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Properties

Label 8-4018e4-1.1-c1e4-0-0
Degree $8$
Conductor $2.606\times 10^{14}$
Sign $1$
Analytic cond. $1.05961\times 10^{6}$
Root an. cond. $5.66426$
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  − 4·2-s + 3-s + 10·4-s − 3·5-s − 4·6-s − 20·8-s − 9-s + 12·10-s + 4·11-s + 10·12-s + 6·13-s − 3·15-s + 35·16-s + 17-s + 4·18-s − 2·19-s − 30·20-s − 16·22-s + 6·23-s − 20·24-s + 25-s − 24·26-s + 2·27-s + 17·29-s + 12·30-s − 5·31-s − 56·32-s + ⋯
L(s)  = 1  − 2.82·2-s + 0.577·3-s + 5·4-s − 1.34·5-s − 1.63·6-s − 7.07·8-s − 1/3·9-s + 3.79·10-s + 1.20·11-s + 2.88·12-s + 1.66·13-s − 0.774·15-s + 35/4·16-s + 0.242·17-s + 0.942·18-s − 0.458·19-s − 6.70·20-s − 3.41·22-s + 1.25·23-s − 4.08·24-s + 1/5·25-s − 4.70·26-s + 0.384·27-s + 3.15·29-s + 2.19·30-s − 0.898·31-s − 9.89·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{8} \cdot 41^{4}\)
Sign: $1$
Analytic conductor: \(1.05961\times 10^{6}\)
Root analytic conductor: \(5.66426\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 7^{8} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9334462909\)
\(L(\frac12)\) \(\approx\) \(0.9334462909\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
7 \( 1 \)
41$C_1$ \( ( 1 + T )^{4} \)
good3$C_2^3:S_4$ \( 1 - T + 2 T^{2} - 5 T^{3} + 2 T^{4} - 5 p T^{5} + 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 3 T + 8 T^{2} + p T^{3} + 2 T^{4} + p^{2} T^{5} + 8 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 4 T + 36 T^{2} - 104 T^{3} + 534 T^{4} - 104 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 6 T + 44 T^{2} - 170 T^{3} + 774 T^{4} - 170 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - T + 10 T^{2} - 23 T^{3} - 86 T^{4} - 23 p T^{5} + 10 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 2 T + 56 T^{2} + 106 T^{3} + 1438 T^{4} + 106 p T^{5} + 56 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 6 T + 84 T^{2} - 350 T^{3} + 2774 T^{4} - 350 p T^{5} + 84 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 17 T + 178 T^{2} - 1231 T^{3} + 7294 T^{4} - 1231 p T^{5} + 178 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 5 T + 68 T^{2} + 289 T^{3} + 2326 T^{4} + 289 p T^{5} + 68 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 6 T + 120 T^{2} + 546 T^{3} + 6174 T^{4} + 546 p T^{5} + 120 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 13 T + 220 T^{2} + 1709 T^{3} + 15190 T^{4} + 1709 p T^{5} + 220 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 6 T + 108 T^{2} + 190 T^{3} + 4550 T^{4} + 190 p T^{5} + 108 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 29 T + 362 T^{2} - 2619 T^{3} + 16622 T^{4} - 2619 p T^{5} + 362 p^{2} T^{6} - 29 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 16 T + 264 T^{2} + 2444 T^{3} + 23526 T^{4} + 2444 p T^{5} + 264 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 21 T + 276 T^{2} + 2979 T^{3} + 26922 T^{4} + 2979 p T^{5} + 276 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 4 T + 172 T^{2} - 384 T^{3} + 14822 T^{4} - 384 p T^{5} + 172 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 17 T + 84 T^{2} + 1019 T^{3} - 16170 T^{4} + 1019 p T^{5} + 84 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 12 T + 116 T^{2} + 276 T^{3} + 3382 T^{4} + 276 p T^{5} + 116 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 27 T + 548 T^{2} + 7039 T^{3} + 74358 T^{4} + 7039 p T^{5} + 548 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 18 T + 412 T^{2} - 4590 T^{3} + 54630 T^{4} - 4590 p T^{5} + 412 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 37 T + 814 T^{2} + 11891 T^{3} + 130434 T^{4} + 11891 p T^{5} + 814 p^{2} T^{6} + 37 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 3 T + 290 T^{2} - 621 T^{3} + 37786 T^{4} - 621 p T^{5} + 290 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.31049082799439623051231251599, −5.77039032199070057510986749065, −5.61304250300732347965019361747, −5.56696454125059157065238001362, −5.44560630142283148860319609686, −4.78305747720690970591650361224, −4.75308875136026694655275149255, −4.62032592422151729768338473786, −4.29980430719397935466485742730, −3.93469637650030537118840728836, −3.91470874707373979676909715338, −3.54389359036476183754981947264, −3.42402980181061592095421217470, −3.12386742764345828814866657257, −2.93446818272853507398965394544, −2.88869070423227516131653320100, −2.70172766854052546479075477054, −2.07795094725416629976571564243, −1.95890264744611950581604308450, −1.71195429302917401116177430183, −1.32912524011129025951592918382, −1.20988649380053312958506922556, −1.01354704744812328210347048937, −0.48808914470777147247224152632, −0.33251683220817613587042227718, 0.33251683220817613587042227718, 0.48808914470777147247224152632, 1.01354704744812328210347048937, 1.20988649380053312958506922556, 1.32912524011129025951592918382, 1.71195429302917401116177430183, 1.95890264744611950581604308450, 2.07795094725416629976571564243, 2.70172766854052546479075477054, 2.88869070423227516131653320100, 2.93446818272853507398965394544, 3.12386742764345828814866657257, 3.42402980181061592095421217470, 3.54389359036476183754981947264, 3.91470874707373979676909715338, 3.93469637650030537118840728836, 4.29980430719397935466485742730, 4.62032592422151729768338473786, 4.75308875136026694655275149255, 4.78305747720690970591650361224, 5.44560630142283148860319609686, 5.56696454125059157065238001362, 5.61304250300732347965019361747, 5.77039032199070057510986749065, 6.31049082799439623051231251599

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