Properties

Label 8-8030e4-1.1-c1e4-0-0
Degree $8$
Conductor $4.158\times 10^{15}$
Sign $1$
Analytic cond. $1.69032\times 10^{7}$
Root an. cond. $8.00748$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3-s + 10·4-s + 4·5-s − 4·6-s − 12·7-s + 20·8-s − 6·9-s + 16·10-s − 4·11-s − 10·12-s + 7·13-s − 48·14-s − 4·15-s + 35·16-s + 4·17-s − 24·18-s − 19·19-s + 40·20-s + 12·21-s − 16·22-s − 20·24-s + 10·25-s + 28·26-s + 7·27-s − 120·28-s + 7·29-s + ⋯
L(s)  = 1  + 2.82·2-s − 0.577·3-s + 5·4-s + 1.78·5-s − 1.63·6-s − 4.53·7-s + 7.07·8-s − 2·9-s + 5.05·10-s − 1.20·11-s − 2.88·12-s + 1.94·13-s − 12.8·14-s − 1.03·15-s + 35/4·16-s + 0.970·17-s − 5.65·18-s − 4.35·19-s + 8.94·20-s + 2.61·21-s − 3.41·22-s − 4.08·24-s + 2·25-s + 5.49·26-s + 1.34·27-s − 22.6·28-s + 1.29·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{4} \cdot 73^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{4} \cdot 73^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 11^{4} \cdot 73^{4}\)
Sign: $1$
Analytic conductor: \(1.69032\times 10^{7}\)
Root analytic conductor: \(8.00748\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 11^{4} \cdot 73^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 - T )^{4} \)
5$C_1$ \( ( 1 - T )^{4} \)
11$C_1$ \( ( 1 + T )^{4} \)
73$C_1$ \( ( 1 + T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + T + 7 T^{2} + 2 p T^{3} + 29 T^{4} + 2 p^{2} T^{5} + 7 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
13$C_2 \wr S_4$ \( 1 - 7 T + 56 T^{2} - 243 T^{3} + 1127 T^{4} - 243 p T^{5} + 56 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 4 T + 46 T^{2} - 189 T^{3} + 1055 T^{4} - 189 p T^{5} + 46 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + p T + 193 T^{2} + 1318 T^{3} + 6613 T^{4} + 1318 p T^{5} + 193 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 34 T^{2} - 128 T^{3} + 643 T^{4} - 128 p T^{5} + 34 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 7 T + 82 T^{2} - 637 T^{3} + 3069 T^{4} - 637 p T^{5} + 82 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 2 T + 68 T^{2} - 138 T^{3} + 3014 T^{4} - 138 p T^{5} + 68 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 9 T + 71 T^{2} - 564 T^{3} + 4263 T^{4} - 564 p T^{5} + 71 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 14 T + 204 T^{2} + 1740 T^{3} + 13341 T^{4} + 1740 p T^{5} + 204 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 3 T + 136 T^{2} + 333 T^{3} + 8079 T^{4} + 333 p T^{5} + 136 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 5 T + 122 T^{2} + 413 T^{3} + 7483 T^{4} + 413 p T^{5} + 122 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 11 T + 240 T^{2} - 1713 T^{3} + 19687 T^{4} - 1713 p T^{5} + 240 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 11 T + 43 T^{2} + 300 T^{3} + 4787 T^{4} + 300 p T^{5} + 43 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 4 T + 62 T^{2} + 441 T^{3} + 455 T^{4} + 441 p T^{5} + 62 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 23 T + 254 T^{2} + 1335 T^{3} + 6583 T^{4} + 1335 p T^{5} + 254 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 10 T + 176 T^{2} + 979 T^{3} + 13849 T^{4} + 979 p T^{5} + 176 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 34 T + 701 T^{2} + 9558 T^{3} + 98779 T^{4} + 9558 p T^{5} + 701 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 13 T + 309 T^{2} + 2790 T^{3} + 37399 T^{4} + 2790 p T^{5} + 309 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 21 T + 424 T^{2} - 5003 T^{3} + 57283 T^{4} - 5003 p T^{5} + 424 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 23 T + 415 T^{2} + 5594 T^{3} + 57745 T^{4} + 5594 p T^{5} + 415 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.94179170636110308537916161066, −5.72628169618438204171861617799, −5.55975820234688107427822656969, −5.53722273365726469662803847436, −5.32584035231292581580031468272, −4.93489002791820908987678479008, −4.68197395728721812875452552232, −4.56465106598960938582161675675, −4.52466000864416888751218829168, −4.20628254067348339469562475012, −3.80392356536836248685368621867, −3.75173066371490858975046943819, −3.68510315283618261535685365772, −3.39197018959142877180114824376, −3.16031039465349505841599879755, −3.01377958526714335655594172408, −2.81869688111315920307314155906, −2.72935851969472907526536802035, −2.49949219414079633750069639515, −2.49095609553244317827581180444, −2.36269477359805208419732159983, −1.69275863612017914628669453738, −1.55404987853656920165683165383, −1.24727639976758481196995493971, −1.12561518475448939515660116170, 0, 0, 0, 0, 1.12561518475448939515660116170, 1.24727639976758481196995493971, 1.55404987853656920165683165383, 1.69275863612017914628669453738, 2.36269477359805208419732159983, 2.49095609553244317827581180444, 2.49949219414079633750069639515, 2.72935851969472907526536802035, 2.81869688111315920307314155906, 3.01377958526714335655594172408, 3.16031039465349505841599879755, 3.39197018959142877180114824376, 3.68510315283618261535685365772, 3.75173066371490858975046943819, 3.80392356536836248685368621867, 4.20628254067348339469562475012, 4.52466000864416888751218829168, 4.56465106598960938582161675675, 4.68197395728721812875452552232, 4.93489002791820908987678479008, 5.32584035231292581580031468272, 5.53722273365726469662803847436, 5.55975820234688107427822656969, 5.72628169618438204171861617799, 5.94179170636110308537916161066

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