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Properties

Label 8-1300e4-1.1-c3e4-0-4
Degree $8$
Conductor $2.856\times 10^{12}$
Sign $1$
Analytic cond. $3.46128\times 10^{7}$
Root an. cond. $8.75799$
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  − 4·3-s + 8·7-s − 28·9-s + 28·11-s − 52·13-s − 200·17-s + 132·19-s − 32·21-s − 244·23-s + 4·27-s − 8·29-s + 292·31-s − 112·33-s − 168·37-s + 208·39-s + 280·41-s − 452·43-s + 280·47-s − 484·49-s + 800·51-s − 584·53-s − 528·57-s − 708·59-s + 1.12e3·61-s − 224·63-s − 176·67-s + 976·69-s + ⋯
L(s)  = 1  − 0.769·3-s + 0.431·7-s − 1.03·9-s + 0.767·11-s − 1.10·13-s − 2.85·17-s + 1.59·19-s − 0.332·21-s − 2.21·23-s + 0.0285·27-s − 0.0512·29-s + 1.69·31-s − 0.590·33-s − 0.746·37-s + 0.854·39-s + 1.06·41-s − 1.60·43-s + 0.868·47-s − 1.41·49-s + 2.19·51-s − 1.51·53-s − 1.22·57-s − 1.56·59-s + 2.36·61-s − 0.447·63-s − 0.320·67-s + 1.70·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3.46128\times 10^{7}\)
Root analytic conductor: \(8.75799\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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 \( 1 \)
5 \( 1 \)
13$C_1$ \( ( 1 + p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 4 T + 44 T^{2} + 284 T^{3} + 382 p T^{4} + 284 p^{3} T^{5} + 44 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 8 T + 548 T^{2} - 5640 T^{3} + 148262 T^{4} - 5640 p^{3} T^{5} + 548 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 28 T + 1460 T^{2} - 45876 T^{3} + 2444250 T^{4} - 45876 p^{3} T^{5} + 1460 p^{6} T^{6} - 28 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 200 T + 20252 T^{2} + 1728888 T^{3} + 135769926 T^{4} + 1728888 p^{3} T^{5} + 20252 p^{6} T^{6} + 200 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 132 T + 23300 T^{2} - 1902828 T^{3} + 217002618 T^{4} - 1902828 p^{3} T^{5} + 23300 p^{6} T^{6} - 132 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 244 T + 49164 T^{2} + 7236924 T^{3} + 929456362 T^{4} + 7236924 p^{3} T^{5} + 49164 p^{6} T^{6} + 244 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 8 T + 53108 T^{2} - 745032 T^{3} + 1693862550 T^{4} - 745032 p^{3} T^{5} + 53108 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 292 T + 87940 T^{2} - 14635868 T^{3} + 2822357642 T^{4} - 14635868 p^{3} T^{5} + 87940 p^{6} T^{6} - 292 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 168 T + 119188 T^{2} + 13477016 T^{3} + 7748407542 T^{4} + 13477016 p^{3} T^{5} + 119188 p^{6} T^{6} + 168 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 280 T + 199484 T^{2} - 56775144 T^{3} + 18493601958 T^{4} - 56775144 p^{3} T^{5} + 199484 p^{6} T^{6} - 280 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 452 T + 237468 T^{2} + 93060700 T^{3} + 25072263706 T^{4} + 93060700 p^{3} T^{5} + 237468 p^{6} T^{6} + 452 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 280 T + 5500 p T^{2} - 84018648 T^{3} + 33765858822 T^{4} - 84018648 p^{3} T^{5} + 5500 p^{7} T^{6} - 280 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 584 T + 493004 T^{2} + 249609624 T^{3} + 102296721846 T^{4} + 249609624 p^{3} T^{5} + 493004 p^{6} T^{6} + 584 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 12 p T + 873412 T^{2} + 409457100 T^{3} + 275177160410 T^{4} + 409457100 p^{3} T^{5} + 873412 p^{6} T^{6} + 12 p^{10} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 1128 T + 1092404 T^{2} - 676342872 T^{3} + 371359528854 T^{4} - 676342872 p^{3} T^{5} + 1092404 p^{6} T^{6} - 1128 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 176 T + 799956 T^{2} + 123238768 T^{3} + 334189723030 T^{4} + 123238768 p^{3} T^{5} + 799956 p^{6} T^{6} + 176 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 1028 T + 1442420 T^{2} + 1026436668 T^{3} + 772928292906 T^{4} + 1026436668 p^{3} T^{5} + 1442420 p^{6} T^{6} + 1028 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 664 T + 555076 T^{2} + 245757704 T^{3} + 196923454310 T^{4} + 245757704 p^{3} T^{5} + 555076 p^{6} T^{6} + 664 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 728 T + 839740 T^{2} + 650486584 T^{3} + 587293110278 T^{4} + 650486584 p^{3} T^{5} + 839740 p^{6} T^{6} + 728 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 552 T + 1918100 T^{2} + 988230024 T^{3} + 1539297094422 T^{4} + 988230024 p^{3} T^{5} + 1918100 p^{6} T^{6} + 552 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 2824 T + 4314684 T^{2} + 5260451832 T^{3} + 5135993645734 T^{4} + 5260451832 p^{3} T^{5} + 4314684 p^{6} T^{6} + 2824 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 1160 T + 3178908 T^{2} + 2304702136 T^{3} + 3910117945798 T^{4} + 2304702136 p^{3} T^{5} + 3178908 p^{6} T^{6} + 1160 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.00615199520779676460777483786, −6.56354311347496772818273803884, −6.37756479246738278629915973144, −6.31871667881977237551652526356, −6.18699515616047713293065973022, −5.82855252994222136439429124443, −5.52220699803535105231134594974, −5.50021017610919310265660791505, −5.29543571511023016040909279611, −4.82215560439650395596295089696, −4.62328111209903494085468289801, −4.61340167591874484222303924121, −4.35647619933101088396683775603, −4.00405867706546305358071431560, −3.91112619350551806133422167813, −3.43263377096127506917174998947, −3.26033373319469723120104630625, −2.82966986774275889486750409753, −2.73432906549188481910749124029, −2.35653734676815555805048238882, −2.20906600521850189073298205847, −1.71003214824510173079896075151, −1.66852308971000821794856266716, −1.13599994990650031821105427494, −0.974900445829079120738164488920, 0, 0, 0, 0, 0.974900445829079120738164488920, 1.13599994990650031821105427494, 1.66852308971000821794856266716, 1.71003214824510173079896075151, 2.20906600521850189073298205847, 2.35653734676815555805048238882, 2.73432906549188481910749124029, 2.82966986774275889486750409753, 3.26033373319469723120104630625, 3.43263377096127506917174998947, 3.91112619350551806133422167813, 4.00405867706546305358071431560, 4.35647619933101088396683775603, 4.61340167591874484222303924121, 4.62328111209903494085468289801, 4.82215560439650395596295089696, 5.29543571511023016040909279611, 5.50021017610919310265660791505, 5.52220699803535105231134594974, 5.82855252994222136439429124443, 6.18699515616047713293065973022, 6.31871667881977237551652526356, 6.37756479246738278629915973144, 6.56354311347496772818273803884, 7.00615199520779676460777483786

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