Properties

Label 8-1176e4-1.1-c3e4-0-0
Degree $8$
Conductor $1.913\times 10^{12}$
Sign $1$
Analytic cond. $2.31789\times 10^{7}$
Root an. cond. $8.32984$
Motivic weight $3$
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  − 12·3-s + 8·5-s + 90·9-s − 40·11-s + 48·13-s − 96·15-s − 152·17-s + 224·19-s − 8·23-s − 232·25-s − 540·27-s − 144·29-s + 400·31-s + 480·33-s − 304·37-s − 576·39-s − 152·41-s + 160·43-s + 720·45-s + 544·47-s + 1.82e3·51-s − 1.32e3·53-s − 320·55-s − 2.68e3·57-s + 1.04e3·59-s + 896·61-s + 384·65-s + ⋯
L(s)  = 1  − 2.30·3-s + 0.715·5-s + 10/3·9-s − 1.09·11-s + 1.02·13-s − 1.65·15-s − 2.16·17-s + 2.70·19-s − 0.0725·23-s − 1.85·25-s − 3.84·27-s − 0.922·29-s + 2.31·31-s + 2.53·33-s − 1.35·37-s − 2.36·39-s − 0.578·41-s + 0.567·43-s + 2.38·45-s + 1.68·47-s + 5.00·51-s − 3.42·53-s − 0.784·55-s − 6.24·57-s + 2.29·59-s + 1.88·61-s + 0.732·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.31789\times 10^{7}\)
Root analytic conductor: \(8.32984\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1176} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(3.034366403\)
\(L(\frac12)\) \(\approx\) \(3.034366403\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + p T )^{4} \)
7 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 296 T^{2} - 2312 T^{3} + 48194 T^{4} - 2312 p^{3} T^{5} + 296 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 40 T + 2404 T^{2} + 79752 T^{3} + 4895510 T^{4} + 79752 p^{3} T^{5} + 2404 p^{6} T^{6} + 40 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 48 T + 4080 T^{2} - 163056 T^{3} + 7581746 T^{4} - 163056 p^{3} T^{5} + 4080 p^{6} T^{6} - 48 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 152 T + 15832 T^{2} + 1323320 T^{3} + 98297586 T^{4} + 1323320 p^{3} T^{5} + 15832 p^{6} T^{6} + 152 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 224 T + 42396 T^{2} - 4864224 T^{3} + 486114742 T^{4} - 4864224 p^{3} T^{5} + 42396 p^{6} T^{6} - 224 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 9780 T^{2} + 1509576 T^{3} + 54064134 T^{4} + 1509576 p^{3} T^{5} + 9780 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 144 T + 70628 T^{2} + 7042032 T^{3} + 2292651510 T^{4} + 7042032 p^{3} T^{5} + 70628 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 400 T + 171340 T^{2} - 38254288 T^{3} + 8472479270 T^{4} - 38254288 p^{3} T^{5} + 171340 p^{6} T^{6} - 400 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 304 T + 169172 T^{2} + 40635984 T^{3} + 11843462518 T^{4} + 40635984 p^{3} T^{5} + 169172 p^{6} T^{6} + 304 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 152 T + 250232 T^{2} + 28847096 T^{3} + 25157924114 T^{4} + 28847096 p^{3} T^{5} + 250232 p^{6} T^{6} + 152 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 160 T + 147340 T^{2} - 16282528 T^{3} + 15975380470 T^{4} - 16282528 p^{3} T^{5} + 147340 p^{6} T^{6} - 160 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 544 T + 378668 T^{2} - 154041760 T^{3} + 58194085286 T^{4} - 154041760 p^{3} T^{5} + 378668 p^{6} T^{6} - 544 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 1320 T + 959116 T^{2} + 462674104 T^{3} + 189646882294 T^{4} + 462674104 p^{3} T^{5} + 959116 p^{6} T^{6} + 1320 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 1040 T + 962396 T^{2} - 555546128 T^{3} + 297086579030 T^{4} - 555546128 p^{3} T^{5} + 962396 p^{6} T^{6} - 1040 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 896 T + 1188912 T^{2} - 644590848 T^{3} + 437823850738 T^{4} - 644590848 p^{3} T^{5} + 1188912 p^{6} T^{6} - 896 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 416 T + 995500 T^{2} + 335375264 T^{3} + 430142403478 T^{4} + 335375264 p^{3} T^{5} + 995500 p^{6} T^{6} + 416 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 248 T + 996916 T^{2} - 246307512 T^{3} + 457673544134 T^{4} - 246307512 p^{3} T^{5} + 996916 p^{6} T^{6} - 248 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 752 T + 1272832 T^{2} + 641462384 T^{3} + 655763261282 T^{4} + 641462384 p^{3} T^{5} + 1272832 p^{6} T^{6} + 752 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 864 T + 1864956 T^{2} - 1235423200 T^{3} + 1357772054598 T^{4} - 1235423200 p^{3} T^{5} + 1864956 p^{6} T^{6} - 864 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 1456 T + 2297228 T^{2} - 2130405040 T^{3} + 2028397026326 T^{4} - 2130405040 p^{3} T^{5} + 2297228 p^{6} T^{6} - 1456 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 2936 T + 5403960 T^{2} - 6890733528 T^{3} + 6651967591762 T^{4} - 6890733528 p^{3} T^{5} + 5403960 p^{6} T^{6} - 2936 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 144 T + 2591968 T^{2} - 824150160 T^{3} + 3033197727234 T^{4} - 824150160 p^{3} T^{5} + 2591968 p^{6} T^{6} - 144 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.50476380347729766275051045196, −6.26132700207302473191366763184, −5.96876986041445407481915089934, −5.96436667969269227134938034229, −5.93193658694550236658827524522, −5.29298008941376845191641812522, −5.25076559377468637918874509963, −5.11869965497779920503891244566, −5.10341602566226007006706319083, −4.64629378547434903352906280493, −4.25659714257570027835040484075, −4.19321956922146612971512865053, −4.08632158761759094452867784328, −3.39846815736164214529371856514, −3.28003340764181301950034563965, −3.27829915966571566884209773419, −2.71570203186089850101471631962, −2.20651834931504747098513911918, −1.99193214617218706797531556679, −1.82290130031543820500956281915, −1.72321501766681537817668643494, −0.913338161695122326305420130911, −0.69044925793705892987760497981, −0.67330610387098018844290457915, −0.33504412875355610270142194740, 0.33504412875355610270142194740, 0.67330610387098018844290457915, 0.69044925793705892987760497981, 0.913338161695122326305420130911, 1.72321501766681537817668643494, 1.82290130031543820500956281915, 1.99193214617218706797531556679, 2.20651834931504747098513911918, 2.71570203186089850101471631962, 3.27829915966571566884209773419, 3.28003340764181301950034563965, 3.39846815736164214529371856514, 4.08632158761759094452867784328, 4.19321956922146612971512865053, 4.25659714257570027835040484075, 4.64629378547434903352906280493, 5.10341602566226007006706319083, 5.11869965497779920503891244566, 5.25076559377468637918874509963, 5.29298008941376845191641812522, 5.93193658694550236658827524522, 5.96436667969269227134938034229, 5.96876986041445407481915089934, 6.26132700207302473191366763184, 6.50476380347729766275051045196

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