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Properties

Label 8-3822e4-1.1-c1e4-0-4
Degree $8$
Conductor $2.134\times 10^{14}$
Sign $1$
Analytic cond. $867503.$
Root an. cond. $5.52438$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 4·3-s + 10·4-s + 2·5-s − 16·6-s + 20·8-s + 10·9-s + 8·10-s + 6·11-s − 40·12-s − 4·13-s − 8·15-s + 35·16-s − 2·17-s + 40·18-s + 2·19-s + 20·20-s + 24·22-s + 10·23-s − 80·24-s − 5·25-s − 16·26-s − 20·27-s + 10·29-s − 32·30-s + 56·32-s − 24·33-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s + 0.894·5-s − 6.53·6-s + 7.07·8-s + 10/3·9-s + 2.52·10-s + 1.80·11-s − 11.5·12-s − 1.10·13-s − 2.06·15-s + 35/4·16-s − 0.485·17-s + 9.42·18-s + 0.458·19-s + 4.47·20-s + 5.11·22-s + 2.08·23-s − 16.3·24-s − 25-s − 3.13·26-s − 3.84·27-s + 1.85·29-s − 5.84·30-s + 9.89·32-s − 4.17·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(867503.\)
Root analytic conductor: \(5.52438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(45.75292696\)
\(L(\frac12)\) \(\approx\) \(45.75292696\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
3$C_1$ \( ( 1 + T )^{4} \)
7 \( 1 \)
13$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 9 T^{2} - 18 T^{3} + 68 T^{4} - 18 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 35 T^{2} - 118 T^{3} + 464 T^{4} - 118 p T^{5} + 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 47 T^{2} + 106 T^{3} + 1048 T^{4} + 106 p T^{5} + 47 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 65 T^{2} - 102 T^{3} + 1776 T^{4} - 102 p T^{5} + 65 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 117 T^{2} - 30 p T^{3} + 4316 T^{4} - 30 p^{2} T^{5} + 117 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 123 T^{2} - 858 T^{3} + 5450 T^{4} - 858 p T^{5} + 123 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 98 T^{2} + 16 T^{3} + 4266 T^{4} + 16 p T^{5} + 98 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 113 T^{2} - 486 T^{3} + 5578 T^{4} - 486 p T^{5} + 113 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 138 T^{2} - 16 T^{3} + 8066 T^{4} - 16 p T^{5} + 138 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 125 T^{2} - 986 T^{3} + 8068 T^{4} - 986 p T^{5} + 125 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 162 T^{2} - 880 T^{3} + 222 p T^{4} - 880 p T^{5} + 162 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 274 T^{2} - 2576 T^{3} + 23362 T^{4} - 2576 p T^{5} + 274 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 210 T^{2} + 16 T^{3} + 17930 T^{4} + 16 p T^{5} + 210 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 167 T^{2} + 658 T^{3} + 12616 T^{4} + 658 p T^{5} + 167 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 28 T + 508 T^{2} - 6308 T^{3} + 59698 T^{4} - 6308 p T^{5} + 508 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 218 T^{2} - 2424 T^{3} + 24770 T^{4} - 2424 p T^{5} + 218 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 353 T^{2} - 3150 T^{3} + 40908 T^{4} - 3150 p T^{5} + 353 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 234 T^{2} - 1360 T^{3} + 24546 T^{4} - 1360 p T^{5} + 234 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 224 T^{2} + 516 T^{3} + 22958 T^{4} + 516 p T^{5} + 224 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 260 T^{2} + 452 T^{3} + 29794 T^{4} + 452 p T^{5} + 260 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 304 T^{2} + 1196 T^{3} + 40606 T^{4} + 1196 p T^{5} + 304 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.17284725479883470925084804295, −5.55664750840795254506148088790, −5.43638924120621481159733778818, −5.36973946290723262766465273050, −5.34815394825281932592410213184, −5.11278571845715007707861114164, −4.94189064669873916257626590074, −4.57035822118453322741560789824, −4.51702115158761676213667177071, −4.11967471918098950727491202218, −4.10873266406056957212760319474, −3.99396059435873970067842337852, −3.92137392736663518868893100973, −3.29443738874146201499626690293, −3.20550610759013880416895448579, −3.01609503226436218307250225305, −2.71828073404801136441865315978, −2.31344761716593241203663534609, −2.11291562468870715672935349228, −1.98301042689242168813696265506, −1.92726654729030969633296578717, −1.17415510914794452098151642848, −0.914675554553736638307062121361, −0.861042283911803961196655504857, −0.68896770970961699940493671310, 0.68896770970961699940493671310, 0.861042283911803961196655504857, 0.914675554553736638307062121361, 1.17415510914794452098151642848, 1.92726654729030969633296578717, 1.98301042689242168813696265506, 2.11291562468870715672935349228, 2.31344761716593241203663534609, 2.71828073404801136441865315978, 3.01609503226436218307250225305, 3.20550610759013880416895448579, 3.29443738874146201499626690293, 3.92137392736663518868893100973, 3.99396059435873970067842337852, 4.10873266406056957212760319474, 4.11967471918098950727491202218, 4.51702115158761676213667177071, 4.57035822118453322741560789824, 4.94189064669873916257626590074, 5.11278571845715007707861114164, 5.34815394825281932592410213184, 5.36973946290723262766465273050, 5.43638924120621481159733778818, 5.55664750840795254506148088790, 6.17284725479883470925084804295

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