Properties

Label 8-2200e4-1.1-c1e4-0-7
Degree $8$
Conductor $2.343\times 10^{13}$
Sign $1$
Analytic cond. $95235.5$
Root an. cond. $4.19131$
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  − 3-s − 7-s − 2·9-s + 4·11-s − 16·13-s − 7·17-s − 9·19-s + 21-s − 6·23-s − 27-s + 3·29-s − 15·31-s − 4·33-s − 5·37-s + 16·39-s + 10·41-s − 8·43-s + 10·47-s − 9·49-s + 7·51-s − 5·53-s + 9·57-s + 6·59-s + 29·61-s + 2·63-s − 6·67-s + 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.20·11-s − 4.43·13-s − 1.69·17-s − 2.06·19-s + 0.218·21-s − 1.25·23-s − 0.192·27-s + 0.557·29-s − 2.69·31-s − 0.696·33-s − 0.821·37-s + 2.56·39-s + 1.56·41-s − 1.21·43-s + 1.45·47-s − 9/7·49-s + 0.980·51-s − 0.686·53-s + 1.19·57-s + 0.781·59-s + 3.71·61-s + 0.251·63-s − 0.733·67-s + 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(95235.5\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + T + p T^{2} + 2 p T^{3} + 2 T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + T + 10 T^{2} + 9 T^{3} + 50 T^{4} + 9 p T^{5} + 10 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
17$C_2 \wr S_4$ \( 1 + 7 T + 24 T^{2} + 65 T^{3} + 270 T^{4} + 65 p T^{5} + 24 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 9 T + 56 T^{2} + 257 T^{3} + 1278 T^{4} + 257 p T^{5} + 56 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 6 T + 33 T^{2} + 196 T^{3} + 1316 T^{4} + 196 p T^{5} + 33 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 3 T + 78 T^{2} - 9 p T^{3} + 2874 T^{4} - 9 p^{2} T^{5} + 78 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 15 T + 167 T^{2} + 1312 T^{3} + 8448 T^{4} + 1312 p T^{5} + 167 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 5 T + 93 T^{2} + 364 T^{3} + 108 p T^{4} + 364 p T^{5} + 93 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 10 T + 116 T^{2} - 654 T^{3} + 5126 T^{4} - 654 p T^{5} + 116 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 8 T + 108 T^{2} + 808 T^{3} + 6726 T^{4} + 808 p T^{5} + 108 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 10 T + 176 T^{2} - 1098 T^{3} + 11486 T^{4} - 1098 p T^{5} + 176 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 5 T + 136 T^{2} + 667 T^{3} + 10078 T^{4} + 667 p T^{5} + 136 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 6 T + 189 T^{2} - 1138 T^{3} + 15308 T^{4} - 1138 p T^{5} + 189 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 29 T + 518 T^{2} - 6171 T^{3} + 55914 T^{4} - 6171 p T^{5} + 518 p^{2} T^{6} - 29 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 6 T + 185 T^{2} + 908 T^{3} + 16380 T^{4} + 908 p T^{5} + 185 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + T + 147 T^{2} - 652 T^{3} + 9456 T^{4} - 652 p T^{5} + 147 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 18 T + 340 T^{2} + 3622 T^{3} + 37958 T^{4} + 3622 p T^{5} + 340 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 20 T + 224 T^{2} + 868 T^{3} + 3454 T^{4} + 868 p T^{5} + 224 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 26 T + 408 T^{2} + 4754 T^{3} + 47774 T^{4} + 4754 p T^{5} + 408 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 21 T + 477 T^{2} - 5682 T^{3} + 68718 T^{4} - 5682 p T^{5} + 477 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 4 T + 241 T^{2} - 384 T^{3} + 28448 T^{4} - 384 p T^{5} + 241 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.80630808319915914153503474575, −6.60362038565822703757732471735, −6.44478529920959154475673841054, −6.36697357534299642309768864298, −6.07584260094966885907579237243, −5.57524344787703236403055972389, −5.56505090267812574780012507775, −5.54600712040268113191777841465, −5.23263767041411721412702085866, −4.81093360040561861007006999544, −4.76401531178458325042431270961, −4.69351417183543095157122392324, −4.20518472634716369482539319664, −4.15091864461663081840018832570, −3.84386620845906273388540555579, −3.69171221823000735682311820468, −3.64521479203511252444868787194, −2.80270801898650924544405580750, −2.64698260416310719369938513572, −2.62250040110565721775495980004, −2.42536656326177769020911188109, −2.11271391215941515826837604719, −1.89512849568786469430926259679, −1.52223992181878773680088038731, −1.16733172319793113448250980886, 0, 0, 0, 0, 1.16733172319793113448250980886, 1.52223992181878773680088038731, 1.89512849568786469430926259679, 2.11271391215941515826837604719, 2.42536656326177769020911188109, 2.62250040110565721775495980004, 2.64698260416310719369938513572, 2.80270801898650924544405580750, 3.64521479203511252444868787194, 3.69171221823000735682311820468, 3.84386620845906273388540555579, 4.15091864461663081840018832570, 4.20518472634716369482539319664, 4.69351417183543095157122392324, 4.76401531178458325042431270961, 4.81093360040561861007006999544, 5.23263767041411721412702085866, 5.54600712040268113191777841465, 5.56505090267812574780012507775, 5.57524344787703236403055972389, 6.07584260094966885907579237243, 6.36697357534299642309768864298, 6.44478529920959154475673841054, 6.60362038565822703757732471735, 6.80630808319915914153503474575

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