Properties

Label 8-1560e4-1.1-c3e4-0-2
Degree $8$
Conductor $5.922\times 10^{12}$
Sign $1$
Analytic cond. $7.17732\times 10^{7}$
Root an. cond. $9.59390$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·3-s + 20·5-s − 34·7-s + 90·9-s − 38·11-s + 52·13-s + 240·15-s − 50·17-s − 180·19-s − 408·21-s − 170·23-s + 250·25-s + 540·27-s − 84·29-s − 408·31-s − 456·33-s − 680·35-s − 38·37-s + 624·39-s − 90·41-s − 732·43-s + 1.80e3·45-s − 520·47-s + 7·49-s − 600·51-s − 338·53-s − 760·55-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.78·5-s − 1.83·7-s + 10/3·9-s − 1.04·11-s + 1.10·13-s + 4.13·15-s − 0.713·17-s − 2.17·19-s − 4.23·21-s − 1.54·23-s + 2·25-s + 3.84·27-s − 0.537·29-s − 2.36·31-s − 2.40·33-s − 3.28·35-s − 0.168·37-s + 2.56·39-s − 0.342·41-s − 2.59·43-s + 5.96·45-s − 1.61·47-s + 1/49·49-s − 1.64·51-s − 0.875·53-s − 1.86·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(7.17732\times 10^{7}\)
Root analytic conductor: \(9.59390\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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$C_1$ \( ( 1 - p T )^{4} \)
5$C_1$ \( ( 1 - p T )^{4} \)
13$C_1$ \( ( 1 - p T )^{4} \)
good7$C_2 \wr S_4$ \( 1 + 34 T + 1149 T^{2} + 29370 T^{3} + 581332 T^{4} + 29370 p^{3} T^{5} + 1149 p^{6} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 38 T + 3565 T^{2} + 108318 T^{3} + 612868 p T^{4} + 108318 p^{3} T^{5} + 3565 p^{6} T^{6} + 38 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 50 T + 9561 T^{2} + 544314 T^{3} + 48306004 T^{4} + 544314 p^{3} T^{5} + 9561 p^{6} T^{6} + 50 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 180 T + 32304 T^{2} + 3485716 T^{3} + 350633070 T^{4} + 3485716 p^{3} T^{5} + 32304 p^{6} T^{6} + 180 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 170 T + 20929 T^{2} + 1408434 T^{3} + 97368812 T^{4} + 1408434 p^{3} T^{5} + 20929 p^{6} T^{6} + 170 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 84 T + 6372 T^{2} + 4821084 T^{3} + 762239462 T^{4} + 4821084 p^{3} T^{5} + 6372 p^{6} T^{6} + 84 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 408 T + 169112 T^{2} + 38339832 T^{3} + 8310105422 T^{4} + 38339832 p^{3} T^{5} + 169112 p^{6} T^{6} + 408 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 38 T + 103085 T^{2} - 8458554 T^{3} + 5157917716 T^{4} - 8458554 p^{3} T^{5} + 103085 p^{6} T^{6} + 38 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 90 T + 135597 T^{2} + 3155250 T^{3} + 11184348092 T^{4} + 3155250 p^{3} T^{5} + 135597 p^{6} T^{6} + 90 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 732 T + 255372 T^{2} + 90144956 T^{3} + 30846890742 T^{4} + 90144956 p^{3} T^{5} + 255372 p^{6} T^{6} + 732 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 520 T + 441976 T^{2} + 144455080 T^{3} + 68214677838 T^{4} + 144455080 p^{3} T^{5} + 441976 p^{6} T^{6} + 520 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 338 T + 415249 T^{2} + 62164922 T^{3} + 69740841708 T^{4} + 62164922 p^{3} T^{5} + 415249 p^{6} T^{6} + 338 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 232 T + 747164 T^{2} + 138094120 T^{3} + 223465577110 T^{4} + 138094120 p^{3} T^{5} + 747164 p^{6} T^{6} + 232 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 326 T + 530153 T^{2} - 220183086 T^{3} + 151600608428 T^{4} - 220183086 p^{3} T^{5} + 530153 p^{6} T^{6} - 326 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 1040 T + 1422156 T^{2} + 886802960 T^{3} + 659253665366 T^{4} + 886802960 p^{3} T^{5} + 1422156 p^{6} T^{6} + 1040 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 702 T + 993053 T^{2} + 461948022 T^{3} + 435890554644 T^{4} + 461948022 p^{3} T^{5} + 993053 p^{6} T^{6} + 702 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 368 T + 1038588 T^{2} + 243814416 T^{3} + 516012908518 T^{4} + 243814416 p^{3} T^{5} + 1038588 p^{6} T^{6} + 368 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 678 T + 841553 T^{2} - 67621842 T^{3} + 140141073404 T^{4} - 67621842 p^{3} T^{5} + 841553 p^{6} T^{6} + 678 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 1888 T + 2263868 T^{2} + 1987352032 T^{3} + 1474198930486 T^{4} + 1987352032 p^{3} T^{5} + 2263868 p^{6} T^{6} + 1888 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 2070 T + 2663397 T^{2} + 2383070550 T^{3} + 1964568333164 T^{4} + 2383070550 p^{3} T^{5} + 2663397 p^{6} T^{6} + 2070 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 1126 T + 3031341 T^{2} - 2844302494 T^{3} + 3853478524652 T^{4} - 2844302494 p^{3} T^{5} + 3031341 p^{6} T^{6} - 1126 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.85914603911816816107030070940, −6.48837168814449081755851458182, −6.37806477762911473147331114695, −6.20804622333751156887636899405, −6.08772275309199311849767752079, −5.74327854640301010346055229222, −5.34161911440084286926372407755, −5.34110595224487665283659628317, −5.30188548077868048681934551547, −4.53740138147714388866472792721, −4.46505012928887216778930210016, −4.28333898440720103814747400703, −4.15458750426137820473979509498, −3.64245304620427438416073953112, −3.46188728105427891209651968631, −3.41179852419118532223301766889, −3.11054341301676490799985330037, −2.78468347581808261602305180278, −2.68310646449846113402485127350, −2.33293124724066647675523968363, −2.21987922325415503628728471179, −1.68779430789988648162840868370, −1.63047349787470971457486970262, −1.42350135201902743962023027999, −1.39068828216685182347203874719, 0, 0, 0, 0, 1.39068828216685182347203874719, 1.42350135201902743962023027999, 1.63047349787470971457486970262, 1.68779430789988648162840868370, 2.21987922325415503628728471179, 2.33293124724066647675523968363, 2.68310646449846113402485127350, 2.78468347581808261602305180278, 3.11054341301676490799985330037, 3.41179852419118532223301766889, 3.46188728105427891209651968631, 3.64245304620427438416073953112, 4.15458750426137820473979509498, 4.28333898440720103814747400703, 4.46505012928887216778930210016, 4.53740138147714388866472792721, 5.30188548077868048681934551547, 5.34110595224487665283659628317, 5.34161911440084286926372407755, 5.74327854640301010346055229222, 6.08772275309199311849767752079, 6.20804622333751156887636899405, 6.37806477762911473147331114695, 6.48837168814449081755851458182, 6.85914603911816816107030070940

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