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Properties

Label 8-10e16-1.1-c1e4-0-13
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  + 4·3-s + 8·7-s + 3·9-s + 2·11-s − 14·13-s − 8·17-s + 32·21-s + 4·23-s − 10·27-s − 10·29-s − 8·31-s + 8·33-s − 18·37-s − 56·39-s − 12·41-s + 14·43-s + 8·47-s + 22·49-s − 32·51-s − 4·53-s − 2·61-s + 24·63-s − 2·67-s + 16·69-s − 8·71-s − 24·73-s + 16·77-s + ⋯
L(s)  = 1  + 2.30·3-s + 3.02·7-s + 9-s + 0.603·11-s − 3.88·13-s − 1.94·17-s + 6.98·21-s + 0.834·23-s − 1.92·27-s − 1.85·29-s − 1.43·31-s + 1.39·33-s − 2.95·37-s − 8.96·39-s − 1.87·41-s + 2.13·43-s + 1.16·47-s + 22/7·49-s − 4.48·51-s − 0.549·53-s − 0.256·61-s + 3.02·63-s − 0.244·67-s + 1.92·69-s − 0.949·71-s − 2.80·73-s + 1.82·77-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$C_4\times C_2$ \( 1 - 4 T + 13 T^{2} - 10 p T^{3} + 61 T^{4} - 10 p^{2} T^{5} + 13 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$((C_8 : C_2):C_2):C_2$ \( 1 - 8 T + 6 p T^{2} - 160 T^{3} + 471 T^{4} - 160 p T^{5} + 6 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 18 T^{2} + 16 T^{3} + 95 T^{4} + 16 p T^{5} + 18 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 118 T^{2} + 660 T^{3} + 2771 T^{4} + 660 p T^{5} + 118 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 87 T^{2} + 420 T^{3} + 2381 T^{4} + 420 p T^{5} + 87 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 61 T^{2} - 10 T^{3} + 1601 T^{4} - 10 p T^{5} + 61 p^{2} T^{6} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 78 T^{2} - 280 T^{3} + 2531 T^{4} - 280 p T^{5} + 78 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 126 T^{2} + 780 T^{3} + 5531 T^{4} + 780 p T^{5} + 126 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 118 T^{2} + 696 T^{3} + 5375 T^{4} + 696 p T^{5} + 118 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 18 T + 257 T^{2} + 2270 T^{3} + 16521 T^{4} + 2270 p T^{5} + 257 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 78 T^{2} - 136 T^{3} - 2305 T^{4} - 136 p T^{5} + 78 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - 14 T + 158 T^{2} - 1080 T^{3} + 7971 T^{4} - 1080 p T^{5} + 158 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 - 8 T + 82 T^{2} + 80 T^{3} + 351 T^{4} + 80 p T^{5} + 82 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 188 T^{2} + 540 T^{3} + 14246 T^{4} + 540 p T^{5} + 188 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 + 156 T^{2} + 12726 T^{4} + 156 p^{2} T^{6} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 173 T^{2} + 354 T^{3} + 13845 T^{4} + 354 p T^{5} + 173 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 242 T^{2} + 320 T^{3} + 23391 T^{4} + 320 p T^{5} + 242 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 198 T^{2} + 1416 T^{3} + 19055 T^{4} + 1416 p T^{5} + 198 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 24 T + 428 T^{2} + 5160 T^{3} + 51526 T^{4} + 5160 p T^{5} + 428 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 20 T + 356 T^{2} + 3700 T^{3} + 39046 T^{4} + 3700 p T^{5} + 356 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 - 14 T + 218 T^{2} - 1720 T^{3} + 18271 T^{4} - 1720 p T^{5} + 218 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 + 28 T + 647 T^{2} + 8980 T^{3} + 107061 T^{4} + 8980 p T^{5} + 647 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.69071994772527548044522923895, −5.27554826189452585016745957068, −5.23907322104600457135878809950, −5.18468115784635574174418280573, −5.05670055602145480865872208202, −4.70086172230301452927358641641, −4.65201010316484631010841261452, −4.50852772784117167717229913783, −4.32705027160962191000822327838, −4.07652339994339046593784549568, −3.77941738855599631114176496059, −3.68441607652278824439681792769, −3.68290850287279539871216217420, −3.01847508471059411039616714907, −2.96077944764696368725351059580, −2.85555429767071356232906783541, −2.83648911034918967478770821877, −2.29449478878004448351848100234, −2.26998509880343413035137259810, −2.15261073330484499052287953324, −2.08786410526564441342316688834, −1.62583349073276121352817550545, −1.55869672217997812151427372102, −1.25425003832147834456420268235, −1.23360274248310744954016506614, 0, 0, 0, 0, 1.23360274248310744954016506614, 1.25425003832147834456420268235, 1.55869672217997812151427372102, 1.62583349073276121352817550545, 2.08786410526564441342316688834, 2.15261073330484499052287953324, 2.26998509880343413035137259810, 2.29449478878004448351848100234, 2.83648911034918967478770821877, 2.85555429767071356232906783541, 2.96077944764696368725351059580, 3.01847508471059411039616714907, 3.68290850287279539871216217420, 3.68441607652278824439681792769, 3.77941738855599631114176496059, 4.07652339994339046593784549568, 4.32705027160962191000822327838, 4.50852772784117167717229913783, 4.65201010316484631010841261452, 4.70086172230301452927358641641, 5.05670055602145480865872208202, 5.18468115784635574174418280573, 5.23907322104600457135878809950, 5.27554826189452585016745957068, 5.69071994772527548044522923895

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