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Properties

Label 8-10e16-1.1-c1e4-0-6
Degree $8$
Conductor $1.000\times 10^{16}$
Sign $1$
Analytic cond. $4.06544\times 10^{7}$
Root an. cond. $8.93590$
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  + 3-s + 2·7-s − 7·9-s + 2·11-s − 6·13-s − 7·17-s − 15·19-s + 2·21-s + 6·23-s − 10·27-s + 10·29-s − 8·31-s + 2·33-s − 12·37-s − 6·39-s + 3·41-s − 14·43-s + 2·47-s − 8·49-s − 7·51-s + 4·53-s − 15·57-s + 20·59-s + 18·61-s − 14·63-s − 23·67-s + 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s − 7/3·9-s + 0.603·11-s − 1.66·13-s − 1.69·17-s − 3.44·19-s + 0.436·21-s + 1.25·23-s − 1.92·27-s + 1.85·29-s − 1.43·31-s + 0.348·33-s − 1.97·37-s − 0.960·39-s + 0.468·41-s − 2.13·43-s + 0.291·47-s − 8/7·49-s − 0.980·51-s + 0.549·53-s − 1.98·57-s + 2.60·59-s + 2.30·61-s − 1.76·63-s − 2.80·67-s + 0.722·69-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(4.06544\times 10^{7}\)
Root analytic conductor: \(8.93590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
5 \( 1 \)
good3$((C_8 : C_2):C_2):C_2$ \( 1 - T + 8 T^{2} - 5 T^{3} + 31 T^{4} - 5 p T^{5} + 8 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 12 T^{2} - 10 T^{3} + 86 T^{4} - 10 p T^{5} + 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 28 T^{2} - 64 T^{3} + 405 T^{4} - 64 p T^{5} + 28 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 + 6 T + 28 T^{2} + 90 T^{3} + 246 T^{4} + 90 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 + 7 T + 52 T^{2} + 245 T^{3} + 1251 T^{4} + 245 p T^{5} + 52 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 15 T + 126 T^{2} + 705 T^{3} + 3311 T^{4} + 705 p T^{5} + 126 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 6 T + 28 T^{2} - 30 T^{3} - 234 T^{4} - 30 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 56 T^{2} + 50 T^{3} - 914 T^{4} + 50 p T^{5} + 56 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 108 T^{2} + 16 p T^{3} + 4310 T^{4} + 16 p^{2} T^{5} + 108 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 132 T^{2} + 780 T^{3} + 5606 T^{4} + 780 p T^{5} + 132 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 - 3 T + 88 T^{2} + 39 T^{3} + 3495 T^{4} + 39 p T^{5} + 88 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 193 T^{2} + 1430 T^{3} + 11611 T^{4} + 1430 p T^{5} + 193 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 112 T^{2} - 370 T^{3} + 6126 T^{4} - 370 p T^{5} + 112 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 148 T^{2} - 380 T^{3} + 10326 T^{4} - 380 p T^{5} + 148 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 20 T + 331 T^{2} - 3430 T^{3} + 31191 T^{4} - 3430 p T^{5} + 331 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 - 18 T + 168 T^{2} - 726 T^{3} + 3470 T^{4} - 726 p T^{5} + 168 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 23 T + 6 p T^{2} + 4465 T^{3} + 42851 T^{4} + 4465 p T^{5} + 6 p^{3} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 208 T^{2} + 1576 T^{3} + 19470 T^{4} + 1576 p T^{5} + 208 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + T + 168 T^{2} + 695 T^{3} + 13811 T^{4} + 695 p T^{5} + 168 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 256 T^{2} + 1450 T^{3} + 25486 T^{4} + 1450 p T^{5} + 256 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 268 T^{2} + 2320 T^{3} + 27621 T^{4} + 2320 p T^{5} + 268 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 - 5 T + 196 T^{2} - 1435 T^{3} + 19071 T^{4} - 1435 p T^{5} + 196 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + T + 93 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
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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

−5.68254725266675559290435130522, −5.40599351581268074662833137522, −5.33209857658778144321253028207, −5.25395755443087821251713957739, −5.02058975638148341802481070712, −4.74075226626595515097200567101, −4.57308806433134448687512005383, −4.55902595493980836895408145629, −4.39834623433462909624416478202, −3.90991215570129072276014412788, −3.84579340510966312382096552156, −3.77273382445347431153514876946, −3.73909104352849250297726942067, −3.02400340807536334503951706841, −3.01714468284746035457896822918, −2.88832825046697099302548728220, −2.77132367440770578086231861088, −2.50274199042082219934868581026, −2.22751460009284248172785804254, −2.10913909927224515991681090862, −2.00622170823406328024269743689, −1.63060637789403223867709657680, −1.59182443954800620228198271212, −1.06343121186708167212508155866, −0.830917986569368485232958910208, 0, 0, 0, 0, 0.830917986569368485232958910208, 1.06343121186708167212508155866, 1.59182443954800620228198271212, 1.63060637789403223867709657680, 2.00622170823406328024269743689, 2.10913909927224515991681090862, 2.22751460009284248172785804254, 2.50274199042082219934868581026, 2.77132367440770578086231861088, 2.88832825046697099302548728220, 3.01714468284746035457896822918, 3.02400340807536334503951706841, 3.73909104352849250297726942067, 3.77273382445347431153514876946, 3.84579340510966312382096552156, 3.90991215570129072276014412788, 4.39834623433462909624416478202, 4.55902595493980836895408145629, 4.57308806433134448687512005383, 4.74075226626595515097200567101, 5.02058975638148341802481070712, 5.25395755443087821251713957739, 5.33209857658778144321253028207, 5.40599351581268074662833137522, 5.68254725266675559290435130522

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