Properties

Label 8-2016e4-1.1-c1e4-0-9
Degree $8$
Conductor $1.652\times 10^{13}$
Sign $1$
Analytic cond. $67153.7$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 16·13-s + 2·17-s + 19·25-s + 16·29-s + 10·37-s − 32·41-s + 14·49-s + 14·53-s + 10·61-s + 96·65-s + 18·73-s − 12·85-s − 18·89-s − 32·97-s − 30·101-s − 2·109-s + 15·121-s − 66·125-s + 127-s + 131-s + 137-s + 139-s − 96·145-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 2.68·5-s − 4.43·13-s + 0.485·17-s + 19/5·25-s + 2.97·29-s + 1.64·37-s − 4.99·41-s + 2·49-s + 1.92·53-s + 1.28·61-s + 11.9·65-s + 2.10·73-s − 1.30·85-s − 1.90·89-s − 3.24·97-s − 2.98·101-s − 0.191·109-s + 1.36·121-s − 5.90·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7.97·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(67153.7\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2016} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8713350557\)
\(L(\frac12)\) \(\approx\) \(0.8713350557\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 - 15 T^{2} + 104 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^3$ \( 1 + 25 T^{2} + 264 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^3$ \( 1 - 39 T^{2} + 992 T^{4} - 39 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
31$C_2^3$ \( 1 - 55 T^{2} + 2064 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 87 T^{2} + 5360 T^{4} - 87 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 7 T - 4 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 111 T^{2} + 8840 T^{4} - 111 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 - 127 T^{2} + 11640 T^{4} - 127 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^3$ \( 1 - 151 T^{2} + 16560 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
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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

−6.59126505120635934808026150108, −6.58938785579002202947655824047, −6.02336448158187654924155466510, −5.95108204420574895342606301619, −5.44571403775347421711467932631, −5.13946341350393984729852944803, −5.11830150325297986493532674103, −5.07376860927659775474446478432, −4.93534479187327599463394063256, −4.47701646489852219843341050654, −4.38412475831685560382062743471, −4.07475957722014555136878686291, −3.96186882139899164203014525417, −3.79976341094494879058115148836, −3.45311620282571182134148518719, −3.02439394334540451085005121174, −2.83927779993662222109080614014, −2.63986301812814163669629027071, −2.62785971823998841128579185016, −2.33063206810995252516373757938, −1.70485322844858430998255331844, −1.53350288105084016813867059960, −0.828907793163875102638107506399, −0.46202895887234430552831230870, −0.35929399108056446612544159993, 0.35929399108056446612544159993, 0.46202895887234430552831230870, 0.828907793163875102638107506399, 1.53350288105084016813867059960, 1.70485322844858430998255331844, 2.33063206810995252516373757938, 2.62785971823998841128579185016, 2.63986301812814163669629027071, 2.83927779993662222109080614014, 3.02439394334540451085005121174, 3.45311620282571182134148518719, 3.79976341094494879058115148836, 3.96186882139899164203014525417, 4.07475957722014555136878686291, 4.38412475831685560382062743471, 4.47701646489852219843341050654, 4.93534479187327599463394063256, 5.07376860927659775474446478432, 5.11830150325297986493532674103, 5.13946341350393984729852944803, 5.44571403775347421711467932631, 5.95108204420574895342606301619, 6.02336448158187654924155466510, 6.58938785579002202947655824047, 6.59126505120635934808026150108

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