Properties

Label 8-91e4-1.1-c1e4-0-1
Degree $8$
Conductor $68574961$
Sign $1$
Analytic cond. $0.278787$
Root an. cond. $0.852431$
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 + 5·4-s − 8·7-s + 6·8-s + 9-s + 6·11-s − 4·13-s − 24·14-s + 4·16-s + 6·17-s + 3·18-s − 6·19-s + 18·22-s + 12·23-s + 5·25-s − 12·26-s − 40·28-s − 10·31-s + 18·34-s + 5·36-s + 4·37-s − 18·38-s − 32·43-s + 30·44-s + 36·46-s + 6·47-s + 34·49-s + ⋯
L(s)  = 1  + 2.12·2-s + 5/2·4-s − 3.02·7-s + 2.12·8-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 6.41·14-s + 16-s + 1.45·17-s + 0.707·18-s − 1.37·19-s + 3.83·22-s + 2.50·23-s + 25-s − 2.35·26-s − 7.55·28-s − 1.79·31-s + 3.08·34-s + 5/6·36-s + 0.657·37-s − 2.91·38-s − 4.87·43-s + 4.52·44-s + 5.30·46-s + 0.875·47-s + 34/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(68574961\)    =    \(7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(0.278787\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 68574961,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.126927175\)
\(L(\frac12)\) \(\approx\) \(2.126927175\)
\(L(\frac{3}{2})\) 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)$
bad7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_1$ \( ( 1 + T )^{4} \)
good2$C_2^2$$\times$$C_2^2$ \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 - T^{2} + p^{2} T^{4} ) \)
3$C_2^3$ \( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
5$C_2$$\times$$C_2^2$ \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \)
11$C_2^2$ \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 6 T + 13 T^{2} + 66 T^{3} - 372 T^{4} + 66 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 12 T + 67 T^{2} - 372 T^{3} + 2088 T^{4} - 372 p T^{5} + 67 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 4 T - 17 T^{2} + 164 T^{3} - 872 T^{4} + 164 p T^{5} - 17 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
47$D_4\times C_2$ \( 1 - 6 T - p T^{2} + 66 T^{3} + 2988 T^{4} + 66 p T^{5} - p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 6 T - 59 T^{2} + 66 T^{3} + 4308 T^{4} + 66 p T^{5} - 59 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 6 T - 71 T^{2} - 66 T^{3} + 5844 T^{4} - 66 p T^{5} - 71 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 8 T - 53 T^{2} - 232 T^{3} + 3688 T^{4} - 232 p T^{5} - 53 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 8 T - 65 T^{2} - 232 T^{3} + 5344 T^{4} - 232 p T^{5} - 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2^3$ \( 1 - 173 T^{2} + 22008 T^{4} - 173 p^{2} T^{6} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−10.41507271308888056682771179061, −10.19511879307132675294626325601, −9.963985434852701662601982436292, −9.576703550189450260791544118253, −9.365255247777837370764012818202, −9.025745998523202722389118705428, −8.946273652550477929618178825074, −8.458333016419311493913696830101, −7.974674709273874811639719904103, −7.28707580737416211507804133607, −7.00073560932273874586189697192, −6.98263698841547473536537244743, −6.77337655772861857255230814779, −6.37522657177174766914774324039, −5.96738277904419845478407978227, −5.94049446538056836979371913842, −5.09779491066717041247569614310, −5.05436121462349775984850677814, −4.64406321105409139521068003200, −3.97646047461772695807843743915, −3.77734419038093641113834944118, −3.35327604878179764201177913968, −3.00255312100191105218792907301, −2.82081953332938905818375982573, −1.69057622283079173193073362078, 1.69057622283079173193073362078, 2.82081953332938905818375982573, 3.00255312100191105218792907301, 3.35327604878179764201177913968, 3.77734419038093641113834944118, 3.97646047461772695807843743915, 4.64406321105409139521068003200, 5.05436121462349775984850677814, 5.09779491066717041247569614310, 5.94049446538056836979371913842, 5.96738277904419845478407978227, 6.37522657177174766914774324039, 6.77337655772861857255230814779, 6.98263698841547473536537244743, 7.00073560932273874586189697192, 7.28707580737416211507804133607, 7.974674709273874811639719904103, 8.458333016419311493913696830101, 8.946273652550477929618178825074, 9.025745998523202722389118705428, 9.365255247777837370764012818202, 9.576703550189450260791544118253, 9.963985434852701662601982436292, 10.19511879307132675294626325601, 10.41507271308888056682771179061

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