Properties

Label 8-3920e4-1.1-c1e4-0-0
Degree $8$
Conductor $2.361\times 10^{14}$
Sign $1$
Analytic cond. $959959.$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·5-s − 9-s − 2·11-s + 10·13-s − 8·15-s + 6·17-s + 4·23-s + 10·25-s + 2·27-s − 2·29-s + 12·31-s + 4·33-s − 20·39-s + 12·41-s + 8·43-s − 4·45-s + 2·47-s − 12·51-s − 4·53-s − 8·55-s − 8·59-s + 20·61-s + 40·65-s + 8·67-s − 8·69-s − 4·71-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s − 1/3·9-s − 0.603·11-s + 2.77·13-s − 2.06·15-s + 1.45·17-s + 0.834·23-s + 2·25-s + 0.384·27-s − 0.371·29-s + 2.15·31-s + 0.696·33-s − 3.20·39-s + 1.87·41-s + 1.21·43-s − 0.596·45-s + 0.291·47-s − 1.68·51-s − 0.549·53-s − 1.07·55-s − 1.04·59-s + 2.56·61-s + 4.96·65-s + 0.977·67-s − 0.963·69-s − 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(959959.\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.04376613\)
\(L(\frac12)\) \(\approx\) \(10.04376613\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 26 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 3 p T^{2} + 46 T^{3} + 480 T^{4} + 46 p T^{5} + 3 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 71 T^{2} - 2 p^{2} T^{3} + 1384 T^{4} - 2 p^{3} T^{5} + 71 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + p T^{2} - 62 T^{3} + 434 T^{4} - 62 p T^{5} + p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 50 T^{2} - 24 T^{3} + 1186 T^{4} - 24 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 54 T^{2} - 324 T^{3} + 1442 T^{4} - 324 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 73 T^{2} + 154 T^{3} + 2740 T^{4} + 154 p T^{5} + 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 134 T^{2} - 828 T^{3} + 5602 T^{4} - 828 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 106 T^{2} - 40 T^{3} + 5114 T^{4} - 40 p T^{5} + 106 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 68 T^{2} + 52 T^{3} - 2014 T^{4} + 52 p T^{5} + 68 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 124 T^{2} - 840 T^{3} + 7718 T^{4} - 840 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 147 T^{2} - 378 T^{3} + 9344 T^{4} - 378 p T^{5} + 147 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + 8866 T^{4} + 4 p^{2} T^{5} + 142 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 288 T^{2} - 2572 T^{3} + 22350 T^{4} - 2572 p T^{5} + 288 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 154 T^{2} - 1176 T^{3} + 13994 T^{4} - 1176 p T^{5} + 154 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 32 T^{2} + 12 p T^{3} + 5438 T^{4} + 12 p^{2} T^{5} + 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 122 T^{2} + 240 T^{3} - 7038 T^{4} + 240 p T^{5} + 122 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 389 T^{2} + 4726 T^{3} + 48924 T^{4} + 4726 p T^{5} + 389 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 36 T + 750 T^{2} + 10564 T^{3} + 110978 T^{4} + 10564 p T^{5} + 750 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 40 T + 914 T^{2} - 13800 T^{3} + 152258 T^{4} - 13800 p T^{5} + 914 p^{2} T^{6} - 40 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 26 T + 545 T^{2} - 7602 T^{3} + 85890 T^{4} - 7602 p T^{5} + 545 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.06605111410860563218184223885, −5.72927914034252776565065219838, −5.63976296750575586509221207143, −5.46621850197378854173999521811, −5.38138677550356775741188374995, −5.12703677309654570573831727374, −4.87849174985891845520854061899, −4.77329998434688235527370987883, −4.37577151104616233967181705752, −4.15393535370905240442541746576, −4.00057873257058881550515434115, −3.82886981208098516003163893817, −3.54478230772689711873937403038, −3.04884443364127360126484763651, −3.00047224673161124470679969896, −2.91533558926522983221710761063, −2.86023475852019576797050357065, −2.14858716271025459932986164317, −2.10186040082033967235576274976, −1.86838455247738737990225607903, −1.58335000728241232489185476295, −1.08342841746518962716229099511, −0.918599883763213567705693806145, −0.74151820273403776564704796328, −0.57903530085571787698534637711, 0.57903530085571787698534637711, 0.74151820273403776564704796328, 0.918599883763213567705693806145, 1.08342841746518962716229099511, 1.58335000728241232489185476295, 1.86838455247738737990225607903, 2.10186040082033967235576274976, 2.14858716271025459932986164317, 2.86023475852019576797050357065, 2.91533558926522983221710761063, 3.00047224673161124470679969896, 3.04884443364127360126484763651, 3.54478230772689711873937403038, 3.82886981208098516003163893817, 4.00057873257058881550515434115, 4.15393535370905240442541746576, 4.37577151104616233967181705752, 4.77329998434688235527370987883, 4.87849174985891845520854061899, 5.12703677309654570573831727374, 5.38138677550356775741188374995, 5.46621850197378854173999521811, 5.63976296750575586509221207143, 5.72927914034252776565065219838, 6.06605111410860563218184223885

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