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Properties

Label 8-5746e4-1.1-c1e4-0-4
Degree $8$
Conductor $1.090\times 10^{15}$
Sign $1$
Analytic cond. $4.43170\times 10^{6}$
Root an. cond. $6.77362$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3-s + 10·4-s − 3·5-s − 4·6-s − 5·7-s + 20·8-s − 2·9-s − 12·10-s − 3·11-s − 10·12-s − 20·14-s + 3·15-s + 35·16-s + 4·17-s − 8·18-s + 2·19-s − 30·20-s + 5·21-s − 12·22-s + 7·23-s − 20·24-s − 5·25-s + 3·27-s − 50·28-s − 3·29-s + 12·30-s + ⋯
L(s)  = 1  + 2.82·2-s − 0.577·3-s + 5·4-s − 1.34·5-s − 1.63·6-s − 1.88·7-s + 7.07·8-s − 2/3·9-s − 3.79·10-s − 0.904·11-s − 2.88·12-s − 5.34·14-s + 0.774·15-s + 35/4·16-s + 0.970·17-s − 1.88·18-s + 0.458·19-s − 6.70·20-s + 1.09·21-s − 2.55·22-s + 1.45·23-s − 4.08·24-s − 25-s + 0.577·27-s − 9.44·28-s − 0.557·29-s + 2.19·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 13^{8} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(4.43170\times 10^{6}\)
Root analytic conductor: \(6.77362\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 13^{8} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
13 \( 1 \)
17$C_1$ \( ( 1 - T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + T + p T^{2} + 2 T^{3} + 5 p T^{4} + 2 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 3 T + 14 T^{2} + 7 p T^{3} + 103 T^{4} + 7 p^{2} T^{5} + 14 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 5 T + 16 T^{2} + 51 T^{3} + 153 T^{4} + 51 p T^{5} + 16 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 3 T + 37 T^{2} + 74 T^{3} + 51 p T^{4} + 74 p T^{5} + 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 2 T + 25 T^{2} - 58 T^{3} + 783 T^{4} - 58 p T^{5} + 25 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 7 T + 64 T^{2} - 297 T^{3} + 1789 T^{4} - 297 p T^{5} + 64 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 3 T + 98 T^{2} + 209 T^{3} + 4015 T^{4} + 209 p T^{5} + 98 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 3 T + 27 T^{2} - 46 T^{3} - 199 T^{4} - 46 p T^{5} + 27 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 11 T + 110 T^{2} + 823 T^{3} + 5585 T^{4} + 823 p T^{5} + 110 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 35 T + 602 T^{2} + 6589 T^{3} + 50079 T^{4} + 6589 p T^{5} + 602 p^{2} T^{6} + 35 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 12 T + 102 T^{2} + 521 T^{3} + 3071 T^{4} + 521 p T^{5} + 102 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 9 T + 76 T^{2} + 641 T^{3} + 6747 T^{4} + 641 p T^{5} + 76 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - T + 129 T^{2} - 140 T^{3} + 8875 T^{4} - 140 p T^{5} + 129 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - T + 100 T^{2} + 21 T^{3} + 8071 T^{4} + 21 p T^{5} + 100 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 14 T + 240 T^{2} + 2420 T^{3} + 21825 T^{4} + 2420 p T^{5} + 240 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 21 T + 299 T^{2} + 2936 T^{3} + 26707 T^{4} + 2936 p T^{5} + 299 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 3 T + 109 T^{2} - 286 T^{3} + 11313 T^{4} - 286 p T^{5} + 109 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 18 T + 361 T^{2} + 3830 T^{3} + 41947 T^{4} + 3830 p T^{5} + 361 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 6 T + 42 T^{2} - 299 T^{3} + 6725 T^{4} - 299 p T^{5} + 42 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 19 T + 328 T^{2} + 3913 T^{3} + 41045 T^{4} + 3913 p T^{5} + 328 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 13 T + 192 T^{2} - 359 T^{3} + 6571 T^{4} - 359 p T^{5} + 192 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 3 T + 215 T^{2} - 134 T^{3} + 22159 T^{4} - 134 p T^{5} + 215 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.21707211199882150420906897951, −5.62765560133617457043634580683, −5.60183885356068944884831083483, −5.42788828461529132204609062467, −5.36868057284605372614730513470, −5.16991325542199445195813674536, −5.01390917411152653256073978464, −4.79199564283010677233864544332, −4.55791846299138164567256249743, −4.52930481511957199947197218287, −4.13037932094771198283572760032, −3.83810380029972299025340226113, −3.69152633798987785390984957439, −3.44233790783549723417260174047, −3.38412624118760638289150225364, −3.33829925082152589887194268419, −3.22110033884078361812136658573, −2.89760045532121684265435680455, −2.66340961161866601253521189391, −2.61636673773839284366531210593, −2.10355768826687903840202863667, −1.77295165356389005955130634656, −1.51728548505050439234104095248, −1.45486250230005629251177470154, −1.17534974312954118788322975328, 0, 0, 0, 0, 1.17534974312954118788322975328, 1.45486250230005629251177470154, 1.51728548505050439234104095248, 1.77295165356389005955130634656, 2.10355768826687903840202863667, 2.61636673773839284366531210593, 2.66340961161866601253521189391, 2.89760045532121684265435680455, 3.22110033884078361812136658573, 3.33829925082152589887194268419, 3.38412624118760638289150225364, 3.44233790783549723417260174047, 3.69152633798987785390984957439, 3.83810380029972299025340226113, 4.13037932094771198283572760032, 4.52930481511957199947197218287, 4.55791846299138164567256249743, 4.79199564283010677233864544332, 5.01390917411152653256073978464, 5.16991325542199445195813674536, 5.36868057284605372614730513470, 5.42788828461529132204609062467, 5.60183885356068944884831083483, 5.62765560133617457043634580683, 6.21707211199882150420906897951

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