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Properties

Label 8-483e4-1.1-c1e4-0-3
Degree $8$
Conductor $54423757521$
Sign $1$
Analytic cond. $221.256$
Root an. cond. $1.96386$
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  − 4·3-s − 2·4-s + 5·5-s + 4·7-s − 8-s + 10·9-s − 5·11-s + 8·12-s + 7·13-s − 20·15-s + 2·16-s + 12·17-s + 3·19-s − 10·20-s − 16·21-s − 4·23-s + 4·24-s + 6·25-s − 20·27-s − 8·28-s + 6·29-s + 4·31-s + 2·32-s + 20·33-s + 20·35-s − 20·36-s + 20·37-s + ⋯
L(s)  = 1  − 2.30·3-s − 4-s + 2.23·5-s + 1.51·7-s − 0.353·8-s + 10/3·9-s − 1.50·11-s + 2.30·12-s + 1.94·13-s − 5.16·15-s + 1/2·16-s + 2.91·17-s + 0.688·19-s − 2.23·20-s − 3.49·21-s − 0.834·23-s + 0.816·24-s + 6/5·25-s − 3.84·27-s − 1.51·28-s + 1.11·29-s + 0.718·31-s + 0.353·32-s + 3.48·33-s + 3.38·35-s − 3.33·36-s + 3.28·37-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(221.256\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.302310129\)
\(L(\frac12)\) \(\approx\) \(2.302310129\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
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

−8.066018200417150483720349876454, −7.78139986995150934986569728195, −7.37391490704649656117541285021, −7.33753146973104786101675728606, −6.77979494219787976788203380379, −6.58120269401366277617222339993, −6.08533705941490201296588103865, −6.00430580177124077695535307342, −5.88872965936954736190969989440, −5.73534552075346854950231378122, −5.40938755206536319707534680853, −5.33080459027650471886209431621, −5.31379707727142957389546270355, −4.67908851655242500978606384479, −4.47409140690282135644917560508, −4.15548279852060529630866474558, −4.13830050850588763243702257286, −3.46742188186919482396754776148, −3.15232867067735294678145979384, −2.62631240960185589906561375320, −2.32691015753255618927708196658, −1.87548980401712755916836128567, −1.19695069786031820778788067855, −1.08236163856229215760881166669, −0.853408866777955455298082469435, 0.853408866777955455298082469435, 1.08236163856229215760881166669, 1.19695069786031820778788067855, 1.87548980401712755916836128567, 2.32691015753255618927708196658, 2.62631240960185589906561375320, 3.15232867067735294678145979384, 3.46742188186919482396754776148, 4.13830050850588763243702257286, 4.15548279852060529630866474558, 4.47409140690282135644917560508, 4.67908851655242500978606384479, 5.31379707727142957389546270355, 5.33080459027650471886209431621, 5.40938755206536319707534680853, 5.73534552075346854950231378122, 5.88872965936954736190969989440, 6.00430580177124077695535307342, 6.08533705941490201296588103865, 6.58120269401366277617222339993, 6.77979494219787976788203380379, 7.33753146973104786101675728606, 7.37391490704649656117541285021, 7.78139986995150934986569728195, 8.066018200417150483720349876454

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