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Properties

Label 8-7500e4-1.1-c1e4-0-1
Degree $8$
Conductor $3.164\times 10^{15}$
Sign $1$
Analytic cond. $1.28633\times 10^{7}$
Root an. cond. $7.73872$
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 + 4·7-s + 10·9-s − 11-s − 5·13-s + 4·17-s − 5·19-s − 16·21-s + 9·23-s − 20·27-s − 4·29-s − 9·31-s + 4·33-s + 2·37-s + 20·39-s + 34·43-s + 6·47-s − 8·49-s − 16·51-s − 2·53-s + 20·57-s + 59-s − 2·61-s + 40·63-s − 4·67-s − 36·69-s − 20·71-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.51·7-s + 10/3·9-s − 0.301·11-s − 1.38·13-s + 0.970·17-s − 1.14·19-s − 3.49·21-s + 1.87·23-s − 3.84·27-s − 0.742·29-s − 1.61·31-s + 0.696·33-s + 0.328·37-s + 3.20·39-s + 5.18·43-s + 0.875·47-s − 8/7·49-s − 2.24·51-s − 0.274·53-s + 2.64·57-s + 0.130·59-s − 0.256·61-s + 5.03·63-s − 0.488·67-s − 4.33·69-s − 2.37·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.28633\times 10^{7}\)
Root analytic conductor: \(7.73872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.456179347\)
\(L(\frac12)\) \(\approx\) \(3.456179347\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{4} \)
5 \( 1 \)
good7$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 24 T^{2} - 83 T^{3} + 239 T^{4} - 83 p T^{5} + 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 + T + 40 T^{2} + 29 T^{3} + 639 T^{4} + 29 p T^{5} + 40 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 42 T^{2} + 175 T^{3} + 749 T^{4} + 175 p T^{5} + 42 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 2 p T^{2} - 143 T^{3} + 849 T^{4} - 143 p T^{5} + 2 p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 51 T^{2} + 10 p T^{3} + 1361 T^{4} + 10 p^{2} T^{5} + 51 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 9 T + 88 T^{2} - 495 T^{3} + 3021 T^{4} - 495 p T^{5} + 88 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 22 T^{2} - 223 T^{3} - 1005 T^{4} - 223 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 9 T + 130 T^{2} + 711 T^{3} + 189 p T^{4} + 711 p T^{5} + 130 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 102 T^{2} - 205 T^{3} + 4991 T^{4} - 205 p T^{5} + 102 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 94 T^{2} + 240 T^{3} + 4191 T^{4} + 240 p T^{5} + 94 p^{2} T^{6} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - 34 T + 573 T^{2} - 6230 T^{3} + 47951 T^{4} - 6230 p T^{5} + 573 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 - 6 T + 2 p T^{2} - 507 T^{3} + 6039 T^{4} - 507 p T^{5} + 2 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 166 T^{2} + 121 T^{3} + 11799 T^{4} + 121 p T^{5} + 166 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - T + 97 T^{2} - 218 T^{3} + 6825 T^{4} - 218 p T^{5} + 97 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 148 T^{2} + 764 T^{3} + 10165 T^{4} + 764 p T^{5} + 148 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 109 T^{2} - 512 T^{3} + 3169 T^{4} - 512 p T^{5} + 109 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 20 T + 334 T^{2} + 3385 T^{3} + 33471 T^{4} + 3385 p T^{5} + 334 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 - 12 T + 151 T^{2} - 1566 T^{3} + 9759 T^{4} - 1566 p T^{5} + 151 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 3 T + 130 T^{2} - 567 T^{3} + 6699 T^{4} - 567 p T^{5} + 130 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 - 14 T + 388 T^{2} - 3520 T^{3} + 50541 T^{4} - 3520 p T^{5} + 388 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 386 T^{2} - 3915 T^{3} + 52911 T^{4} - 3915 p T^{5} + 386 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 28 T + 6 p T^{2} - 7880 T^{3} + 89831 T^{4} - 7880 p T^{5} + 6 p^{3} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.62336920935959765305586611991, −5.28065905191659798312502575913, −5.06404917216838292080614759272, −5.00831141876622118018709305688, −4.83589406701961187282582026775, −4.72818947000473391014263849979, −4.62542324069847480491221056547, −4.38616272023069216675257502412, −4.24891942016214359273407616742, −3.68935844665057733038193319011, −3.63902355173159990801115560220, −3.63741568964270023959291400334, −3.58480545762820453162152105392, −2.77839747587303799720871767113, −2.63742428438704496048532887571, −2.56516334850384817411971906076, −2.53332260651296412867119156117, −1.89659819478581458218814480479, −1.84407812410726899417911309507, −1.58979888255520539395432994602, −1.46855612731239195679941252030, −0.983696698699724962757799453656, −0.70915133880790601358737891796, −0.47788715665751704372982538522, −0.45119408251613334591219694519, 0.45119408251613334591219694519, 0.47788715665751704372982538522, 0.70915133880790601358737891796, 0.983696698699724962757799453656, 1.46855612731239195679941252030, 1.58979888255520539395432994602, 1.84407812410726899417911309507, 1.89659819478581458218814480479, 2.53332260651296412867119156117, 2.56516334850384817411971906076, 2.63742428438704496048532887571, 2.77839747587303799720871767113, 3.58480545762820453162152105392, 3.63741568964270023959291400334, 3.63902355173159990801115560220, 3.68935844665057733038193319011, 4.24891942016214359273407616742, 4.38616272023069216675257502412, 4.62542324069847480491221056547, 4.72818947000473391014263849979, 4.83589406701961187282582026775, 5.00831141876622118018709305688, 5.06404917216838292080614759272, 5.28065905191659798312502575913, 5.62336920935959765305586611991

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