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Properties

Label 8-1815e4-1.1-c1e4-0-8
Degree $8$
Conductor $1.085\times 10^{13}$
Sign $1$
Analytic cond. $44117.9$
Root an. cond. $3.80694$
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  − 2-s − 4·3-s − 3·4-s + 4·5-s + 4·6-s + 5·8-s + 10·9-s − 4·10-s + 12·12-s − 7·13-s − 16·15-s − 8·17-s − 10·18-s − 11·19-s − 12·20-s + 5·23-s − 20·24-s + 10·25-s + 7·26-s − 20·27-s − 17·29-s + 16·30-s − 5·31-s − 9·32-s + 8·34-s − 30·36-s + 15·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s − 3/2·4-s + 1.78·5-s + 1.63·6-s + 1.76·8-s + 10/3·9-s − 1.26·10-s + 3.46·12-s − 1.94·13-s − 4.13·15-s − 1.94·17-s − 2.35·18-s − 2.52·19-s − 2.68·20-s + 1.04·23-s − 4.08·24-s + 2·25-s + 1.37·26-s − 3.84·27-s − 3.15·29-s + 2.92·30-s − 0.898·31-s − 1.59·32-s + 1.37·34-s − 5·36-s + 2.46·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(44117.9\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad3$C_1$ \( ( 1 + T )^{4} \)
5$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$C_4\times C_2$ \( 1 + T + p^{2} T^{2} + p T^{3} + 9 T^{4} + p^{2} T^{5} + p^{4} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$((C_8 : C_2):C_2):C_2$ \( 1 + 18 T^{2} - 15 T^{3} + 149 T^{4} - 15 p T^{5} + 18 p^{2} T^{6} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 + 7 T + 36 T^{2} + 71 T^{3} + 239 T^{4} + 71 p T^{5} + 36 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 62 T^{2} + 320 T^{3} + 1471 T^{4} + 320 p T^{5} + 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 11 T + 82 T^{2} + 443 T^{3} + 2155 T^{4} + 443 p T^{5} + 82 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 5 T + 67 T^{2} - 370 T^{3} + 2019 T^{4} - 370 p T^{5} + 67 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 17 T + 200 T^{2} + 1517 T^{3} + 9499 T^{4} + 1517 p T^{5} + 200 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 79 T^{2} + 230 T^{3} + 2971 T^{4} + 230 p T^{5} + 79 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 213 T^{2} - 1710 T^{3} + 12869 T^{4} - 1710 p T^{5} + 213 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 154 T^{2} + 1085 T^{3} + 9261 T^{4} + 1085 p T^{5} + 154 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 48 T^{2} + 25 T^{3} - 139 T^{4} + 25 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 172 T^{2} + 985 T^{3} + 11871 T^{4} + 985 p T^{5} + 172 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 202 T^{2} - 1565 T^{3} + 15819 T^{4} - 1565 p T^{5} + 202 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 9 T + 167 T^{2} - 1332 T^{3} + 14085 T^{4} - 1332 p T^{5} + 167 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 37 T + 748 T^{2} + 9769 T^{3} + 90385 T^{4} + 9769 p T^{5} + 748 p^{2} T^{6} + 37 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 - 3 T + 222 T^{2} - 675 T^{3} + 20801 T^{4} - 675 p T^{5} + 222 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 - 13 T + 323 T^{2} - 2686 T^{3} + 35395 T^{4} - 2686 p T^{5} + 323 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 15 T + 252 T^{2} + 1965 T^{3} + 21929 T^{4} + 1965 p T^{5} + 252 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 20 T + 411 T^{2} + 4750 T^{3} + 52151 T^{4} + 4750 p T^{5} + 411 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 - 17 T + 71 T^{2} + 1114 T^{3} - 16661 T^{4} + 1114 p T^{5} + 71 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 + 24 T + 227 T^{2} + 342 T^{3} - 7305 T^{4} + 342 p T^{5} + 227 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 + 32 T + 672 T^{2} + 9775 T^{3} + 111611 T^{4} + 9775 p T^{5} + 672 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.79678833726397805884860766880, −6.76034157932096237910370243624, −6.59882288832572563522381454426, −6.26367961542371636310772053651, −6.25961712749791999200539461563, −6.00794701843054049894846885531, −5.62068043040605456068710498241, −5.45402389665555081068113712362, −5.42196085768046293126951715447, −4.98638046203039118126084499831, −4.89871491952611647606943988365, −4.78167118083930833149628549367, −4.68254151170728931209631205976, −4.16025699978740079074135126882, −4.16004443863565372472483165457, −3.94765003195956566975458661409, −3.88024502142333517577971749098, −2.97653847604399589698193462725, −2.95233754042748636846284159134, −2.39476343814207383041999735409, −2.27034422803523504265391519129, −2.08981928893916971942967115368, −1.48351716031531517278873444559, −1.44447752437342648619362969884, −1.29028749541621741307100730713, 0, 0, 0, 0, 1.29028749541621741307100730713, 1.44447752437342648619362969884, 1.48351716031531517278873444559, 2.08981928893916971942967115368, 2.27034422803523504265391519129, 2.39476343814207383041999735409, 2.95233754042748636846284159134, 2.97653847604399589698193462725, 3.88024502142333517577971749098, 3.94765003195956566975458661409, 4.16004443863565372472483165457, 4.16025699978740079074135126882, 4.68254151170728931209631205976, 4.78167118083930833149628549367, 4.89871491952611647606943988365, 4.98638046203039118126084499831, 5.42196085768046293126951715447, 5.45402389665555081068113712362, 5.62068043040605456068710498241, 6.00794701843054049894846885531, 6.25961712749791999200539461563, 6.26367961542371636310772053651, 6.59882288832572563522381454426, 6.76034157932096237910370243624, 6.79678833726397805884860766880

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