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Properties

Label 8-8470e4-1.1-c1e4-0-5
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

Downloads

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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 + 5·9-s + 16·10-s − 40·12-s − 14·13-s + 16·14-s − 16·15-s + 35·16-s − 12·17-s + 20·18-s + 2·19-s + 40·20-s − 16·21-s − 80·24-s + 10·25-s − 56·26-s − 2·27-s + 40·28-s − 10·29-s − 64·30-s − 18·31-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 + 5/3·9-s + 5.05·10-s − 11.5·12-s − 3.88·13-s + 4.27·14-s − 4.13·15-s + 35/4·16-s − 2.91·17-s + 4.71·18-s + 0.458·19-s + 8.94·20-s − 3.49·21-s − 16.3·24-s + 2·25-s − 10.9·26-s − 0.384·27-s + 7.55·28-s − 1.85·29-s − 11.6·30-s − 3.23·31-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: Trivial
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$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 11 T^{2} + 26 T^{3} + 53 T^{4} + 26 p T^{5} + 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 106 T^{2} + 42 p T^{3} + 2198 T^{4} + 42 p^{2} T^{5} + 106 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 109 T^{2} + 642 T^{3} + 3103 T^{4} + 642 p T^{5} + 109 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 47 T^{2} - 104 T^{3} + 1209 T^{4} - 104 p T^{5} + 47 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2 \wr C_2$ \( 1 + 60 T^{2} + 1878 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 78 T^{2} + 430 T^{3} + 2678 T^{4} + 430 p T^{5} + 78 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 138 T^{2} + 546 T^{3} + 1950 T^{4} + 546 p T^{5} + 138 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 194 T^{2} - 1558 T^{3} + 10518 T^{4} - 1558 p T^{5} + 194 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 367 T^{2} + 3660 T^{3} + 27571 T^{4} + 3660 p T^{5} + 367 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 119 T^{2} + 1010 T^{3} + 7507 T^{4} + 1010 p T^{5} + 119 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 120 T^{2} + 392 T^{3} + 5246 T^{4} + 392 p T^{5} + 120 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 290 T^{2} - 2680 T^{3} + 22318 T^{4} - 2680 p T^{5} + 290 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 215 T^{2} + 2188 T^{3} + 18253 T^{4} + 2188 p T^{5} + 215 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 260 T^{2} + 2406 T^{3} + 23994 T^{4} + 2406 p T^{5} + 260 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 245 T^{2} - 50 T^{3} + 23823 T^{4} - 50 p T^{5} + 245 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 322 T^{2} + 2830 T^{3} + 35086 T^{4} + 2830 p T^{5} + 322 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 30 T + 395 T^{2} + 2990 T^{3} + 20763 T^{4} + 2990 p T^{5} + 395 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 242 T^{2} + 1596 T^{3} + 27174 T^{4} + 1596 p T^{5} + 242 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 203 T^{2} + 1142 T^{3} + 18865 T^{4} + 1142 p T^{5} + 203 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 187 T^{2} - 328 T^{3} + 14695 T^{4} - 328 p T^{5} + 187 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 311 T^{2} + 2150 T^{3} + 40087 T^{4} + 2150 p T^{5} + 311 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.73412871584645390868738769951, −5.54028479925752803193308430390, −5.41471964014715489249757851130, −5.36588050435770427081491629219, −5.19261527188803267219173095132, −4.89094808876283128385410572679, −4.70966029075402287298705759612, −4.70044080302775976862849784172, −4.57637722316193880092238966626, −4.40141239893183279096390549541, −4.26137037331529387689004716571, −3.88457755226910797492272548745, −3.87522166610605210012124727601, −3.19425624484420317891396926597, −3.15717419070781357221977292004, −3.01040462017020156443432917075, −2.98303629038929464459908827341, −2.33064970069936460721967796597, −2.26913551880451943458853095703, −2.19372507023194033620817594476, −2.06798211995058781350165300317, −1.75182864066965601498160346613, −1.70752430301576476376700665745, −1.23465302267431994723338321951, −1.21652336148100754141193278792, 0, 0, 0, 0, 1.21652336148100754141193278792, 1.23465302267431994723338321951, 1.70752430301576476376700665745, 1.75182864066965601498160346613, 2.06798211995058781350165300317, 2.19372507023194033620817594476, 2.26913551880451943458853095703, 2.33064970069936460721967796597, 2.98303629038929464459908827341, 3.01040462017020156443432917075, 3.15717419070781357221977292004, 3.19425624484420317891396926597, 3.87522166610605210012124727601, 3.88457755226910797492272548745, 4.26137037331529387689004716571, 4.40141239893183279096390549541, 4.57637722316193880092238966626, 4.70044080302775976862849784172, 4.70966029075402287298705759612, 4.89094808876283128385410572679, 5.19261527188803267219173095132, 5.36588050435770427081491629219, 5.41471964014715489249757851130, 5.54028479925752803193308430390, 5.73412871584645390868738769951

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