^{3} T^{10} + 2721 p^{4} T^{11} + 463 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 + 15 T + 252 T^{2} + 2335 T^{3} + 40204 T^{4} + 384018 T^{5} + 4378733 T^{6} + 30821664 T^{7} + 4378733 p T^{8} + 384018 p^{2} T^{9} + 40204 p^{3} T^{10} + 2335 p^{4} T^{11} + 252 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 - 14 T + 532 T^{2} - 6878 T^{3} + 130172 T^{4} - 1444175 T^{5} + 18617595 T^{6} - 167849250 T^{7} + 18617595 p T^{8} - 1444175 p^{2} T^{9} + 130172 p^{3} T^{10} - 6878 p^{4} T^{11} + 532 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 3 T + 475 T^{2} - 1207 T^{3} + 104508 T^{4} - 227430 T^{5} + 14447348 T^{6} - 26865048 T^{7} + 14447348 p T^{8} - 227430 p^{2} T^{9} + 104508 p^{3} T^{10} - 1207 p^{4} T^{11} + 475 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.76447512690177841282856997774, −3.68101255315542346444958968872, −3.65755248688920706377624264153, −3.58037185492773101395328845762, −3.44081000747088007639815164738, −3.30494778613244330455351009633, −3.20261921964188226402211756420, −3.13841879255035874565313779878, −3.11682998625577729140061692263, −3.06234103453882132118329169449, −2.93727634776662109083862804418, −2.64053315201547211358656128563, −2.34503180134762693604375030588, −2.29704730953931081890847018502, −2.22205001649635053201401665423, −2.22164552089705022849597504044, −2.21243375175487858723331149449, −2.20712594402098554688672285988, −1.58933089237121421702711929855, −1.46277656236414483271641951737, −1.43864294588264191294012905043, −1.41001037265649908373912207860, −1.32844109365442934920119690437, −1.31103109483073760969153543524, −1.04699263015907031249581359895, 0, 0, 0, 0, 0, 0, 0, 1.04699263015907031249581359895, 1.31103109483073760969153543524, 1.32844109365442934920119690437, 1.41001037265649908373912207860, 1.43864294588264191294012905043, 1.46277656236414483271641951737, 1.58933089237121421702711929855, 2.20712594402098554688672285988, 2.21243375175487858723331149449, 2.22164552089705022849597504044, 2.22205001649635053201401665423, 2.29704730953931081890847018502, 2.34503180134762693604375030588, 2.64053315201547211358656128563, 2.93727634776662109083862804418, 3.06234103453882132118329169449, 3.11682998625577729140061692263, 3.13841879255035874565313779878, 3.20261921964188226402211756420, 3.30494778613244330455351009633, 3.44081000747088007639815164738, 3.58037185492773101395328845762, 3.65755248688920706377624264153, 3.68101255315542346444958968872, 3.76447512690177841282856997774

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

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