Properties

Degree $8$
Conductor $2.044\times 10^{15}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·5-s + 2·9-s − 4·11-s − 8·15-s + 4·17-s − 6·19-s − 12·23-s + 4·25-s + 4·27-s + 4·29-s − 8·31-s + 8·33-s + 16·37-s + 4·43-s + 8·45-s + 6·47-s − 6·49-s − 8·51-s + 16·53-s − 16·55-s + 12·57-s + 12·59-s + 24·61-s − 28·67-s + 24·69-s + 2·71-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 2/3·9-s − 1.20·11-s − 2.06·15-s + 0.970·17-s − 1.37·19-s − 2.50·23-s + 4/5·25-s + 0.769·27-s + 0.742·29-s − 1.43·31-s + 1.39·33-s + 2.63·37-s + 0.609·43-s + 1.19·45-s + 0.875·47-s − 6/7·49-s − 1.12·51-s + 2.19·53-s − 2.15·55-s + 1.58·57-s + 1.56·59-s + 3.07·61-s − 3.42·67-s + 2.88·69-s + 0.237·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 41^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{6724} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 41^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.196365546\)
\(L(\frac12)\) \(\approx\) \(2.196365546\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3$C_2 \wr S_4$ \( 1 + 2 T + 2 T^{2} - 4 T^{3} - 8 T^{4} - 4 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 4 T + 12 T^{2} - 16 T^{3} + 34 T^{4} - 16 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 6 T^{2} - 26 T^{3} + 24 T^{4} - 26 p T^{5} + 6 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 4 T + 26 T^{2} + 114 T^{3} + 384 T^{4} + 114 p T^{5} + 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 12 T^{2} + 48 T^{3} + 118 T^{4} + 48 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 4 T + 20 T^{2} - 124 T^{3} + 534 T^{4} - 124 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 6 T + 62 T^{2} + 208 T^{3} + 1448 T^{4} + 208 p T^{5} + 62 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 12 T + 108 T^{2} + 700 T^{3} + 3718 T^{4} + 700 p T^{5} + 108 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 4 T + 76 T^{2} - 12 p T^{3} + 2870 T^{4} - 12 p^{2} T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 8 T + 92 T^{2} + 712 T^{3} + 3846 T^{4} + 712 p T^{5} + 92 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 16 T + 212 T^{2} - 1740 T^{3} + 12626 T^{4} - 1740 p T^{5} + 212 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 4 T + 124 T^{2} - 244 T^{3} + 6678 T^{4} - 244 p T^{5} + 124 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 6 T + 126 T^{2} - 640 T^{3} + 8608 T^{4} - 640 p T^{5} + 126 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 16 T + 4 p T^{2} - 1824 T^{3} + 15558 T^{4} - 1824 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 12 T + 252 T^{2} - 1996 T^{3} + 22582 T^{4} - 1996 p T^{5} + 252 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 24 T + 420 T^{2} - 4824 T^{3} + 44086 T^{4} - 4824 p T^{5} + 420 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 28 T + 538 T^{2} + 6638 T^{3} + 64208 T^{4} + 6638 p T^{5} + 538 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 2 T + 98 T^{2} + 268 T^{3} + 48 p T^{4} + 268 p T^{5} + 98 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 8 T + 212 T^{2} - 1060 T^{3} + 19890 T^{4} - 1060 p T^{5} + 212 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 18 T + 366 T^{2} - 4224 T^{3} + 45328 T^{4} - 4224 p T^{5} + 366 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 12 T + 252 T^{2} + 1644 T^{3} + 24598 T^{4} + 1644 p T^{5} + 252 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 4 T + 228 T^{2} + 796 T^{3} + 326 p T^{4} + 796 p T^{5} + 228 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 16 T + 268 T^{2} + 3376 T^{3} + 38118 T^{4} + 3376 p T^{5} + 268 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.73696685594156586422736324292, −5.45659966577216050396294590331, −5.38017910054005335566593201670, −5.21971392893576506600918901737, −4.99474670608161952494673428685, −4.68537200409089096738632466319, −4.39027270593753717763490358459, −4.32628592666445124257855700736, −4.24904367547780011846844204732, −3.89458075633052435173962312151, −3.83847923318943175042546290765, −3.55904857843142462112418169405, −3.23238423637044424725660932578, −3.03798911416873315915142834386, −2.55265154960699881896491146697, −2.54778747143231244901315907242, −2.29014701559842378351435942418, −2.26734061779770868420806215508, −2.15201121867836451447185762515, −1.66649080093027739177911646094, −1.36040404009701131156739285299, −1.15205766768069528571859093686, −0.983130730068416537128204228536, −0.42628324180809372064517322649, −0.26740920364919483376776983794, 0.26740920364919483376776983794, 0.42628324180809372064517322649, 0.983130730068416537128204228536, 1.15205766768069528571859093686, 1.36040404009701131156739285299, 1.66649080093027739177911646094, 2.15201121867836451447185762515, 2.26734061779770868420806215508, 2.29014701559842378351435942418, 2.54778747143231244901315907242, 2.55265154960699881896491146697, 3.03798911416873315915142834386, 3.23238423637044424725660932578, 3.55904857843142462112418169405, 3.83847923318943175042546290765, 3.89458075633052435173962312151, 4.24904367547780011846844204732, 4.32628592666445124257855700736, 4.39027270593753717763490358459, 4.68537200409089096738632466319, 4.99474670608161952494673428685, 5.21971392893576506600918901737, 5.38017910054005335566593201670, 5.45659966577216050396294590331, 5.73696685594156586422736324292

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