Properties

Degree 8
Conductor $ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 10·4-s + 4·5-s + 7-s + 20·8-s + 16·10-s + 3·11-s + 2·13-s + 4·14-s + 35·16-s + 2·17-s + 2·19-s + 40·20-s + 12·22-s + 12·23-s + 10·25-s + 8·26-s + 10·28-s − 2·29-s + 10·31-s + 56·32-s + 8·34-s + 4·35-s + 3·37-s + 8·38-s + 80·40-s − 6·43-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s + 1.78·5-s + 0.377·7-s + 7.07·8-s + 5.05·10-s + 0.904·11-s + 0.554·13-s + 1.06·14-s + 35/4·16-s + 0.485·17-s + 0.458·19-s + 8.94·20-s + 2.55·22-s + 2.50·23-s + 2·25-s + 1.56·26-s + 1.88·28-s − 0.371·29-s + 1.79·31-s + 9.89·32-s + 1.37·34-s + 0.676·35-s + 0.493·37-s + 1.29·38-s + 12.6·40-s − 0.914·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6030} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(200.6831135\)
\(L(\frac12)\)  \(\approx\)  \(200.6831135\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;67\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{4} \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
67$C_1$ \( ( 1 + T )^{4} \)
good7$C_2 \wr S_4$ \( 1 - T + 8 T^{2} - 9 T^{3} + 66 T^{4} - 9 p T^{5} + 8 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 3 T + 24 T^{2} - 67 T^{3} + 398 T^{4} - 67 p T^{5} + 24 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 2 T + 32 T^{2} - 50 T^{3} + 486 T^{4} - 50 p T^{5} + 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 2 T + 36 T^{2} - 22 T^{3} + 678 T^{4} - 22 p T^{5} + 36 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 2 T + 44 T^{2} - 34 T^{3} + 982 T^{4} - 34 p T^{5} + 44 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 12 T + 100 T^{2} - 624 T^{3} + 3094 T^{4} - 624 p T^{5} + 100 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 2 T + 64 T^{2} + 322 T^{3} + 1918 T^{4} + 322 p T^{5} + 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 10 T + 140 T^{2} - 878 T^{3} + 6646 T^{4} - 878 p T^{5} + 140 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 3 T + 122 T^{2} - 273 T^{3} + 6346 T^{4} - 273 p T^{5} + 122 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 92 T^{2} - 16 T^{3} + 4358 T^{4} - 16 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 6 T + 92 T^{2} + 518 T^{3} + 6006 T^{4} + 518 p T^{5} + 92 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 14 T + 116 T^{2} - 762 T^{3} + 4862 T^{4} - 762 p T^{5} + 116 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 10 T + 168 T^{2} - 1230 T^{3} + 12542 T^{4} - 1230 p T^{5} + 168 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 164 T^{2} - 16 T^{3} + 12566 T^{4} - 16 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 9 T + 254 T^{2} - 1599 T^{3} + 23518 T^{4} - 1599 p T^{5} + 254 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 9 T + 242 T^{2} - 1313 T^{3} + 22814 T^{4} - 1313 p T^{5} + 242 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 14 T + 248 T^{2} + 2274 T^{3} + 25134 T^{4} + 2274 p T^{5} + 248 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 12 T + 200 T^{2} - 992 T^{3} + 13686 T^{4} - 992 p T^{5} + 200 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 25 T + 324 T^{2} - 3793 T^{3} + 40262 T^{4} - 3793 p T^{5} + 324 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 9 T + 122 T^{2} - 783 T^{3} + 10410 T^{4} - 783 p T^{5} + 122 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 3 T + 368 T^{2} + 841 T^{3} + 52686 T^{4} + 841 p T^{5} + 368 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.66844285181768270962852543711, −5.38299390647734274583868840495, −5.27855666441404565704171590217, −5.10104356570960563610654813863, −4.99715441297207004980174738992, −4.78954313608282756113983000815, −4.52128669086471931970452027593, −4.38168343531070270312768135012, −4.37912011577070927089298769787, −3.92593429429863213739557066390, −3.60768236894160735178602150946, −3.56751063087084859015278106674, −3.52598702621087233571197751013, −3.12906884707229505517340483580, −2.83083279810865154989203454986, −2.82408507323631665595631495747, −2.69559758127604620805874421086, −2.23041256326920294202043202255, −2.03177459569741288596429218517, −1.88089965247911845607076969805, −1.83442378977609818111514535246, −1.16785206999067924713004166765, −1.03682371434504502760165639563, −0.973024210035254159172176342806, −0.72780526784434483781654196629, 0.72780526784434483781654196629, 0.973024210035254159172176342806, 1.03682371434504502760165639563, 1.16785206999067924713004166765, 1.83442378977609818111514535246, 1.88089965247911845607076969805, 2.03177459569741288596429218517, 2.23041256326920294202043202255, 2.69559758127604620805874421086, 2.82408507323631665595631495747, 2.83083279810865154989203454986, 3.12906884707229505517340483580, 3.52598702621087233571197751013, 3.56751063087084859015278106674, 3.60768236894160735178602150946, 3.92593429429863213739557066390, 4.37912011577070927089298769787, 4.38168343531070270312768135012, 4.52128669086471931970452027593, 4.78954313608282756113983000815, 4.99715441297207004980174738992, 5.10104356570960563610654813863, 5.27855666441404565704171590217, 5.38299390647734274583868840495, 5.66844285181768270962852543711

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