Properties

Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more about

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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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$C_2 \wr C_2\wr C_2$ \( 1 - 144 T + 2591968 T^{2} - 824150160 T^{3} + 3033197727234 T^{4} - 824150160 p^{3} T^{5} + 2591968 p^{6} T^{6} - 144 p^{9} T^{7} + p^{12} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.46827721600026290806898644002, −6.26288260955320306579980909000, −6.13125032868207808789385002721, −5.88417439212052389323744446981, −5.84231163429599317515092400655, −5.21277158964052701423457356170, −5.09514815667377460172346453153, −4.89513119663858676631966708263, −4.71576735326991678127664231498, −4.37102320149255210143048192190, −4.12810189244668091782891846759, −4.01224083911137744005391565121, −3.96684409416604842116762658666, −3.46861034537734163450884498036, −3.41039046916349511291395381925, −3.32446503282435461266908869954, −3.11419673252730239734338411325, −2.35894350103838170068575492100, −2.35789718480854725632291779520, −2.21890908793913834133819503056, −2.03445090727117358991757432236, −1.75800553960802472349861135790, −1.49978067305721876004197705330, −1.31498927224312596647926966662, −1.25212773238234968325918313147, 0, 0, 0, 0, 1.25212773238234968325918313147, 1.31498927224312596647926966662, 1.49978067305721876004197705330, 1.75800553960802472349861135790, 2.03445090727117358991757432236, 2.21890908793913834133819503056, 2.35789718480854725632291779520, 2.35894350103838170068575492100, 3.11419673252730239734338411325, 3.32446503282435461266908869954, 3.41039046916349511291395381925, 3.46861034537734163450884498036, 3.96684409416604842116762658666, 4.01224083911137744005391565121, 4.12810189244668091782891846759, 4.37102320149255210143048192190, 4.71576735326991678127664231498, 4.89513119663858676631966708263, 5.09514815667377460172346453153, 5.21277158964052701423457356170, 5.84231163429599317515092400655, 5.88417439212052389323744446981, 6.13125032868207808789385002721, 6.26288260955320306579980909000, 6.46827721600026290806898644002

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