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Properties

Label 8-1449e4-1.1-c1e4-0-3
Degree $8$
Conductor $4.408\times 10^{12}$
Sign $1$
Analytic cond. $17921.8$
Root an. cond. $3.40151$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5·5-s + 4·7-s + 8-s + 5·11-s + 7·13-s + 2·16-s − 12·17-s + 3·19-s + 10·20-s + 4·23-s + 6·25-s − 8·28-s − 6·29-s + 4·31-s − 2·32-s − 20·35-s + 20·37-s − 5·40-s − 3·41-s + 9·43-s − 10·44-s − 7·47-s + 10·49-s − 14·52-s + 6·53-s − 25·55-s + ⋯
L(s)  = 1  − 4-s − 2.23·5-s + 1.51·7-s + 0.353·8-s + 1.50·11-s + 1.94·13-s + 1/2·16-s − 2.91·17-s + 0.688·19-s + 2.23·20-s + 0.834·23-s + 6/5·25-s − 1.51·28-s − 1.11·29-s + 0.718·31-s − 0.353·32-s − 3.38·35-s + 3.28·37-s − 0.790·40-s − 0.468·41-s + 1.37·43-s − 1.50·44-s − 1.02·47-s + 10/7·49-s − 1.94·52-s + 0.824·53-s − 3.37·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(17921.8\)
Root analytic conductor: \(3.40151\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.213325244\)
\(L(\frac12)\) \(\approx\) \(3.213325244\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
23$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + p T^{2} - T^{3} + p T^{4} - p T^{5} + p^{3} T^{6} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 + p T + 19 T^{2} + 61 T^{3} + 148 T^{4} + 61 p T^{5} + 19 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 5 T + 35 T^{2} - 100 T^{3} + 460 T^{4} - 100 p T^{5} + 35 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 7 T + 41 T^{2} - 121 T^{3} + 492 T^{4} - 121 p T^{5} + 41 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 12 T + 105 T^{2} + 602 T^{3} + 2888 T^{4} + 602 p T^{5} + 105 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 3 T + 39 T^{2} - 120 T^{3} + 812 T^{4} - 120 p T^{5} + 39 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 6 T + 85 T^{2} + 258 T^{3} + 2812 T^{4} + 258 p T^{5} + 85 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 4 T + 113 T^{2} - 326 T^{3} + 5068 T^{4} - 326 p T^{5} + 113 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 20 T + 239 T^{2} - 1940 T^{3} + 13080 T^{4} - 1940 p T^{5} + 239 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 3 T + 137 T^{2} + 298 T^{3} + 7838 T^{4} + 298 p T^{5} + 137 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 9 T + 133 T^{2} - 977 T^{3} + 8012 T^{4} - 977 p T^{5} + 133 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 7 T + 194 T^{2} + 959 T^{3} + 13786 T^{4} + 959 p T^{5} + 194 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 6 T + 144 T^{2} - 705 T^{3} + 9444 T^{4} - 705 p T^{5} + 144 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 2 T - 42 T^{2} - 91 T^{3} + 5646 T^{4} - 91 p T^{5} - 42 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 24 T + 416 T^{2} - 4583 T^{3} + 42024 T^{4} - 4583 p T^{5} + 416 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - T + 127 T^{2} - 225 T^{3} + 12248 T^{4} - 225 p T^{5} + 127 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 17 T + 363 T^{2} - 3749 T^{3} + 41528 T^{4} - 3749 p T^{5} + 363 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 16 T + 197 T^{2} - 1234 T^{3} + 11448 T^{4} - 1234 p T^{5} + 197 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 10 T + 299 T^{2} + 2156 T^{3} + 34888 T^{4} + 2156 p T^{5} + 299 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 8 T + 137 T^{2} - 458 T^{3} + 1708 T^{4} - 458 p T^{5} + 137 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 3 T + 243 T^{2} - 1181 T^{3} + 27320 T^{4} - 1181 p T^{5} + 243 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 2 T + 301 T^{2} + 226 T^{3} + 39580 T^{4} + 226 p T^{5} + 301 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.75482314270277489328946716004, −6.54188593670510091225262653045, −6.26527625400123745309812827712, −6.21593634501715843295262002119, −6.18857255069453120421485758447, −5.61542048682272155027672847041, −5.33389264968437426012314506799, −5.21730686776054707791465178814, −4.87612297281730863126342357551, −4.67677228673703421201738031280, −4.44001302761881934070506937183, −4.24006939781376570562638045585, −4.14542886113682980717603561464, −3.88364053689209564595947871843, −3.82074467726599430102288110264, −3.47262107097778259514272468392, −3.44467052638250643823348991228, −2.66325502998751394349493692560, −2.60555125049593158330926400402, −2.17753329320047194668244540053, −1.76591910803781584328621050263, −1.61667524175534746634977226468, −0.857669742248525348815589827704, −0.828487297195982993622951530321, −0.55750122907402016844331512209, 0.55750122907402016844331512209, 0.828487297195982993622951530321, 0.857669742248525348815589827704, 1.61667524175534746634977226468, 1.76591910803781584328621050263, 2.17753329320047194668244540053, 2.60555125049593158330926400402, 2.66325502998751394349493692560, 3.44467052638250643823348991228, 3.47262107097778259514272468392, 3.82074467726599430102288110264, 3.88364053689209564595947871843, 4.14542886113682980717603561464, 4.24006939781376570562638045585, 4.44001302761881934070506937183, 4.67677228673703421201738031280, 4.87612297281730863126342357551, 5.21730686776054707791465178814, 5.33389264968437426012314506799, 5.61542048682272155027672847041, 6.18857255069453120421485758447, 6.21593634501715843295262002119, 6.26527625400123745309812827712, 6.54188593670510091225262653045, 6.75482314270277489328946716004

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