Properties

Label 8-882e4-1.1-c4e4-0-6
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $6.90958\times 10^{7}$
Root an. cond. $9.54841$
Motivic weight $4$
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  − 16·4-s − 472·13-s + 192·16-s + 40·19-s + 452·25-s − 136·31-s − 4.19e3·37-s + 9.58e3·43-s + 7.55e3·52-s − 4.52e3·61-s − 2.04e3·64-s − 1.94e3·67-s + 2.13e4·73-s − 640·76-s + 1.03e4·79-s + 4.48e4·97-s − 7.23e3·100-s + 2.74e4·103-s + 3.32e4·109-s + 5.30e4·121-s + 2.17e3·124-s + 127-s + 131-s + 137-s + 139-s + 6.70e4·148-s + 149-s + ⋯
L(s)  = 1  − 4-s − 2.79·13-s + 3/4·16-s + 0.110·19-s + 0.723·25-s − 0.141·31-s − 3.06·37-s + 5.18·43-s + 2.79·52-s − 1.21·61-s − 1/2·64-s − 0.433·67-s + 4.00·73-s − 0.110·76-s + 1.65·79-s + 4.76·97-s − 0.723·100-s + 2.59·103-s + 2.79·109-s + 3.62·121-s + 0.141·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 3.06·148-s + 4.50e−5·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6.90958\times 10^{7}\)
Root analytic conductor: \(9.54841\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(6.675323722\)
\(L(\frac12)\) \(\approx\) \(6.675323722\)
\(L(3)\) 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$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 - 452 T^{2} - 95146 T^{4} - 452 p^{8} T^{6} + p^{16} T^{8} \)
11$D_4\times C_2$ \( 1 - 53072 T^{2} + 1132855106 T^{4} - 53072 p^{8} T^{6} + p^{16} T^{8} \)
13$D_{4}$ \( ( 1 + 236 T + 65558 T^{2} + 236 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 229220 T^{2} + 24556773974 T^{4} - 229220 p^{8} T^{6} + p^{16} T^{8} \)
19$D_{4}$ \( ( 1 - 20 T - 47958 T^{2} - 20 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 918512 T^{2} + 366465886466 T^{4} - 918512 p^{8} T^{6} + p^{16} T^{8} \)
29$D_4\times C_2$ \( 1 - 2538896 T^{2} + 2606258922434 T^{4} - 2538896 p^{8} T^{6} + p^{16} T^{8} \)
31$D_{4}$ \( ( 1 + 68 T + 1846826 T^{2} + 68 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 2096 T + 3611826 T^{2} + 2096 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 1782916 T^{2} - 5802193925994 T^{4} - 1782916 p^{8} T^{6} + p^{16} T^{8} \)
43$D_{4}$ \( ( 1 - 4792 T + 12529026 T^{2} - 4792 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 9480916 T^{2} + 51658338912486 T^{4} - 9480916 p^{8} T^{6} + p^{16} T^{8} \)
53$D_4\times C_2$ \( 1 + 7270304 T^{2} + 133516740466626 T^{4} + 7270304 p^{8} T^{6} + p^{16} T^{8} \)
59$D_4\times C_2$ \( 1 - 22659476 T^{2} + 261789383129894 T^{4} - 22659476 p^{8} T^{6} + p^{16} T^{8} \)
61$D_{4}$ \( ( 1 + 2264 T + 28247318 T^{2} + 2264 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 972 T + 38557270 T^{2} + 972 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 33883024 T^{2} + 1543886951212034 T^{4} + 33883024 p^{8} T^{6} + p^{16} T^{8} \)
73$D_{4}$ \( ( 1 - 10680 T + 75860374 T^{2} - 10680 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 5156 T + 84195014 T^{2} - 5156 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 34049348 T^{2} + 133007755977158 T^{4} - 34049348 p^{8} T^{6} + p^{16} T^{8} \)
89$D_4\times C_2$ \( 1 - 233538340 T^{2} + 21473360777771862 T^{4} - 233538340 p^{8} T^{6} + p^{16} T^{8} \)
97$D_{4}$ \( ( 1 - 22416 T + 291316294 T^{2} - 22416 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
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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

−6.97345411707353046288743193319, −6.34551003377952647624909869444, −6.01520243252842056183403715447, −5.98115850298076525226681259874, −5.90582453203365722017802061909, −5.30935771411475774677653902725, −5.26515933391943943123905912232, −5.02725731155908034716826315400, −4.73341770655116558013476834912, −4.58149026319138810643348099756, −4.48516289364960025967976462634, −4.22408399998958224883177943976, −3.62601003971422177890403148845, −3.58363503430963726138029239701, −3.32563388274456776546077006368, −3.01210955182277509452244842725, −2.79558757312527540002404534677, −2.16890464346543091723583440205, −2.14772847651052362917711736272, −1.98680005856877487764195063177, −1.71949123440869040485593288208, −0.78031739048957045513073554313, −0.63896836374341721298578457755, −0.57187434762008391122358607575, −0.50513230128296233782673868012, 0.50513230128296233782673868012, 0.57187434762008391122358607575, 0.63896836374341721298578457755, 0.78031739048957045513073554313, 1.71949123440869040485593288208, 1.98680005856877487764195063177, 2.14772847651052362917711736272, 2.16890464346543091723583440205, 2.79558757312527540002404534677, 3.01210955182277509452244842725, 3.32563388274456776546077006368, 3.58363503430963726138029239701, 3.62601003971422177890403148845, 4.22408399998958224883177943976, 4.48516289364960025967976462634, 4.58149026319138810643348099756, 4.73341770655116558013476834912, 5.02725731155908034716826315400, 5.26515933391943943123905912232, 5.30935771411475774677653902725, 5.90582453203365722017802061909, 5.98115850298076525226681259874, 6.01520243252842056183403715447, 6.34551003377952647624909869444, 6.97345411707353046288743193319

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