Properties

Label 8-7623e4-1.1-c1e4-0-1
Degree $8$
Conductor $3.377\times 10^{15}$
Sign $1$
Analytic cond. $1.37281\times 10^{7}$
Root an. cond. $7.80192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 6·5-s − 4·7-s − 4·8-s − 12·10-s + 2·13-s − 8·14-s − 6·16-s + 2·17-s − 6·19-s − 10·23-s + 8·25-s + 4·26-s + 14·29-s + 4·34-s + 24·35-s − 12·37-s − 12·38-s + 24·40-s + 16·41-s − 20·43-s − 20·46-s − 2·47-s + 10·49-s + 16·50-s + 16·53-s + 16·56-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.68·5-s − 1.51·7-s − 1.41·8-s − 3.79·10-s + 0.554·13-s − 2.13·14-s − 3/2·16-s + 0.485·17-s − 1.37·19-s − 2.08·23-s + 8/5·25-s + 0.784·26-s + 2.59·29-s + 0.685·34-s + 4.05·35-s − 1.97·37-s − 1.94·38-s + 3.79·40-s + 2.49·41-s − 3.04·43-s − 2.94·46-s − 0.291·47-s + 10/7·49-s + 2.26·50-s + 2.19·53-s + 2.13·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.37281\times 10^{7}\)
Root analytic conductor: \(7.80192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.561462380\)
\(L(\frac12)\) \(\approx\) \(1.561462380\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + p^{2} T^{2} - p^{2} T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 28 T^{2} + 84 T^{3} + 219 T^{4} + 84 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 12 T^{2} - 46 T^{3} + 332 T^{4} - 46 p T^{5} + 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 56 T^{2} - 80 T^{3} + 1339 T^{4} - 80 p T^{5} + 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 72 T^{2} + 330 T^{3} + 2012 T^{4} + 330 p T^{5} + 72 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 116 T^{2} + 694 T^{3} + 4276 T^{4} + 694 p T^{5} + 116 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 164 T^{2} - 1238 T^{3} + 7756 T^{4} - 1238 p T^{5} + 164 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 108 T^{2} - 24 T^{3} + 4766 T^{4} - 24 p T^{5} + 108 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 168 T^{2} + 1284 T^{3} + 9650 T^{4} + 1284 p T^{5} + 168 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 196 T^{2} - 1712 T^{3} + 11994 T^{4} - 1712 p T^{5} + 196 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 306 T^{2} + 2920 T^{3} + 22895 T^{4} + 2920 p T^{5} + 306 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 88 T^{2} + 568 T^{3} + 4023 T^{4} + 568 p T^{5} + 88 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 244 T^{2} - 2096 T^{3} + 18582 T^{4} - 2096 p T^{5} + 244 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 2 T - 20 T^{2} + 136 T^{3} + 105 p T^{4} + 136 p T^{5} - 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 180 T^{2} + 418 T^{3} + 14804 T^{4} + 418 p T^{5} + 180 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 282 T^{2} + 2776 T^{3} + 24707 T^{4} + 2776 p T^{5} + 282 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 76 T^{2} + 790 T^{3} - 8172 T^{4} + 790 p T^{5} + 76 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 52 T^{2} - 406 T^{3} + 7828 T^{4} - 406 p T^{5} + 52 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 312 T^{2} + 3272 T^{3} + 37490 T^{4} + 3272 p T^{5} + 312 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 400 T^{2} - 4404 T^{3} + 52635 T^{4} - 4404 p T^{5} + 400 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 244 T^{2} - 2764 T^{3} + 32067 T^{4} - 2764 p T^{5} + 244 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 38 T + 844 T^{2} - 12458 T^{3} + 140980 T^{4} - 12458 p T^{5} + 844 p^{2} T^{6} - 38 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.56849521000129267876799655955, −5.21022517842721892741453919909, −5.06203427997415907779799764274, −4.83122637000696085352616898885, −4.72802026302015625126871834626, −4.42067479445041096948942444673, −4.25549216870891828786498681340, −4.24908275847505138178801115853, −4.18180333290181656561342726348, −3.70203626875090808629778713414, −3.66056891330454772585855315355, −3.64968970014405889627604218605, −3.33598831039095057394173645111, −3.24599389704391618927408378523, −2.95678964609476940120315718375, −2.93609902946943926167987702600, −2.38544360101993884856538465336, −2.18440187574299482339432440833, −2.10943688428549132221393850007, −1.76808187883311624707727331352, −1.48779132614420047696544846698, −0.805649533454790470065871473691, −0.68153743558677321703945468756, −0.35165767904842287183835154178, −0.31936397473343321203614815351, 0.31936397473343321203614815351, 0.35165767904842287183835154178, 0.68153743558677321703945468756, 0.805649533454790470065871473691, 1.48779132614420047696544846698, 1.76808187883311624707727331352, 2.10943688428549132221393850007, 2.18440187574299482339432440833, 2.38544360101993884856538465336, 2.93609902946943926167987702600, 2.95678964609476940120315718375, 3.24599389704391618927408378523, 3.33598831039095057394173645111, 3.64968970014405889627604218605, 3.66056891330454772585855315355, 3.70203626875090808629778713414, 4.18180333290181656561342726348, 4.24908275847505138178801115853, 4.25549216870891828786498681340, 4.42067479445041096948942444673, 4.72802026302015625126871834626, 4.83122637000696085352616898885, 5.06203427997415907779799764274, 5.21022517842721892741453919909, 5.56849521000129267876799655955

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