Properties

Label 8-1176e4-1.1-c3e4-0-5
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 $4$

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: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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

−7.25988689615512523068351413976, −6.86383336665716731692873359340, −6.65073168208905586420550762933, −6.54721823259642549511789963260, −6.27636174206909303143141342084, −5.70799437683524723906896896286, −5.64375578432339556583543828273, −5.48734439152422017381029548396, −5.45142217699757072872860119756, −4.71222214433961972985644538853, −4.51809539444083450175164286242, −4.50776680227796427312519357078, −4.48120742904490138475773559026, −3.71101155857432334560636547150, −3.66246165329950508112361535809, −3.53858928648541481004021645437, −3.51504485878016903196191823134, −2.96303401178177584700312212808, −2.69536731447413397551578748237, −2.46505167281548663375535744322, −2.42865276816430869734631087357, −1.76010687222635587330981890187, −1.66957071693829084101383883529, −1.47361748154992834824946703195, −1.31517955957237040639765132981, 0, 0, 0, 0, 1.31517955957237040639765132981, 1.47361748154992834824946703195, 1.66957071693829084101383883529, 1.76010687222635587330981890187, 2.42865276816430869734631087357, 2.46505167281548663375535744322, 2.69536731447413397551578748237, 2.96303401178177584700312212808, 3.51504485878016903196191823134, 3.53858928648541481004021645437, 3.66246165329950508112361535809, 3.71101155857432334560636547150, 4.48120742904490138475773559026, 4.50776680227796427312519357078, 4.51809539444083450175164286242, 4.71222214433961972985644538853, 5.45142217699757072872860119756, 5.48734439152422017381029548396, 5.64375578432339556583543828273, 5.70799437683524723906896896286, 6.27636174206909303143141342084, 6.54721823259642549511789963260, 6.65073168208905586420550762933, 6.86383336665716731692873359340, 7.25988689615512523068351413976

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