Properties

Label 8-31e8-1.1-c1e4-0-13
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  − 3·2-s + 6·3-s + 7·4-s + 4·5-s − 18·6-s + 7·7-s − 15·8-s + 23·9-s − 12·10-s − 2·11-s + 42·12-s + 6·13-s − 21·14-s + 24·15-s + 30·16-s − 8·17-s − 69·18-s + 5·19-s + 28·20-s + 42·21-s + 6·22-s + 16·23-s − 90·24-s − 10·25-s − 18·26-s + 70·27-s + 49·28-s + ⋯
L(s)  = 1  − 2.12·2-s + 3.46·3-s + 7/2·4-s + 1.78·5-s − 7.34·6-s + 2.64·7-s − 5.30·8-s + 23/3·9-s − 3.79·10-s − 0.603·11-s + 12.1·12-s + 1.66·13-s − 5.61·14-s + 6.19·15-s + 15/2·16-s − 1.94·17-s − 16.2·18-s + 1.14·19-s + 6.26·20-s + 9.16·21-s + 1.27·22-s + 3.33·23-s − 18.3·24-s − 2·25-s − 3.53·26-s + 13.4·27-s + 9.26·28-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

\(L(1)\) \(\approx\) \(16.41027078\)
\(L(\frac12)\) \(\approx\) \(16.41027078\)
\(L(\frac{3}{2})\) 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$C_2^2:C_4$ \( 1 + 3 T + p T^{2} + T^{4} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
3$C_4\times C_2$ \( 1 - 2 p T + 13 T^{2} - 10 T^{3} + T^{4} - 10 p T^{5} + 13 p^{2} T^{6} - 2 p^{4} T^{7} + p^{4} T^{8} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
7$C_2^2:C_4$ \( 1 - p T + 12 T^{2} + 25 T^{3} - 139 T^{4} + 25 p T^{5} + 12 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
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$C_2^2:C_4$ \( 1 - 6 T + 3 T^{2} + 10 T^{3} + 81 T^{4} + 10 p T^{5} + 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 + 8 T + 7 T^{2} - 110 T^{3} - 579 T^{4} - 110 p T^{5} + 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
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$C_4\times C_2$ \( 1 - 16 T + 113 T^{2} - 570 T^{3} + 2741 T^{4} - 570 p T^{5} + 113 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2:C_4$ \( 1 + 11 T^{2} - 90 T^{3} + 661 T^{4} - 90 p T^{5} + 11 p^{2} T^{6} + p^{4} T^{8} \)
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$C_2^2:C_4$ \( 1 + 4 T - 27 T^{2} + 110 T^{3} + 2381 T^{4} + 110 p T^{5} - 27 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2:C_4$ \( 1 + 8 T + 17 T^{2} + 380 T^{3} + 4721 T^{4} + 380 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 + 4 T + 43 T^{2} + 380 T^{3} + 4761 T^{4} + 380 p T^{5} + 43 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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$C_2^2:C_4$ \( 1 + 27 T + 308 T^{2} + 2499 T^{3} + 20605 T^{4} + 2499 p T^{5} + 308 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 - 6 T + 3 T^{2} - 640 T^{3} + 9141 T^{4} - 640 p T^{5} + 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 + 10 T + 81 T^{2} + 1190 T^{3} + 16121 T^{4} + 1190 p T^{5} + 81 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 - 26 T + 193 T^{2} + 380 T^{3} - 13419 T^{4} + 380 p T^{5} + 193 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2^2:C_4$ \( 1 - 10 T + 71 T^{2} - 1290 T^{3} + 19001 T^{4} - 1290 p T^{5} + 71 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 + 13 T + 192 T^{2} + 2195 T^{3} + 31031 T^{4} + 2195 p T^{5} + 192 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.39842390412272489311843357691, −7.36142114813468399901678521624, −6.86302740328696827884120494235, −6.61466823048207455485651315814, −6.58815086601572815643132917973, −6.28737921662002837996173981848, −6.03375274792184105842349317906, −5.52542716040919943424019060214, −5.33882776261672230325793429951, −5.07202129814812930945793239500, −5.06842605522713379975144396411, −4.57151728250865097609197468681, −4.15484235630669890523425135490, −3.80378675687783404769838451730, −3.76947257642346899843851376142, −3.30518752253978700249630687729, −2.97929343863950431148806272387, −2.77811334991902784030277994456, −2.70218855299484832086021477386, −1.97002962773525574962501570552, −1.91503200531188728632955611173, −1.83744716131505506394403416607, −1.73906740857049114381448411304, −1.22141152546496166374695409077, −1.01403136590181789897243143693, 1.01403136590181789897243143693, 1.22141152546496166374695409077, 1.73906740857049114381448411304, 1.83744716131505506394403416607, 1.91503200531188728632955611173, 1.97002962773525574962501570552, 2.70218855299484832086021477386, 2.77811334991902784030277994456, 2.97929343863950431148806272387, 3.30518752253978700249630687729, 3.76947257642346899843851376142, 3.80378675687783404769838451730, 4.15484235630669890523425135490, 4.57151728250865097609197468681, 5.06842605522713379975144396411, 5.07202129814812930945793239500, 5.33882776261672230325793429951, 5.52542716040919943424019060214, 6.03375274792184105842349317906, 6.28737921662002837996173981848, 6.58815086601572815643132917973, 6.61466823048207455485651315814, 6.86302740328696827884120494235, 7.36142114813468399901678521624, 7.39842390412272489311843357691

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