Properties

Degree 16
Conductor $ 3^{16} \cdot 7^{8} \cdot 11^{16} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 8

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·4-s − 10·5-s + 8·7-s + 5·8-s + 10·10-s − 6·13-s − 8·14-s + 4·16-s + 5·17-s − 13·19-s + 40·20-s − 16·23-s + 38·25-s + 6·26-s − 32·28-s − 9·29-s + 9·31-s − 12·32-s − 5·34-s − 80·35-s + 7·37-s + 13·38-s − 50·40-s + 10·41-s − 4·43-s + 16·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 2·4-s − 4.47·5-s + 3.02·7-s + 1.76·8-s + 3.16·10-s − 1.66·13-s − 2.13·14-s + 16-s + 1.21·17-s − 2.98·19-s + 8.94·20-s − 3.33·23-s + 38/5·25-s + 1.17·26-s − 6.04·28-s − 1.67·29-s + 1.61·31-s − 2.12·32-s − 0.857·34-s − 13.5·35-s + 1.15·37-s + 2.10·38-s − 7.90·40-s + 1.56·41-s − 0.609·43-s + 2.35·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{16} \cdot 7^{8} \cdot 11^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  8
Selberg data  =  $(16,\ 3^{16} \cdot 7^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( ( 1 - T )^{8} \)
11 \( 1 \)
good2 \( 1 + T + 5 T^{2} + p^{2} T^{3} + 15 T^{4} + 7 p T^{5} + 19 p T^{6} + 15 p T^{7} + 75 T^{8} + 15 p^{2} T^{9} + 19 p^{3} T^{10} + 7 p^{4} T^{11} + 15 p^{4} T^{12} + p^{7} T^{13} + 5 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
5 \( 1 + 2 p T + 62 T^{2} + 283 T^{3} + 43 p^{2} T^{4} + 3513 T^{5} + 2046 p T^{6} + 26646 T^{7} + 62784 T^{8} + 26646 p T^{9} + 2046 p^{3} T^{10} + 3513 p^{3} T^{11} + 43 p^{6} T^{12} + 283 p^{5} T^{13} + 62 p^{6} T^{14} + 2 p^{8} T^{15} + p^{8} T^{16} \)
13 \( 1 + 6 T + 68 T^{2} + 285 T^{3} + 163 p T^{4} + 7443 T^{5} + 44016 T^{6} + 130942 T^{7} + 656120 T^{8} + 130942 p T^{9} + 44016 p^{2} T^{10} + 7443 p^{3} T^{11} + 163 p^{5} T^{12} + 285 p^{5} T^{13} + 68 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 5 T + 99 T^{2} - 384 T^{3} + 4383 T^{4} - 13616 T^{5} + 119813 T^{6} - 309659 T^{7} + 2344384 T^{8} - 309659 p T^{9} + 119813 p^{2} T^{10} - 13616 p^{3} T^{11} + 4383 p^{4} T^{12} - 384 p^{5} T^{13} + 99 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 + 13 T + 167 T^{2} + 1424 T^{3} + 11403 T^{4} + 72606 T^{5} + 434133 T^{6} + 2182431 T^{7} + 10302272 T^{8} + 2182431 p T^{9} + 434133 p^{2} T^{10} + 72606 p^{3} T^{11} + 11403 p^{4} T^{12} + 1424 p^{5} T^{13} + 167 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 + 16 T + 224 T^{2} + 2234 T^{3} + 19567 T^{4} + 141712 T^{5} + 40290 p T^{6} + 5230592 T^{7} + 26816927 T^{8} + 5230592 p T^{9} + 40290 p^{3} T^{10} + 141712 p^{3} T^{11} + 19567 p^{4} T^{12} + 2234 p^{5} T^{13} + 224 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 + 9 T + 187 T^{2} + 1382 T^{3} + 16028 T^{4} + 97473 T^{5} + 828470 T^{6} + 4208558 T^{7} + 28798593 T^{8} + 4208558 p T^{9} + 828470 p^{2} T^{10} + 97473 p^{3} T^{11} + 16028 p^{4} T^{12} + 1382 p^{5} T^{13} + 187 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 9 T + 142 T^{2} - 963 T^{3} + 9327 T^{4} - 55589 T^{5} + 14166 p T^{6} - 2365595 T^{7} + 15862880 T^{8} - 2365595 p T^{9} + 14166 p^{3} T^{10} - 55589 p^{3} T^{11} + 9327 p^{4} T^{12} - 963 p^{5} T^{13} + 142 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 7 T + 231 T^{2} - 1552 T^{3} + 24962 T^{4} - 154999 T^{5} + 1656240 T^{6} - 9060906 T^{7} + 73868707 T^{8} - 9060906 p T^{9} + 1656240 p^{2} T^{10} - 154999 p^{3} T^{11} + 24962 p^{4} T^{12} - 1552 p^{5} T^{13} + 231 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 10 T + 261 T^{2} - 2176 T^{3} + 30891 T^{4} - 221954 T^{5} + 54771 p T^{6} - 13854084 T^{7} + 110613464 T^{8} - 13854084 p T^{9} + 54771 p^{3} T^{10} - 221954 p^{3} T^{11} + 30891 p^{4} T^{12} - 2176 p^{5} T^{13} + 261 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 + 4 T + 239 T^{2} + 936 T^{3} + 27462 T^{4} + 98508 T^{5} + 2007760 T^{6} + 6271668 T^{7} + 102278657 T^{8} + 6271668 p T^{9} + 2007760 p^{2} T^{10} + 98508 p^{3} T^{11} + 27462 p^{4} T^{12} + 936 p^{5} T^{13} + 239 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 16 T + 320 T^{2} + 3319 T^{3} + 41705 T^{4} + 344729 T^{5} + 3361468 T^{6} + 23054530 T^{7} + 186100580 T^{8} + 23054530 p T^{9} + 3361468 p^{2} T^{10} + 344729 p^{3} T^{11} + 41705 p^{4} T^{12} + 3319 p^{5} T^{13} + 320 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 37 T + 857 T^{2} + 14496 T^{3} + 199396 T^{4} + 2305797 T^{5} + 23105040 T^{6} + 202528658 T^{7} + 1568879259 T^{8} + 202528658 p T^{9} + 23105040 p^{2} T^{10} + 2305797 p^{3} T^{11} + 199396 p^{4} T^{12} + 14496 p^{5} T^{13} + 857 p^{6} T^{14} + 37 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + T + 332 T^{2} + 463 T^{3} + 53293 T^{4} + 80577 T^{5} + 5426300 T^{6} + 7692267 T^{7} + 381400788 T^{8} + 7692267 p T^{9} + 5426300 p^{2} T^{10} + 80577 p^{3} T^{11} + 53293 p^{4} T^{12} + 463 p^{5} T^{13} + 332 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 19 T + 492 T^{2} - 7043 T^{3} + 104067 T^{4} - 1170549 T^{5} + 12549376 T^{6} - 113336285 T^{7} + 952230800 T^{8} - 113336285 p T^{9} + 12549376 p^{2} T^{10} - 1170549 p^{3} T^{11} + 104067 p^{4} T^{12} - 7043 p^{5} T^{13} + 492 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 150444400 p T^{9} + 15702408 p^{2} T^{10} + 1415171 p^{3} T^{11} + 119005 p^{4} T^{12} + 7751 p^{5} T^{13} + 520 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 + 13 T + 398 T^{2} + 3457 T^{3} + 57783 T^{4} + 323737 T^{5} + 4210848 T^{6} + 14543752 T^{7} + 249426533 T^{8} + 14543752 p T^{9} + 4210848 p^{2} T^{10} + 323737 p^{3} T^{11} + 57783 p^{4} T^{12} + 3457 p^{5} T^{13} + 398 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 25 T + 616 T^{2} + 9963 T^{3} + 152093 T^{4} + 1861173 T^{5} + 21394652 T^{6} + 209466223 T^{7} + 1925408604 T^{8} + 209466223 p T^{9} + 21394652 p^{2} T^{10} + 1861173 p^{3} T^{11} + 152093 p^{4} T^{12} + 9963 p^{5} T^{13} + 616 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 337 T^{2} + 1005 T^{3} + 57248 T^{4} + 275700 T^{5} + 6743044 T^{6} + 36745800 T^{7} + 601257265 T^{8} + 36745800 p T^{9} + 6743044 p^{2} T^{10} + 275700 p^{3} T^{11} + 57248 p^{4} T^{12} + 1005 p^{5} T^{13} + 337 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 25 T + 786 T^{2} - 13983 T^{3} + 250863 T^{4} - 3398873 T^{5} + 43717402 T^{6} - 464690723 T^{7} + 4604658344 T^{8} - 464690723 p T^{9} + 43717402 p^{2} T^{10} - 3398873 p^{3} T^{11} + 250863 p^{4} T^{12} - 13983 p^{5} T^{13} + 786 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 + 37 T + 1032 T^{2} + 20901 T^{3} + 357823 T^{4} + 5152639 T^{5} + 737152 p T^{6} + 731972999 T^{7} + 7339853952 T^{8} + 731972999 p T^{9} + 737152 p^{3} T^{10} + 5152639 p^{3} T^{11} + 357823 p^{4} T^{12} + 20901 p^{5} T^{13} + 1032 p^{6} T^{14} + 37 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 15 T + 554 T^{2} - 7041 T^{3} + 141913 T^{4} - 1604409 T^{5} + 23035698 T^{6} - 230491951 T^{7} + 2630434004 T^{8} - 230491951 p T^{9} + 23035698 p^{2} T^{10} - 1604409 p^{3} T^{11} + 141913 p^{4} T^{12} - 7041 p^{5} T^{13} + 554 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.74496763312182612608797484862, −3.44612167428702970535010807098, −3.41098313774011483098330192541, −3.36805351232760595997369310585, −3.24223664499633320035966494965, −3.22225066375480446145569439808, −3.10212392552685890150942198430, −2.91002804611702792917356377945, −2.75170159553824904383223892508, −2.60610662173162115953789937561, −2.48543775955036049435450099712, −2.46454065330327922120755902304, −2.14861279764480641867310182757, −2.05881944265911279756266484057, −2.04678160554319797401908399048, −1.93544757907957635512886099955, −1.92192425088444527438095896253, −1.80473489381734329220556980305, −1.51907028076046816864834453633, −1.43039474556594633313353251883, −1.23326762595996216353734153920, −1.03555223875678682698222945060, −1.02370960388510578666082169785, −0.896429490626231175202944561293, −0.888494675307936829097359807989, 0, 0, 0, 0, 0, 0, 0, 0, 0.888494675307936829097359807989, 0.896429490626231175202944561293, 1.02370960388510578666082169785, 1.03555223875678682698222945060, 1.23326762595996216353734153920, 1.43039474556594633313353251883, 1.51907028076046816864834453633, 1.80473489381734329220556980305, 1.92192425088444527438095896253, 1.93544757907957635512886099955, 2.04678160554319797401908399048, 2.05881944265911279756266484057, 2.14861279764480641867310182757, 2.46454065330327922120755902304, 2.48543775955036049435450099712, 2.60610662173162115953789937561, 2.75170159553824904383223892508, 2.91002804611702792917356377945, 3.10212392552685890150942198430, 3.22225066375480446145569439808, 3.24223664499633320035966494965, 3.36805351232760595997369310585, 3.41098313774011483098330192541, 3.44612167428702970535010807098, 3.74496763312182612608797484862

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

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