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Properties

Label 8-1900e4-1.1-c3e4-0-2
Degree $8$
Conductor $1.303\times 10^{13}$
Sign $1$
Analytic cond. $1.57935\times 10^{8}$
Root an. cond. $10.5879$
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  − 7·3-s − 23·7-s + 7·9-s + 64·11-s − 31·13-s + 17·17-s − 76·19-s + 161·21-s − 183·23-s − 18·27-s + 279·29-s + 270·31-s − 448·33-s − 30·37-s + 217·39-s − 92·41-s − 286·43-s − 576·47-s − 347·49-s − 119·51-s − 771·53-s + 532·57-s + 195·59-s + 614·61-s − 161·63-s − 1.16e3·67-s + 1.28e3·69-s + ⋯
L(s)  = 1  − 1.34·3-s − 1.24·7-s + 7/27·9-s + 1.75·11-s − 0.661·13-s + 0.242·17-s − 0.917·19-s + 1.67·21-s − 1.65·23-s − 0.128·27-s + 1.78·29-s + 1.56·31-s − 2.36·33-s − 0.133·37-s + 0.890·39-s − 0.350·41-s − 1.01·43-s − 1.78·47-s − 1.01·49-s − 0.326·51-s − 1.99·53-s + 1.23·57-s + 0.430·59-s + 1.28·61-s − 0.321·63-s − 2.12·67-s + 2.23·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.57935\times 10^{8}\)
Root analytic conductor: \(10.5879\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 19^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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 \( 1 \)
19$C_1$ \( ( 1 + p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 7 T + 14 p T^{2} + 263 T^{3} + 1478 T^{4} + 263 p^{3} T^{5} + 14 p^{7} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 23 T + 876 T^{2} + 17491 T^{3} + 444406 T^{4} + 17491 p^{3} T^{5} + 876 p^{6} T^{6} + 23 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 64 T + 6308 T^{2} - 22992 p T^{3} + 13150854 T^{4} - 22992 p^{4} T^{5} + 6308 p^{6} T^{6} - 64 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 31 T + 8660 T^{2} + 194529 T^{3} + 28335194 T^{4} + 194529 p^{3} T^{5} + 8660 p^{6} T^{6} + 31 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - p T + 8306 T^{2} - 311839 T^{3} + 40672826 T^{4} - 311839 p^{3} T^{5} + 8306 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 183 T + 44072 T^{2} + 6212107 T^{3} + 770860638 T^{4} + 6212107 p^{3} T^{5} + 44072 p^{6} T^{6} + 183 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 279 T + 107030 T^{2} - 19813933 T^{3} + 4006409298 T^{4} - 19813933 p^{3} T^{5} + 107030 p^{6} T^{6} - 279 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 270 T + 123412 T^{2} - 23676790 T^{3} + 5573667798 T^{4} - 23676790 p^{3} T^{5} + 123412 p^{6} T^{6} - 270 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 30 T + 190384 T^{2} + 3746390 T^{3} + 14143322646 T^{4} + 3746390 p^{3} T^{5} + 190384 p^{6} T^{6} + 30 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 92 T + 110420 T^{2} + 44291236 T^{3} + 4674780278 T^{4} + 44291236 p^{3} T^{5} + 110420 p^{6} T^{6} + 92 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 286 T + 262244 T^{2} + 64602558 T^{3} + 29273489126 T^{4} + 64602558 p^{3} T^{5} + 262244 p^{6} T^{6} + 286 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 576 T + 364268 T^{2} + 152111376 T^{3} + 53077424358 T^{4} + 152111376 p^{3} T^{5} + 364268 p^{6} T^{6} + 576 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 771 T + 520364 T^{2} + 181537533 T^{3} + 81295040946 T^{4} + 181537533 p^{3} T^{5} + 520364 p^{6} T^{6} + 771 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 195 T + 435224 T^{2} - 135322955 T^{3} + 110888854686 T^{4} - 135322955 p^{3} T^{5} + 435224 p^{6} T^{6} - 195 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 614 T + 858528 T^{2} - 372998002 T^{3} + 285152998414 T^{4} - 372998002 p^{3} T^{5} + 858528 p^{6} T^{6} - 614 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 1167 T + 1446466 T^{2} + 952608319 T^{3} + 661132085166 T^{4} + 952608319 p^{3} T^{5} + 1446466 p^{6} T^{6} + 1167 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 632 T + 1123676 T^{2} - 503683032 T^{3} + 550832301414 T^{4} - 503683032 p^{3} T^{5} + 1123676 p^{6} T^{6} - 632 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 611 T + 959230 T^{2} - 808180389 T^{3} + 442005191874 T^{4} - 808180389 p^{3} T^{5} + 959230 p^{6} T^{6} - 611 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 630 T + 1271548 T^{2} - 718904990 T^{3} + 827414522118 T^{4} - 718904990 p^{3} T^{5} + 1271548 p^{6} T^{6} - 630 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 322 T + 1277652 T^{2} - 244515138 T^{3} + 913858850278 T^{4} - 244515138 p^{3} T^{5} + 1277652 p^{6} T^{6} - 322 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 1248 T + 1241628 T^{2} - 329812800 T^{3} + 99058476326 T^{4} - 329812800 p^{3} T^{5} + 1241628 p^{6} T^{6} - 1248 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 1078 T + 3976040 T^{2} + 2960106318 T^{3} + 5586425630054 T^{4} + 2960106318 p^{3} T^{5} + 3976040 p^{6} T^{6} + 1078 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.51570915481537156461344005881, −6.25453209279940966898818197759, −6.19637969215216877604219680961, −6.16629971409548789974257209757, −6.15442721448906666305832335993, −5.42072993005071364502235769049, −5.20062787434475819581991033775, −5.14964210502854020198193663004, −5.12586959077317680482457926744, −4.70260507420683071907180450304, −4.40639131743450123829676971524, −4.30363751425008330086350091210, −4.04292362322353234913019133692, −3.77053306911219437920532176816, −3.45074629563959434235280963253, −3.41788080772827120877922362428, −3.17000450465735561964354380771, −2.72155827067206787977636126336, −2.47668888722095655143045073213, −2.36251919814978727384972012105, −1.98868518377682616045059950370, −1.48479630058249143428848611564, −1.39833646447319725222398502818, −1.06916569929781707186133952545, −0.881197798674465302218715636632, 0, 0, 0, 0, 0.881197798674465302218715636632, 1.06916569929781707186133952545, 1.39833646447319725222398502818, 1.48479630058249143428848611564, 1.98868518377682616045059950370, 2.36251919814978727384972012105, 2.47668888722095655143045073213, 2.72155827067206787977636126336, 3.17000450465735561964354380771, 3.41788080772827120877922362428, 3.45074629563959434235280963253, 3.77053306911219437920532176816, 4.04292362322353234913019133692, 4.30363751425008330086350091210, 4.40639131743450123829676971524, 4.70260507420683071907180450304, 5.12586959077317680482457926744, 5.14964210502854020198193663004, 5.20062787434475819581991033775, 5.42072993005071364502235769049, 6.15442721448906666305832335993, 6.16629971409548789974257209757, 6.19637969215216877604219680961, 6.25453209279940966898818197759, 6.51570915481537156461344005881

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