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Properties

Label 8-1155e4-1.1-c1e4-0-6
Degree $8$
Conductor $1.780\times 10^{12}$
Sign $1$
Analytic cond. $7234.96$
Root an. cond. $3.03689$
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 − 4·3-s + 4-s + 4·5-s − 8·6-s + 4·7-s + 10·9-s + 8·10-s + 4·11-s − 4·12-s + 4·13-s + 8·14-s − 16·15-s − 16-s + 6·17-s + 20·18-s − 6·19-s + 4·20-s − 16·21-s + 8·22-s + 10·23-s + 10·25-s + 8·26-s − 20·27-s + 4·28-s + 6·29-s − 32·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.30·3-s + 1/2·4-s + 1.78·5-s − 3.26·6-s + 1.51·7-s + 10/3·9-s + 2.52·10-s + 1.20·11-s − 1.15·12-s + 1.10·13-s + 2.13·14-s − 4.13·15-s − 1/4·16-s + 1.45·17-s + 4.71·18-s − 1.37·19-s + 0.894·20-s − 3.49·21-s + 1.70·22-s + 2.08·23-s + 2·25-s + 1.56·26-s − 3.84·27-s + 0.755·28-s + 1.11·29-s − 5.84·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(7234.96\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.67997506\)
\(L(\frac12)\) \(\approx\) \(10.67997506\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
7$C_1$ \( ( 1 - T )^{4} \)
11$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + 3 p T^{4} - p^{3} T^{5} + 3 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 38 T^{2} - 132 T^{3} + 642 T^{4} - 132 p T^{5} + 38 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 57 T^{2} - 302 T^{3} + 1364 T^{4} - 302 p T^{5} + 57 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 77 T^{2} + 318 T^{3} + 2188 T^{4} + 318 p T^{5} + 77 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 53 T^{2} - 242 T^{3} + 1252 T^{4} - 242 p T^{5} + 53 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 71 T^{2} - 422 T^{3} + 2720 T^{4} - 422 p T^{5} + 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 106 T^{2} + 552 T^{3} + 4394 T^{4} + 552 p T^{5} + 106 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 118 T^{2} - 988 T^{3} + 6818 T^{4} - 988 p T^{5} + 118 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 58 T^{2} + 264 T^{3} + 2362 T^{4} + 264 p T^{5} + 58 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 11 T^{2} - 238 T^{3} + 3504 T^{4} - 238 p T^{5} + 11 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 42 T^{2} - 352 T^{3} + 1706 T^{4} - 352 p T^{5} + 42 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 233 T^{2} - 2118 T^{3} + 19148 T^{4} - 2118 p T^{5} + 233 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 99 T^{2} - 266 T^{3} + 9056 T^{4} - 266 p T^{5} + 99 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 193 T^{2} - 226 T^{3} + 16300 T^{4} - 226 p T^{5} + 193 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 96 T^{2} - 180 T^{3} - 2706 T^{4} - 180 p T^{5} + 96 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 190 T^{2} - 692 T^{3} + 19074 T^{4} - 692 p T^{5} + 190 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 416 T^{2} - 4604 T^{3} + 50046 T^{4} - 4604 p T^{5} + 416 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 206 T^{2} + 596 T^{3} + 21154 T^{4} + 596 p T^{5} + 206 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 285 T^{2} - 3094 T^{3} + 34668 T^{4} - 3094 p T^{5} + 285 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 231 T^{2} + 1758 T^{3} + 25464 T^{4} + 1758 p T^{5} + 231 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 447 T^{2} + 4150 T^{3} + 67896 T^{4} + 4150 p T^{5} + 447 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.75769397514392743183776224517, −6.60927225879537868889461450791, −6.46043679240819654663026080546, −6.16190391748312735924905363215, −6.08299076107448577976659239887, −5.68352618221532122751564122155, −5.65881203589831497196727500984, −5.40565502374712576390095813362, −5.38287925607375865404541487939, −4.87294389213988524812543160591, −4.76983769111103109061002971676, −4.71641658253762657754636037071, −4.48679035549235084261579423598, −4.01454163044669360701244230685, −3.85799005662132767044643755673, −3.85109913354017221701459968770, −3.39894471799193583221846511621, −2.75609678347945041325150269220, −2.72937683253695944941053018271, −2.25498290488002468647185225546, −1.83352336461956567819928324923, −1.63118793542702921115555211920, −1.18889018108439852641031901082, −0.908877128149224408316155164361, −0.842668062455860957895377063332, 0.842668062455860957895377063332, 0.908877128149224408316155164361, 1.18889018108439852641031901082, 1.63118793542702921115555211920, 1.83352336461956567819928324923, 2.25498290488002468647185225546, 2.72937683253695944941053018271, 2.75609678347945041325150269220, 3.39894471799193583221846511621, 3.85109913354017221701459968770, 3.85799005662132767044643755673, 4.01454163044669360701244230685, 4.48679035549235084261579423598, 4.71641658253762657754636037071, 4.76983769111103109061002971676, 4.87294389213988524812543160591, 5.38287925607375865404541487939, 5.40565502374712576390095813362, 5.65881203589831497196727500984, 5.68352618221532122751564122155, 6.08299076107448577976659239887, 6.16190391748312735924905363215, 6.46043679240819654663026080546, 6.60927225879537868889461450791, 6.75769397514392743183776224517

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