Properties

Label 8-4598e4-1.1-c1e4-0-4
Degree $8$
Conductor $4.470\times 10^{14}$
Sign $1$
Analytic cond. $1.81712\times 10^{6}$
Root an. cond. $6.05930$
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 − 2·3-s + 10·4-s − 2·5-s + 8·6-s + 4·7-s − 20·8-s − 4·9-s + 8·10-s − 20·12-s + 4·13-s − 16·14-s + 4·15-s + 35·16-s + 4·17-s + 16·18-s − 4·19-s − 20·20-s − 8·21-s − 8·23-s + 40·24-s − 12·25-s − 16·26-s + 8·27-s + 40·28-s − 2·29-s − 16·30-s + ⋯
L(s)  = 1  − 2.82·2-s − 1.15·3-s + 5·4-s − 0.894·5-s + 3.26·6-s + 1.51·7-s − 7.07·8-s − 4/3·9-s + 2.52·10-s − 5.77·12-s + 1.10·13-s − 4.27·14-s + 1.03·15-s + 35/4·16-s + 0.970·17-s + 3.77·18-s − 0.917·19-s − 4.47·20-s − 1.74·21-s − 1.66·23-s + 8.16·24-s − 2.39·25-s − 3.13·26-s + 1.53·27-s + 7.55·28-s − 0.371·29-s − 2.92·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 11^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.81712\times 10^{6}\)
Root analytic conductor: \(6.05930\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4598} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 11^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 8 T^{2} + 16 T^{3} + 31 T^{4} + 16 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 16 T^{2} + 22 T^{3} + 108 T^{4} + 22 p T^{5} + 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 36 T^{2} - 140 T^{3} + 614 T^{4} - 140 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 64 T^{2} - 188 T^{3} + 1602 T^{4} - 188 p T^{5} + 64 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 62 T^{2} + 224 T^{3} + 1207 T^{4} + 224 p T^{5} + 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 44 T^{2} + 236 T^{3} + 859 T^{4} + 236 p T^{5} + 44 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 36 T^{2} + 72 T^{3} + 1646 T^{4} + 72 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 132 T^{2} + 1112 T^{3} + 7031 T^{4} + 1112 p T^{5} + 132 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 76 T^{2} - 106 T^{3} + 4476 T^{4} - 106 p T^{5} + 76 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 252 T^{2} + 2168 T^{3} + 17870 T^{4} + 2168 p T^{5} + 252 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 142 T^{2} + 96 T^{3} + 9027 T^{4} + 96 p T^{5} + 142 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 184 T^{2} + 912 T^{3} + 13875 T^{4} + 912 p T^{5} + 184 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 340 T^{2} - 3728 T^{3} + 33051 T^{4} - 3728 p T^{5} + 340 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 76 T^{2} - 926 T^{3} + 2956 T^{4} - 926 p T^{5} + 76 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 300 T^{2} + 2692 T^{3} + 24710 T^{4} + 2692 p T^{5} + 300 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 164 T^{2} + 1522 T^{3} + 17380 T^{4} + 1522 p T^{5} + 164 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 192 T^{2} + 24 T^{3} + 19298 T^{4} + 24 p T^{5} + 192 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 108 T^{2} - 282 T^{3} + 1700 T^{4} - 282 p T^{5} + 108 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 34 T + 8 p T^{2} - 8810 T^{3} + 91044 T^{4} - 8810 p T^{5} + 8 p^{3} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 196 T^{2} + 610 T^{3} + 20724 T^{4} + 610 p T^{5} + 196 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 108 T^{2} - 800 T^{3} - 2098 T^{4} - 800 p T^{5} + 108 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.33829326689684011131067476091, −5.95226531914287664176739347349, −5.90634494275886850279670167207, −5.84991464813146309409342755014, −5.83626149971570827405659741455, −5.35025874919038164279144063586, −5.11273713789340872070425600766, −5.05259037807373650839831161833, −4.92510652540056605999537513745, −4.45147854529290862801050479531, −4.33528591138150974948436921344, −3.95884234092868216880613699520, −3.81205859925938481124190076969, −3.46114613799644296860665328344, −3.34875909715844792522425004192, −3.27053568894918616204758428957, −3.13009240365442165388728266235, −2.47328869430187173104693663807, −2.24620497574495737774455986600, −2.02719851857888953669281908130, −2.00535456403352980783509387756, −1.62635814621842171700154043797, −1.53360365029604361466778599690, −1.00975746967924539269192426675, −0.970249398181954931609648358259, 0, 0, 0, 0, 0.970249398181954931609648358259, 1.00975746967924539269192426675, 1.53360365029604361466778599690, 1.62635814621842171700154043797, 2.00535456403352980783509387756, 2.02719851857888953669281908130, 2.24620497574495737774455986600, 2.47328869430187173104693663807, 3.13009240365442165388728266235, 3.27053568894918616204758428957, 3.34875909715844792522425004192, 3.46114613799644296860665328344, 3.81205859925938481124190076969, 3.95884234092868216880613699520, 4.33528591138150974948436921344, 4.45147854529290862801050479531, 4.92510652540056605999537513745, 5.05259037807373650839831161833, 5.11273713789340872070425600766, 5.35025874919038164279144063586, 5.83626149971570827405659741455, 5.84991464813146309409342755014, 5.90634494275886850279670167207, 5.95226531914287664176739347349, 6.33829326689684011131067476091

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