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Properties

Label 8-95e8-1.1-c1e4-0-3
Degree $8$
Conductor $6.634\times 10^{15}$
Sign $1$
Analytic cond. $2.69710\times 10^{7}$
Root an. cond. $8.48910$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 2·4-s + 6-s + 11·7-s − 8-s − 3·9-s − 2·12-s − 2·13-s + 11·14-s + 16-s + 7·17-s − 3·18-s + 11·21-s + 11·23-s − 24-s − 2·26-s − 9·27-s − 22·28-s + 15·29-s + 31-s − 4·32-s + 7·34-s + 6·36-s − 11·37-s − 2·39-s − 22·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 4-s + 0.408·6-s + 4.15·7-s − 0.353·8-s − 9-s − 0.577·12-s − 0.554·13-s + 2.93·14-s + 1/4·16-s + 1.69·17-s − 0.707·18-s + 2.40·21-s + 2.29·23-s − 0.204·24-s − 0.392·26-s − 1.73·27-s − 4.15·28-s + 2.78·29-s + 0.179·31-s − 0.707·32-s + 1.20·34-s + 36-s − 1.80·37-s − 0.320·39-s − 3.43·41-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(2.69710\times 10^{7}\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(30.80129032\)
\(L(\frac12)\) \(\approx\) \(30.80129032\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
19 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - T + 3 T^{2} - p^{2} T^{3} + p^{3} T^{4} - p^{3} T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr C_2\wr C_2$ \( 1 - T + 4 T^{2} + 2 T^{3} + 7 T^{4} + 2 p T^{5} + 4 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 68 T^{2} - 284 T^{3} + 873 T^{4} - 284 p T^{5} + 68 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 35 T^{2} - 10 T^{3} + 527 T^{4} - 10 p T^{5} + 35 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 20 T^{2} + 100 T^{3} + 253 T^{4} + 100 p T^{5} + 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 43 T^{2} - 101 T^{3} + 448 T^{4} - 101 p T^{5} + 43 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 94 T^{2} - 478 T^{3} + 2557 T^{4} - 478 p T^{5} + 94 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 165 T^{2} - 1255 T^{3} + 268 p T^{4} - 1255 p T^{5} + 165 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - T + 64 T^{2} - 96 T^{3} + 2705 T^{4} - 96 p T^{5} + 64 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 178 T^{2} + 1204 T^{3} + 10333 T^{4} + 1204 p T^{5} + 178 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 264 T^{2} + 2100 T^{3} + 14225 T^{4} + 2100 p T^{5} + 264 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 26 T + 397 T^{2} - 94 p T^{3} + 30823 T^{4} - 94 p^{2} T^{5} + 397 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 26 T + 388 T^{2} - 3904 T^{3} + 30373 T^{4} - 3904 p T^{5} + 388 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 237 T^{2} + 2012 T^{3} + 17613 T^{4} + 2012 p T^{5} + 237 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 195 T^{2} - 1420 T^{3} + 16877 T^{4} - 1420 p T^{5} + 195 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 113 T^{2} + 566 T^{3} + 4600 T^{4} + 566 p T^{5} + 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 158 T^{2} - 524 T^{3} + 14583 T^{4} - 524 p T^{5} + 158 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 384 T^{2} + 4000 T^{3} + 44465 T^{4} + 4000 p T^{5} + 384 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 442 T^{2} - 5168 T^{3} + 51803 T^{4} - 5168 p T^{5} + 442 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 30 T + 637 T^{2} + 8570 T^{3} + 90548 T^{4} + 8570 p T^{5} + 637 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 327 T^{2} - 2692 T^{3} + 40508 T^{4} - 2692 p T^{5} + 327 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 4 p T^{2} + 2390 T^{3} + 47525 T^{4} + 2390 p T^{5} + 4 p^{3} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 19 T + 340 T^{2} - 4300 T^{3} + 46303 T^{4} - 4300 p T^{5} + 340 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.34090141603202049402666747276, −4.99185325500268495975495569105, −4.98187715551303682203450274783, −4.88689302759474788434071978619, −4.74910989754016907291727589216, −4.70785776280992132990129826363, −4.35775732432172182868972609915, −4.22783890749506324098309548139, −4.13718582523150664112369315228, −3.79764895521946885106927332339, −3.61166135159029133980448290047, −3.32235907851241905727257435317, −3.26826308874573911756671857068, −2.87116223854227584392366738266, −2.77501819792343095068304503289, −2.61373096439204824221474921813, −2.34108254138263647883533098843, −1.93294947113707606050709714988, −1.80510993150680264316845239973, −1.71439217026713930965330205453, −1.55240966540368217942159873172, −0.980430970744608418166054487352, −0.945636504506242838558107832417, −0.67687070821380005541659113017, −0.50104893645986292055784407962, 0.50104893645986292055784407962, 0.67687070821380005541659113017, 0.945636504506242838558107832417, 0.980430970744608418166054487352, 1.55240966540368217942159873172, 1.71439217026713930965330205453, 1.80510993150680264316845239973, 1.93294947113707606050709714988, 2.34108254138263647883533098843, 2.61373096439204824221474921813, 2.77501819792343095068304503289, 2.87116223854227584392366738266, 3.26826308874573911756671857068, 3.32235907851241905727257435317, 3.61166135159029133980448290047, 3.79764895521946885106927332339, 4.13718582523150664112369315228, 4.22783890749506324098309548139, 4.35775732432172182868972609915, 4.70785776280992132990129826363, 4.74910989754016907291727589216, 4.88689302759474788434071978619, 4.98187715551303682203450274783, 4.99185325500268495975495569105, 5.34090141603202049402666747276

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