Properties

Label 8-4730e4-1.1-c1e4-0-0
Degree $8$
Conductor $5.005\times 10^{14}$
Sign $1$
Analytic cond. $2.03494\times 10^{6}$
Root an. cond. $6.14566$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3·3-s + 10·4-s + 4·5-s − 12·6-s + 20·8-s + 16·10-s − 4·11-s − 30·12-s − 9·13-s − 12·15-s + 35·16-s − 15·17-s − 2·19-s + 40·20-s − 16·22-s − 14·23-s − 60·24-s + 10·25-s − 36·26-s + 11·27-s − 8·29-s − 48·30-s + 4·31-s + 56·32-s + 12·33-s − 60·34-s + ⋯
L(s)  = 1  + 2.82·2-s − 1.73·3-s + 5·4-s + 1.78·5-s − 4.89·6-s + 7.07·8-s + 5.05·10-s − 1.20·11-s − 8.66·12-s − 2.49·13-s − 3.09·15-s + 35/4·16-s − 3.63·17-s − 0.458·19-s + 8.94·20-s − 3.41·22-s − 2.91·23-s − 12.2·24-s + 2·25-s − 7.06·26-s + 2.11·27-s − 1.48·29-s − 8.76·30-s + 0.718·31-s + 9.89·32-s + 2.08·33-s − 10.2·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{4} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{4} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 11^{4} \cdot 43^{4}\)
Sign: $1$
Analytic conductor: \(2.03494\times 10^{6}\)
Root analytic conductor: \(6.14566\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 11^{4} \cdot 43^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad2$C_1$ \( ( 1 - T )^{4} \)
5$C_1$ \( ( 1 - T )^{4} \)
11$C_1$ \( ( 1 + T )^{4} \)
43$C_1$ \( ( 1 + T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + p T + p^{2} T^{2} + 16 T^{3} + 32 T^{4} + 16 p T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 2 p T^{2} - 2 T^{3} + 141 T^{4} - 2 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 9 T + 72 T^{2} + 345 T^{3} + 1512 T^{4} + 345 p T^{5} + 72 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 15 T + 115 T^{2} + 582 T^{3} + 2488 T^{4} + 582 p T^{5} + 115 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 2 T + 44 T^{2} + 6 T^{3} + 891 T^{4} + 6 p T^{5} + 44 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 14 T + 132 T^{2} + 36 p T^{3} + 4478 T^{4} + 36 p^{2} T^{5} + 132 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 8 T + 120 T^{2} + 678 T^{3} + 5267 T^{4} + 678 p T^{5} + 120 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 4 T + 74 T^{2} - 8 p T^{3} + 3283 T^{4} - 8 p^{2} T^{5} + 74 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 13 T + 141 T^{2} + 1130 T^{3} + 8004 T^{4} + 1130 p T^{5} + 141 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + T + 154 T^{2} + 113 T^{3} + 9280 T^{4} + 113 p T^{5} + 154 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
53$C_2 \wr S_4$ \( 1 + 5 T + 117 T^{2} + 658 T^{3} + 8538 T^{4} + 658 p T^{5} + 117 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 10 T + 160 T^{2} - 1184 T^{3} + 12430 T^{4} - 1184 p T^{5} + 160 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 12 T + 140 T^{2} + 1494 T^{3} + 10387 T^{4} + 1494 p T^{5} + 140 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 17 T + 256 T^{2} + 2917 T^{3} + 25018 T^{4} + 2917 p T^{5} + 256 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 3 T + 143 T^{2} + 200 T^{3} + 9062 T^{4} + 200 p T^{5} + 143 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 21 T + 442 T^{2} + 5027 T^{3} + 54266 T^{4} + 5027 p T^{5} + 442 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 13 T + 142 T^{2} + 1699 T^{3} + 21232 T^{4} + 1699 p T^{5} + 142 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 16 T + 212 T^{2} + 1662 T^{3} + 17126 T^{4} + 1662 p T^{5} + 212 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 11 T + 171 T^{2} - 828 T^{3} + 12944 T^{4} - 828 p T^{5} + 171 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 26 T + 412 T^{2} - 5550 T^{3} + 62678 T^{4} - 5550 p T^{5} + 412 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.14361184301253792975878895009, −5.85943379148380675994214807620, −5.79233466070307124499283108731, −5.77541755866677781339469723496, −5.67857417242281768475744071787, −5.04533749347598407878559987072, −4.90267852061984151339419887687, −4.89905592961351820806281425430, −4.84199805098046230706668941273, −4.65064135012599589194183051629, −4.52736616069317953608813844049, −4.41447750363982924827960223609, −4.06265715166602358346561906426, −3.42715732421380913352780263510, −3.40952431233701584037872339800, −3.37750621522699796006570016695, −3.24054955714549380750515848063, −2.52266329568597306040936601062, −2.46202957448895099094674836265, −2.40832468774646131690744964583, −2.18301760939844646324842446250, −2.06375705659878414225743145159, −1.87212869955207462422956252338, −1.52029851394806000765686990602, −1.22946611126645627911295550013, 0, 0, 0, 0, 1.22946611126645627911295550013, 1.52029851394806000765686990602, 1.87212869955207462422956252338, 2.06375705659878414225743145159, 2.18301760939844646324842446250, 2.40832468774646131690744964583, 2.46202957448895099094674836265, 2.52266329568597306040936601062, 3.24054955714549380750515848063, 3.37750621522699796006570016695, 3.40952431233701584037872339800, 3.42715732421380913352780263510, 4.06265715166602358346561906426, 4.41447750363982924827960223609, 4.52736616069317953608813844049, 4.65064135012599589194183051629, 4.84199805098046230706668941273, 4.89905592961351820806281425430, 4.90267852061984151339419887687, 5.04533749347598407878559987072, 5.67857417242281768475744071787, 5.77541755866677781339469723496, 5.79233466070307124499283108731, 5.85943379148380675994214807620, 6.14361184301253792975878895009

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