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Properties

Label 8-6422e4-1.1-c1e4-0-0
Degree $8$
Conductor $1.701\times 10^{15}$
Sign $1$
Analytic cond. $6.91495\times 10^{6}$
Root an. cond. $7.16100$
Motivic weight $1$
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  − 4·2-s + 2·3-s + 10·4-s − 2·5-s − 8·6-s − 7·7-s − 20·8-s − 3·9-s + 8·10-s − 11-s + 20·12-s + 28·14-s − 4·15-s + 35·16-s − 8·17-s + 12·18-s − 4·19-s − 20·20-s − 14·21-s + 4·22-s + 5·23-s − 40·24-s − 7·25-s − 9·27-s − 70·28-s − 5·29-s + 16·30-s + ⋯
L(s)  = 1  − 2.82·2-s + 1.15·3-s + 5·4-s − 0.894·5-s − 3.26·6-s − 2.64·7-s − 7.07·8-s − 9-s + 2.52·10-s − 0.301·11-s + 5.77·12-s + 7.48·14-s − 1.03·15-s + 35/4·16-s − 1.94·17-s + 2.82·18-s − 0.917·19-s − 4.47·20-s − 3.05·21-s + 0.852·22-s + 1.04·23-s − 8.16·24-s − 7/5·25-s − 1.73·27-s − 13.2·28-s − 0.928·29-s + 2.92·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 13^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(6.91495\times 10^{6}\)
Root analytic conductor: \(7.16100\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 13^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
13 \( 1 \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - 2 T + 7 T^{2} - 11 T^{3} + 29 T^{4} - 11 p T^{5} + 7 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 2 T + 11 T^{2} + 19 T^{3} + 57 T^{4} + 19 p T^{5} + 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + p T + 34 T^{2} + 123 T^{3} + 370 T^{4} + 123 p T^{5} + 34 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + T + 2 p T^{2} - 2 T^{3} + 254 T^{4} - 2 p T^{5} + 2 p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 8 T + 37 T^{2} + 13 p T^{3} + 1229 T^{4} + 13 p^{2} T^{5} + 37 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 5 T + 66 T^{2} - 166 T^{3} + 1732 T^{4} - 166 p T^{5} + 66 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 5 T + 52 T^{2} + 74 T^{3} + 1274 T^{4} + 74 p T^{5} + 52 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 9 T + 72 T^{2} + 458 T^{3} + 2582 T^{4} + 458 p T^{5} + 72 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 5 T + 84 T^{2} + 431 T^{3} + 3438 T^{4} + 431 p T^{5} + 84 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 15 T + 142 T^{2} - 1238 T^{3} + 232 p T^{4} - 1238 p T^{5} + 142 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 9 T + 84 T^{2} + 685 T^{3} + 6350 T^{4} + 685 p T^{5} + 84 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + T + 76 T^{2} + 121 T^{3} + 5438 T^{4} + 121 p T^{5} + 76 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 19 T + 230 T^{2} + 1862 T^{3} + 13920 T^{4} + 1862 p T^{5} + 230 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 4 T + 200 T^{2} - 620 T^{3} + 16590 T^{4} - 620 p T^{5} + 200 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 10 T + 192 T^{2} - 1254 T^{3} + 15294 T^{4} - 1254 p T^{5} + 192 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 16 T + 252 T^{2} - 2640 T^{3} + 25814 T^{4} - 2640 p T^{5} + 252 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 6 T + 149 T^{2} - 1629 T^{3} + 10833 T^{4} - 1629 p T^{5} + 149 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 8 T + 288 T^{2} + 1680 T^{3} + 31454 T^{4} + 1680 p T^{5} + 288 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 12 T + 172 T^{2} + 1628 T^{3} + 18790 T^{4} + 1628 p T^{5} + 172 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - T + 160 T^{2} + 8 T^{3} + 13994 T^{4} + 8 p T^{5} + 160 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 9 T + 254 T^{2} - 1168 T^{3} + 26274 T^{4} - 1168 p T^{5} + 254 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 21 T + 260 T^{2} - 1858 T^{3} + 14620 T^{4} - 1858 p T^{5} + 260 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.26861696017824293409936938668, −5.97334220779593112656794326273, −5.88991875809571780974015586633, −5.52947827530595738487677885718, −5.44438314224854071143986392206, −5.23528684979729914986699980065, −4.81174074725656314706657950957, −4.65361986581578525789071067840, −4.56220063893647559749990132566, −4.02585129104062658092704438416, −3.83222997728095388958304014483, −3.68638065649633940427247521629, −3.66481619932838977726156245313, −3.30565092919114276507219177564, −3.11519464131172298571629461135, −3.06108935233375674808700313990, −2.94182321781733616255776363607, −2.39190920068257433834777405547, −2.23622578690068798871177080494, −2.22777389702264530635845520277, −2.17849775967321817314104137748, −1.72015889904581436138302105625, −1.45764504332031858533750799698, −0.908087936218564451826815956547, −0.73036531727140554863399363278, 0, 0, 0, 0, 0.73036531727140554863399363278, 0.908087936218564451826815956547, 1.45764504332031858533750799698, 1.72015889904581436138302105625, 2.17849775967321817314104137748, 2.22777389702264530635845520277, 2.23622578690068798871177080494, 2.39190920068257433834777405547, 2.94182321781733616255776363607, 3.06108935233375674808700313990, 3.11519464131172298571629461135, 3.30565092919114276507219177564, 3.66481619932838977726156245313, 3.68638065649633940427247521629, 3.83222997728095388958304014483, 4.02585129104062658092704438416, 4.56220063893647559749990132566, 4.65361986581578525789071067840, 4.81174074725656314706657950957, 5.23528684979729914986699980065, 5.44438314224854071143986392206, 5.52947827530595738487677885718, 5.88991875809571780974015586633, 5.97334220779593112656794326273, 6.26861696017824293409936938668

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