Properties

Label 8-31e8-1.1-c1e4-0-12
Degree $8$
Conductor $852891037441$
Sign $1$
Analytic cond. $3467.38$
Root an. cond. $2.77013$
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  + 2·2-s + 4·3-s + 2·4-s + 4·5-s + 8·6-s − 3·7-s + 5·8-s + 3·9-s + 8·10-s + 2·11-s + 8·12-s + 4·13-s − 6·14-s + 16·15-s + 5·16-s − 2·17-s + 6·18-s − 5·19-s + 8·20-s − 12·21-s + 4·22-s + 14·23-s + 20·24-s − 10·25-s + 8·26-s − 10·27-s − 6·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 4-s + 1.78·5-s + 3.26·6-s − 1.13·7-s + 1.76·8-s + 9-s + 2.52·10-s + 0.603·11-s + 2.30·12-s + 1.10·13-s − 1.60·14-s + 4.13·15-s + 5/4·16-s − 0.485·17-s + 1.41·18-s − 1.14·19-s + 1.78·20-s − 2.61·21-s + 0.852·22-s + 2.91·23-s + 4.08·24-s − 2·25-s + 1.56·26-s − 1.92·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31^{8}\)
Sign: $1$
Analytic conductor: \(3467.38\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(26.74057185\)
\(L(\frac12)\) \(\approx\) \(26.74057185\)
\(L(\frac{3}{2})\) 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)$
bad31 \( 1 \)
good2$C_2^2:C_4$ \( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
3$C_4\times C_2$ \( 1 - 4 T + 13 T^{2} - 10 p T^{3} + 61 T^{4} - 10 p^{2} T^{5} + 13 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
7$C_2^2:C_4$ \( 1 + 3 T + 12 T^{2} + 5 p T^{3} + 141 T^{4} + 5 p^{2} T^{5} + 12 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_4\times C_2$ \( 1 - 2 T - 7 T^{2} + 36 T^{3} + 5 T^{4} + 36 p T^{5} - 7 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 + 2 T + 7 T^{2} + 70 T^{3} + 441 T^{4} + 70 p T^{5} + 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_4\times C_2$ \( 1 + 5 T - 4 T^{2} + 25 T^{3} + 481 T^{4} + 25 p T^{5} - 4 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_4\times C_2$ \( 1 - 14 T + 113 T^{2} - 750 T^{3} + 4121 T^{4} - 750 p T^{5} + 113 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2:C_4$ \( 1 - 10 T + 11 T^{2} + 10 p T^{3} - 2239 T^{4} + 10 p^{2} T^{5} + 11 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
41$C_4\times C_2$ \( 1 + 7 T + 8 T^{2} - 231 T^{3} - 1945 T^{4} - 231 p T^{5} + 8 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2^2:C_4$ \( 1 + 6 T - 27 T^{2} - 70 T^{3} + 1521 T^{4} - 70 p T^{5} - 27 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2:C_4$ \( 1 - 12 T + 17 T^{2} + 60 T^{3} + 961 T^{4} + 60 p T^{5} + 17 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 + 16 T + 43 T^{2} - 700 T^{3} - 7959 T^{4} - 700 p T^{5} + 43 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_4\times C_2$ \( 1 - 5 T - 44 T^{2} - 25 T^{3} + 3801 T^{4} - 25 p T^{5} - 44 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 6 T + 6 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
71$C_2^2:C_4$ \( 1 - 23 T + 308 T^{2} - 3551 T^{3} + 34805 T^{4} - 3551 p T^{5} + 308 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 - 14 T + 3 T^{2} + 560 T^{3} - 2539 T^{4} + 560 p T^{5} + 3 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 + 20 T + 81 T^{2} - 590 T^{3} - 7579 T^{4} - 590 p T^{5} + 81 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 - 14 T + 193 T^{2} - 2140 T^{3} + 26421 T^{4} - 2140 p T^{5} + 193 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2^2:C_4$ \( 1 - 20 T + 71 T^{2} + 690 T^{3} - 7699 T^{4} + 690 p T^{5} + 71 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 - 27 T + 192 T^{2} + 955 T^{3} - 23289 T^{4} + 955 p T^{5} + 192 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.91360569925678264444950154751, −6.85621850084825559817728074214, −6.56935197553930657308260628992, −6.53254934391305821727374292371, −6.38235410822394256777712702831, −6.14288195558794695070000762029, −5.83299523154427080516381496308, −5.44881710700127790494832208798, −5.28379696119645725095175794766, −5.14282316188837369649884849285, −4.94413334878732315410590411270, −4.42462566404667766815095419969, −4.38903784271437618710009724654, −3.85917229662276327598615738841, −3.81650384490007736904939329047, −3.49859147740206352123418766848, −3.41804223636609329394888235321, −2.91645269991502459828018569957, −2.89456099495951250840318770413, −2.27370754833561595973934446860, −2.25506520347049065377097434788, −2.06516233778738711783333634009, −1.88141650612529652715743079893, −1.14560757710278423202081249974, −0.72910083841674221146316541974, 0.72910083841674221146316541974, 1.14560757710278423202081249974, 1.88141650612529652715743079893, 2.06516233778738711783333634009, 2.25506520347049065377097434788, 2.27370754833561595973934446860, 2.89456099495951250840318770413, 2.91645269991502459828018569957, 3.41804223636609329394888235321, 3.49859147740206352123418766848, 3.81650384490007736904939329047, 3.85917229662276327598615738841, 4.38903784271437618710009724654, 4.42462566404667766815095419969, 4.94413334878732315410590411270, 5.14282316188837369649884849285, 5.28379696119645725095175794766, 5.44881710700127790494832208798, 5.83299523154427080516381496308, 6.14288195558794695070000762029, 6.38235410822394256777712702831, 6.53254934391305821727374292371, 6.56935197553930657308260628992, 6.85621850084825559817728074214, 6.91360569925678264444950154751

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