Properties

Label 8-8470e4-1.1-c1e4-0-3
Degree $8$
Conductor $5.147\times 10^{15}$
Sign $1$
Analytic cond. $2.09238\times 10^{7}$
Root an. cond. $8.22394$
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 − 4·3-s + 10·4-s + 4·5-s + 16·6-s + 4·7-s − 20·8-s + 2·9-s − 16·10-s − 40·12-s − 16·14-s − 16·15-s + 35·16-s − 8·17-s − 8·18-s + 12·19-s + 40·20-s − 16·21-s − 8·23-s + 80·24-s + 10·25-s + 16·27-s + 40·28-s + 8·29-s + 64·30-s − 56·32-s + 32·34-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s + 1.51·7-s − 7.07·8-s + 2/3·9-s − 5.05·10-s − 11.5·12-s − 4.27·14-s − 4.13·15-s + 35/4·16-s − 1.94·17-s − 1.88·18-s + 2.75·19-s + 8.94·20-s − 3.49·21-s − 1.66·23-s + 16.3·24-s + 2·25-s + 3.07·27-s + 7.55·28-s + 1.48·29-s + 11.6·30-s − 9.89·32-s + 5.48·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.09238\times 10^{7}\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 - T )^{4} \)
7$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good3$D_{4}$ \( ( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 30 T^{2} - 48 T^{3} + 419 T^{4} - 48 p T^{5} + 30 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 12 T + 102 T^{2} - 624 T^{3} + 2987 T^{4} - 624 p T^{5} + 102 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
23$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 82 T^{2} + 544 T^{3} + 2715 T^{4} + 544 p T^{5} + 82 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 8 T + 100 T^{2} - 664 T^{3} + 4134 T^{4} - 664 p T^{5} + 100 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^3$ \( 1 + 194 T^{4} + p^{4} T^{8} \)
37$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 16 T + 204 T^{2} + 1616 T^{3} + 11558 T^{4} + 1616 p T^{5} + 204 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
41$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 80 T^{2} + 8 T^{3} + 802 T^{4} + 8 p T^{5} + 80 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
43$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 8 T + 112 T^{2} - 728 T^{3} + 5650 T^{4} - 728 p T^{5} + 112 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 16 T + 184 T^{2} + 1712 T^{3} + 13650 T^{4} + 1712 p T^{5} + 184 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
53$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 136 T^{2} - 192 T^{3} + 8802 T^{4} - 192 p T^{5} + 136 p^{2} T^{6} + p^{4} T^{8} \)
59$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 218 T^{2} + 1280 T^{3} + 18835 T^{4} + 1280 p T^{5} + 218 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
61$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 8 T + 228 T^{2} - 1336 T^{3} + 20246 T^{4} - 1336 p T^{5} + 228 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 136 T^{2} + 10146 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} \)
71$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 92 T^{2} - 280 T^{3} - 26 T^{4} - 280 p T^{5} + 92 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 + 4 T + 148 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
79$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 24 T + 6 p T^{2} - 5760 T^{3} + 61379 T^{4} - 5760 p T^{5} + 6 p^{3} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
83$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 50 T^{2} - 640 T^{3} - 11285 T^{4} - 640 p T^{5} + 50 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 208 T^{2} + 21858 T^{4} + 208 p^{2} T^{6} + p^{4} T^{8} \)
97$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 244 T^{2} + 824 T^{3} + 25846 T^{4} + 824 p T^{5} + 244 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.01007046520654578385106043448, −5.57917174414246355419363974273, −5.51222972997683029072480915135, −5.43390181694413172166671518348, −5.24760676318334711282216852906, −5.07615203157050540339042062051, −4.86441559747509794422740720450, −4.78438796642455649214697088900, −4.78430375560844719732710112214, −4.04735172450996448913118132339, −3.97981403744323845693379432029, −3.69357892732632858327948080730, −3.67958962363019632123365297718, −2.98333735214154305490834463400, −2.86482842168831714426382725093, −2.81975094012488672893216423052, −2.81047840927210985558782660770, −2.22320173079674280812291776399, −2.16634126418344892400218322139, −1.96033180228123892551143595526, −1.74331570097576150541923258984, −1.26438023683923859283179244277, −1.24165676357429858224899079803, −1.14898872122668600101275366491, −0.998479830940473430376323975256, 0, 0, 0, 0, 0.998479830940473430376323975256, 1.14898872122668600101275366491, 1.24165676357429858224899079803, 1.26438023683923859283179244277, 1.74331570097576150541923258984, 1.96033180228123892551143595526, 2.16634126418344892400218322139, 2.22320173079674280812291776399, 2.81047840927210985558782660770, 2.81975094012488672893216423052, 2.86482842168831714426382725093, 2.98333735214154305490834463400, 3.67958962363019632123365297718, 3.69357892732632858327948080730, 3.97981403744323845693379432029, 4.04735172450996448913118132339, 4.78430375560844719732710112214, 4.78438796642455649214697088900, 4.86441559747509794422740720450, 5.07615203157050540339042062051, 5.24760676318334711282216852906, 5.43390181694413172166671518348, 5.51222972997683029072480915135, 5.57917174414246355419363974273, 6.01007046520654578385106043448

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