Properties

Degree 16
Conductor $ 29^{8} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 2·4-s − 4·6-s − 4·7-s − 10·8-s + 2·9-s − 6·11-s − 4·12-s − 8·14-s − 18·16-s + 12·17-s + 4·18-s − 16·19-s + 8·21-s − 12·22-s + 20·24-s + 152·25-s − 40·27-s − 8·28-s + 128·29-s − 10·31-s − 2·32-s + 12·33-s + 24·34-s + 4·36-s − 84·37-s + ⋯
L(s)  = 1  + 2-s − 2/3·3-s + 1/2·4-s − 2/3·6-s − 4/7·7-s − 5/4·8-s + 2/9·9-s − 0.545·11-s − 1/3·12-s − 4/7·14-s − 9/8·16-s + 0.705·17-s + 2/9·18-s − 0.842·19-s + 8/21·21-s − 0.545·22-s + 5/6·24-s + 6.07·25-s − 1.48·27-s − 2/7·28-s + 4.41·29-s − 0.322·31-s − 0.0625·32-s + 4/11·33-s + 0.705·34-s + 1/9·36-s − 2.27·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(29^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{29} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 29^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.863011\)
\(L(\frac12)\)  \(\approx\)  \(0.863011\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 29$,\(F_p(T)\) is a polynomial of degree 16. If $p = 29$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad29 \( 1 - 128 T + 7400 T^{2} - 320 p^{2} T^{3} + 9518 p^{2} T^{4} - 320 p^{4} T^{5} + 7400 p^{4} T^{6} - 128 p^{6} T^{7} + p^{8} T^{8} \)
good2 \( 1 - p T + p T^{2} + 5 p T^{3} - 13 p T^{4} + 9 p T^{5} + 33 p T^{6} - 45 p T^{7} + 321 T^{8} - 45 p^{3} T^{9} + 33 p^{5} T^{10} + 9 p^{7} T^{11} - 13 p^{9} T^{12} + 5 p^{11} T^{13} + p^{13} T^{14} - p^{15} T^{15} + p^{16} T^{16} \)
3 \( 1 + 2 T + 2 T^{2} + 40 T^{3} + 10 T^{4} - 10 p T^{5} + 80 p^{2} T^{6} + 46 p^{3} T^{7} + 43 p^{4} T^{8} + 46 p^{5} T^{9} + 80 p^{6} T^{10} - 10 p^{7} T^{11} + 10 p^{8} T^{12} + 40 p^{10} T^{13} + 2 p^{12} T^{14} + 2 p^{14} T^{15} + p^{16} T^{16} \)
5 \( 1 - 152 T^{2} + 11042 T^{4} - 495504 T^{6} + 14942291 T^{8} - 495504 p^{4} T^{10} + 11042 p^{8} T^{12} - 152 p^{12} T^{14} + p^{16} T^{16} \)
7 \( ( 1 + 2 T + 118 T^{2} + 262 T^{3} + 6782 T^{4} + 262 p^{2} T^{5} + 118 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
11 \( 1 + 6 T + 18 T^{2} - 984 T^{3} + 5458 T^{4} + 25326 T^{5} + 537840 T^{6} - 21947202 T^{7} - 268164717 T^{8} - 21947202 p^{2} T^{9} + 537840 p^{4} T^{10} + 25326 p^{6} T^{11} + 5458 p^{8} T^{12} - 984 p^{10} T^{13} + 18 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16} \)
13 \( 1 - 900 T^{2} + 416090 T^{4} - 121830768 T^{6} + 24605646387 T^{8} - 121830768 p^{4} T^{10} + 416090 p^{8} T^{12} - 900 p^{12} T^{14} + p^{16} T^{16} \)
17 \( 1 - 12 T + 72 T^{2} - 156 T^{3} - 11400 T^{4} - 788916 T^{5} + 35640 p^{2} T^{6} - 16260516 p T^{7} + 7918453918 T^{8} - 16260516 p^{3} T^{9} + 35640 p^{6} T^{10} - 788916 p^{6} T^{11} - 11400 p^{8} T^{12} - 156 p^{10} T^{13} + 72 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \)
19 \( 1 + 16 T + 128 T^{2} - 9968 T^{3} - 125288 T^{4} + 124848 T^{5} + 67714944 T^{6} - 106114512 T^{7} + 873564318 T^{8} - 106114512 p^{2} T^{9} + 67714944 p^{4} T^{10} + 124848 p^{6} T^{11} - 125288 p^{8} T^{12} - 9968 p^{10} T^{13} + 128 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16} \)
23 \( ( 1 + 1798 T^{2} - 204 T^{3} + 1362542 T^{4} - 204 p^{2} T^{5} + 1798 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( 1 + 10 T + 50 T^{2} - 18752 T^{3} - 59126 T^{4} + 5169114 T^{5} + 230466192 T^{6} - 14955517326 T^{7} - 982612297341 T^{8} - 14955517326 p^{2} T^{9} + 230466192 p^{4} T^{10} + 5169114 p^{6} T^{11} - 59126 p^{8} T^{12} - 18752 p^{10} T^{13} + 50 p^{12} T^{14} + 10 p^{14} T^{15} + p^{16} T^{16} \)
37 \( 1 + 84 T + 3528 T^{2} + 71652 T^{3} - 2089980 T^{4} - 123205908 T^{5} - 408842280 T^{6} + 218820380604 T^{7} + 13702947378118 T^{8} + 218820380604 p^{2} T^{9} - 408842280 p^{4} T^{10} - 123205908 p^{6} T^{11} - 2089980 p^{8} T^{12} + 71652 p^{10} T^{13} + 3528 p^{12} T^{14} + 84 p^{14} T^{15} + p^{16} T^{16} \)
41 \( 1 - 20 T + 200 T^{2} - 65948 T^{3} + 2077432 T^{4} + 115380012 T^{5} - 548517288 T^{6} + 112249030692 T^{7} - 9153258882402 T^{8} + 112249030692 p^{2} T^{9} - 548517288 p^{4} T^{10} + 115380012 p^{6} T^{11} + 2077432 p^{8} T^{12} - 65948 p^{10} T^{13} + 200 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} \)
43 \( 1 + 190 T + 18050 T^{2} + 1310896 T^{3} + 89172690 T^{4} + 5513705486 T^{5} + 297261149248 T^{6} + 14457675945470 T^{7} + 648656884794643 T^{8} + 14457675945470 p^{2} T^{9} + 297261149248 p^{4} T^{10} + 5513705486 p^{6} T^{11} + 89172690 p^{8} T^{12} + 1310896 p^{10} T^{13} + 18050 p^{12} T^{14} + 190 p^{14} T^{15} + p^{16} T^{16} \)
47 \( 1 - 58 T + 1682 T^{2} - 75904 T^{3} + 8447626 T^{4} - 469112154 T^{5} + 7188912 p^{2} T^{6} - 15968051454 p T^{7} + 30625787988195 T^{8} - 15968051454 p^{3} T^{9} + 7188912 p^{6} T^{10} - 469112154 p^{6} T^{11} + 8447626 p^{8} T^{12} - 75904 p^{10} T^{13} + 1682 p^{12} T^{14} - 58 p^{14} T^{15} + p^{16} T^{16} \)
53 \( ( 1 - 126 T + 12120 T^{2} - 805920 T^{3} + 48326113 T^{4} - 805920 p^{2} T^{5} + 12120 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( ( 1 + 20 T + 4038 T^{2} - 142432 T^{3} + 879134 T^{4} - 142432 p^{2} T^{5} + 4038 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
61 \( 1 + 208 T + 21632 T^{2} + 2017096 T^{3} + 184647288 T^{4} + 13664724728 T^{5} + 882310746016 T^{6} + 58889137003472 T^{7} + 3818097286447198 T^{8} + 58889137003472 p^{2} T^{9} + 882310746016 p^{4} T^{10} + 13664724728 p^{6} T^{11} + 184647288 p^{8} T^{12} + 2017096 p^{10} T^{13} + 21632 p^{12} T^{14} + 208 p^{14} T^{15} + p^{16} T^{16} \)
67 \( 1 - 11008 T^{2} + 27285916 T^{4} + 201250654976 T^{6} - 1694675724118778 T^{8} + 201250654976 p^{4} T^{10} + 27285916 p^{8} T^{12} - 11008 p^{12} T^{14} + p^{16} T^{16} \)
71 \( 1 - 21324 T^{2} + 250110380 T^{4} - 1967023408740 T^{6} + 11455271890903830 T^{8} - 1967023408740 p^{4} T^{10} + 250110380 p^{8} T^{12} - 21324 p^{12} T^{14} + p^{16} T^{16} \)
73 \( 1 + 188 T + 17672 T^{2} + 1559180 T^{3} + 182719716 T^{4} + 18486242500 T^{5} + 1461911905048 T^{6} + 113727191207188 T^{7} + 8685700663672390 T^{8} + 113727191207188 p^{2} T^{9} + 1461911905048 p^{4} T^{10} + 18486242500 p^{6} T^{11} + 182719716 p^{8} T^{12} + 1559180 p^{10} T^{13} + 17672 p^{12} T^{14} + 188 p^{14} T^{15} + p^{16} T^{16} \)
79 \( 1 + 382 T + 72962 T^{2} + 10210992 T^{3} + 1201614634 T^{4} + 122408323406 T^{5} + 11219951427216 T^{6} + 967189248125926 T^{7} + 78871519191926211 T^{8} + 967189248125926 p^{2} T^{9} + 11219951427216 p^{4} T^{10} + 122408323406 p^{6} T^{11} + 1201614634 p^{8} T^{12} + 10210992 p^{10} T^{13} + 72962 p^{12} T^{14} + 382 p^{14} T^{15} + p^{16} T^{16} \)
83 \( ( 1 - 140 T + 22718 T^{2} - 2108544 T^{3} + 207732422 T^{4} - 2108544 p^{2} T^{5} + 22718 p^{4} T^{6} - 140 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
89 \( 1 + 64 T + 2048 T^{2} - 960272 T^{3} - 17872712 T^{4} - 244915632 T^{5} + 481989870720 T^{6} - 41121933793344 T^{7} - 1442843088822882 T^{8} - 41121933793344 p^{2} T^{9} + 481989870720 p^{4} T^{10} - 244915632 p^{6} T^{11} - 17872712 p^{8} T^{12} - 960272 p^{10} T^{13} + 2048 p^{12} T^{14} + 64 p^{14} T^{15} + p^{16} T^{16} \)
97 \( 1 + 44 T + 968 T^{2} - 45508 T^{3} + 12434104 T^{4} + 5439303060 T^{5} + 228328611000 T^{6} + 46830606041796 T^{7} + 9294107220541086 T^{8} + 46830606041796 p^{2} T^{9} + 228328611000 p^{4} T^{10} + 5439303060 p^{6} T^{11} + 12434104 p^{8} T^{12} - 45508 p^{10} T^{13} + 968 p^{12} T^{14} + 44 p^{14} T^{15} + p^{16} T^{16} \)
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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

−8.444407355063947988627836793240, −8.434901873588437524862968822168, −8.016473004562881158360423819147, −7.64691071018187867877271855946, −7.42575825059943410272127857067, −6.97323286213651169553456736135, −6.93979564666978871367798565554, −6.92310746468817936861513692497, −6.71000806915674810977560924237, −6.25705537520000261859133951345, −6.16609247298252896762696064335, −6.05667775407692816830534473862, −5.96827886354922175232360218838, −5.21431381505785316141195096004, −5.11422401381085990128893379332, −4.95593808491616919262208247534, −4.89520948468533016460411422647, −4.64144976502120482743189231835, −4.31615943094930326648377703674, −3.64788929232417560239361487827, −3.32172985133567089676984978740, −3.09957726086535191363110545545, −2.83792964823455948155975766554, −2.79299997031339127130910812733, −1.42761129586385661902147002133, 1.42761129586385661902147002133, 2.79299997031339127130910812733, 2.83792964823455948155975766554, 3.09957726086535191363110545545, 3.32172985133567089676984978740, 3.64788929232417560239361487827, 4.31615943094930326648377703674, 4.64144976502120482743189231835, 4.89520948468533016460411422647, 4.95593808491616919262208247534, 5.11422401381085990128893379332, 5.21431381505785316141195096004, 5.96827886354922175232360218838, 6.05667775407692816830534473862, 6.16609247298252896762696064335, 6.25705537520000261859133951345, 6.71000806915674810977560924237, 6.92310746468817936861513692497, 6.93979564666978871367798565554, 6.97323286213651169553456736135, 7.42575825059943410272127857067, 7.64691071018187867877271855946, 8.016473004562881158360423819147, 8.434901873588437524862968822168, 8.444407355063947988627836793240

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

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