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Properties

Label 8-575e4-1.1-c1e4-0-6
Degree $8$
Conductor $109312890625$
Sign $1$
Analytic cond. $444.405$
Root an. cond. $2.14275$
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  − 2·3-s − 3·4-s − 6·7-s − 2·8-s − 2·9-s + 2·11-s + 6·12-s − 14·13-s + 4·16-s − 14·17-s − 4·19-s + 12·21-s + 4·23-s + 4·24-s + 4·27-s + 18·28-s + 4·29-s + 8·32-s − 4·33-s + 6·36-s − 2·37-s + 28·39-s − 8·41-s − 4·43-s − 6·44-s − 2·47-s − 8·48-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s − 2.26·7-s − 0.707·8-s − 2/3·9-s + 0.603·11-s + 1.73·12-s − 3.88·13-s + 16-s − 3.39·17-s − 0.917·19-s + 2.61·21-s + 0.834·23-s + 0.816·24-s + 0.769·27-s + 3.40·28-s + 0.742·29-s + 1.41·32-s − 0.696·33-s + 36-s − 0.328·37-s + 4.48·39-s − 1.24·41-s − 0.609·43-s − 0.904·44-s − 0.291·47-s − 1.15·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(444.405\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad5 \( 1 \)
23$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + 3 T^{2} + p T^{3} + 5 T^{4} + p^{2} T^{5} + 3 p^{2} T^{6} + p^{4} T^{8} \)
3$C_2 \wr S_4$ \( 1 + 2 T + 2 p T^{2} + 4 p T^{3} + 23 T^{4} + 4 p^{2} T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 6 T + 32 T^{2} + 114 T^{3} + 346 T^{4} + 114 p T^{5} + 32 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 2 T + 28 T^{2} - 2 p T^{3} + 346 T^{4} - 2 p^{2} T^{5} + 28 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 14 T + 118 T^{2} + 664 T^{3} + 2791 T^{4} + 664 p T^{5} + 118 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 14 T + 126 T^{2} + 778 T^{3} + 3726 T^{4} + 778 p T^{5} + 126 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 4 T + 70 T^{2} + 200 T^{3} + 1918 T^{4} + 200 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 4 T + 94 T^{2} - 344 T^{3} + 3775 T^{4} - 344 p T^{5} + 94 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 50 T^{2} - 256 T^{3} + 1011 T^{4} - 256 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 2 T + 76 T^{2} - 158 T^{3} + 2410 T^{4} - 158 p T^{5} + 76 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 8 T + 70 T^{2} + 616 T^{3} + 4863 T^{4} + 616 p T^{5} + 70 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 4 T + 54 T^{2} + 80 T^{3} + 2910 T^{4} + 80 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 2 T + 50 T^{2} + 116 T^{3} + 4795 T^{4} + 116 p T^{5} + 50 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 4 T + 108 T^{2} + 524 T^{3} + 8022 T^{4} + 524 p T^{5} + 108 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 216 T^{2} + 16 T^{3} + 18542 T^{4} + 16 p T^{5} + 216 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 8 T + 142 T^{2} + 316 T^{3} + 7126 T^{4} + 316 p T^{5} + 142 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 2 T + 260 T^{2} - 390 T^{3} + 25866 T^{4} - 390 p T^{5} + 260 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 24 T + 350 T^{2} + 3544 T^{3} + 32183 T^{4} + 3544 p T^{5} + 350 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 18 T + 270 T^{2} + 2088 T^{3} + 20423 T^{4} + 2088 p T^{5} + 270 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 24 T + 494 T^{2} - 6100 T^{3} + 65598 T^{4} - 6100 p T^{5} + 494 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 6 T + 62 T^{2} - 518 T^{3} + 11462 T^{4} - 518 p T^{5} + 62 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 8 T + 270 T^{2} + 1764 T^{3} + 34598 T^{4} + 1764 p T^{5} + 270 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 34 T + 736 T^{2} + 10514 T^{3} + 119290 T^{4} + 10514 p T^{5} + 736 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.433111981195191621083343840025, −7.83550137406951632517167433637, −7.81230632146078720605101239489, −7.38735265934259576812546622562, −7.13495112639324830089095307297, −6.87020955186298845006265579010, −6.53075197174651911061530787520, −6.52881991139788445709684722705, −6.46059058115025077899954655516, −6.24298680020580603624180141728, −5.63342474800200471825770591769, −5.61280108071471714174037240422, −5.25873810800949290026357622564, −4.89502501376892766693312428799, −4.73280117456249083193462129310, −4.65808566254475230126659928981, −4.36188884015730104378408567927, −4.10772432859256061901774780325, −3.57944052786861155921600525855, −3.36912112312066648075705241916, −2.81925552567233433081919613707, −2.73056148281291366437688210093, −2.58035597777517470644538672901, −2.11629666377104314734823351759, −1.48363314772937410152673770352, 0, 0, 0, 0, 1.48363314772937410152673770352, 2.11629666377104314734823351759, 2.58035597777517470644538672901, 2.73056148281291366437688210093, 2.81925552567233433081919613707, 3.36912112312066648075705241916, 3.57944052786861155921600525855, 4.10772432859256061901774780325, 4.36188884015730104378408567927, 4.65808566254475230126659928981, 4.73280117456249083193462129310, 4.89502501376892766693312428799, 5.25873810800949290026357622564, 5.61280108071471714174037240422, 5.63342474800200471825770591769, 6.24298680020580603624180141728, 6.46059058115025077899954655516, 6.52881991139788445709684722705, 6.53075197174651911061530787520, 6.87020955186298845006265579010, 7.13495112639324830089095307297, 7.38735265934259576812546622562, 7.81230632146078720605101239489, 7.83550137406951632517167433637, 8.433111981195191621083343840025

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