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Properties

Label 8-75e8-1.1-c1e4-0-0
Degree $8$
Conductor $1.001\times 10^{15}$
Sign $1$
Analytic cond. $4.07003\times 10^{6}$
Root an. cond. $6.70192$
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-s − 3·4-s + 5·7-s + 5·8-s + 6·11-s + 7·13-s − 5·14-s + 7·17-s − 9·19-s − 6·22-s − 10·23-s − 7·26-s − 15·28-s + 28·29-s − 10·31-s − 9·32-s − 7·34-s − 10·37-s + 9·38-s + 43-s − 18·44-s + 10·46-s + 23·47-s − 3·49-s − 21·52-s + 25·56-s − 28·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 3/2·4-s + 1.88·7-s + 1.76·8-s + 1.80·11-s + 1.94·13-s − 1.33·14-s + 1.69·17-s − 2.06·19-s − 1.27·22-s − 2.08·23-s − 1.37·26-s − 2.83·28-s + 5.19·29-s − 1.79·31-s − 1.59·32-s − 1.20·34-s − 1.64·37-s + 1.45·38-s + 0.152·43-s − 2.71·44-s + 1.47·46-s + 3.35·47-s − 3/7·49-s − 2.91·52-s + 3.34·56-s − 3.67·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(4.07003\times 10^{6}\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.483913540\)
\(L(\frac12)\) \(\approx\) \(5.483913540\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
good2$C_4\times C_2$ \( 1 + T + p^{2} T^{2} + p T^{3} + 9 T^{4} + p^{2} T^{5} + p^{4} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$((C_8 : C_2):C_2):C_2$ \( 1 - 5 T + 4 p T^{2} - 95 T^{3} + 289 T^{4} - 95 p T^{5} + 4 p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 - 6 T + 50 T^{2} - 189 T^{3} + 849 T^{4} - 189 p T^{5} + 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 - 7 T + 46 T^{2} - 181 T^{3} + 769 T^{4} - 181 p T^{5} + 46 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - 7 T + 52 T^{2} - 155 T^{3} + 831 T^{4} - 155 p T^{5} + 52 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 9 T + 82 T^{2} + 477 T^{3} + 2385 T^{4} + 477 p T^{5} + 82 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 112 T^{2} + 680 T^{3} + 4089 T^{4} + 680 p T^{5} + 112 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 - 28 T + 400 T^{2} - 3653 T^{3} + 23319 T^{4} - 3653 p T^{5} + 400 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 4 p T^{2} + 805 T^{3} + 5641 T^{4} + 805 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 58 T^{2} + 85 T^{3} + 79 T^{4} + 85 p T^{5} + 58 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 94 T^{2} + 135 T^{3} + 4491 T^{4} + 135 p T^{5} + 94 p^{2} T^{6} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - T + 93 T^{2} - 470 T^{3} + 3881 T^{4} - 470 p T^{5} + 93 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 - 23 T + 277 T^{2} - 2230 T^{3} + 15531 T^{4} - 2230 p T^{5} + 277 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 67 T^{2} + 270 T^{3} + 4479 T^{4} + 270 p T^{5} + 67 p^{2} T^{6} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 82 T^{2} + 287 T^{3} + 4245 T^{4} + 287 p T^{5} + 82 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 43 T + 913 T^{2} + 12286 T^{3} + 114205 T^{4} + 12286 p T^{5} + 913 p^{2} T^{6} + 43 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 - 8 T + 207 T^{2} - 1150 T^{3} + 18911 T^{4} - 1150 p T^{5} + 207 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 - 27 T + 418 T^{2} - 5079 T^{3} + 49545 T^{4} - 5079 p T^{5} + 418 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 232 T^{2} - 2385 T^{3} + 25959 T^{4} - 2385 p T^{5} + 232 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 211 T^{2} - 1000 T^{3} + 17701 T^{4} - 1000 p T^{5} + 211 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 + 3 T + 86 T^{2} + 549 T^{3} + 13989 T^{4} + 549 p T^{5} + 86 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 - 9 T + 352 T^{2} - 2307 T^{3} + 46875 T^{4} - 2307 p T^{5} + 352 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 13 T + 172 T^{2} - 2335 T^{3} + 29251 T^{4} - 2335 p T^{5} + 172 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.85677595694834124753882879135, −5.52968458089545596087798935309, −5.06889168665523756882931780531, −5.02777551560252679902474419487, −5.00405344907279420828765024836, −4.62042383526340905597474430842, −4.59414982066549374111737341582, −4.54990260572386613005976839816, −4.25936342131544478296941557752, −3.81288624515131638895155628402, −3.78892097555125073135990722382, −3.73954130340871449403360981334, −3.67060571282155849032794394061, −3.19165067589391078629246695653, −3.03283451713178317084112675834, −2.63562822129319105066769056392, −2.34829420733133660830467404184, −1.89891314031133822986936543109, −1.89299070221160507341514633484, −1.74864386827374048989704888325, −1.38527188191401878204294840136, −1.20858676443267471070563930296, −0.797921138681153388286984086047, −0.68695642305602218146483759407, −0.42313518584164741259639134421, 0.42313518584164741259639134421, 0.68695642305602218146483759407, 0.797921138681153388286984086047, 1.20858676443267471070563930296, 1.38527188191401878204294840136, 1.74864386827374048989704888325, 1.89299070221160507341514633484, 1.89891314031133822986936543109, 2.34829420733133660830467404184, 2.63562822129319105066769056392, 3.03283451713178317084112675834, 3.19165067589391078629246695653, 3.67060571282155849032794394061, 3.73954130340871449403360981334, 3.78892097555125073135990722382, 3.81288624515131638895155628402, 4.25936342131544478296941557752, 4.54990260572386613005976839816, 4.59414982066549374111737341582, 4.62042383526340905597474430842, 5.00405344907279420828765024836, 5.02777551560252679902474419487, 5.06889168665523756882931780531, 5.52968458089545596087798935309, 5.85677595694834124753882879135

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