Properties

Label 8-98e4-1.1-c1e4-0-0
Degree $8$
Conductor $92236816$
Sign $1$
Analytic cond. $0.374983$
Root an. cond. $0.884609$
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-s + 2·8-s + 4·9-s + 4·11-s − 4·16-s − 8·18-s − 8·22-s + 8·23-s + 2·25-s + 8·29-s + 2·32-s + 4·36-s − 20·37-s + 8·43-s + 4·44-s − 16·46-s − 4·50-s + 4·53-s − 16·58-s + 3·64-s − 24·67-s − 48·71-s + 8·72-s + 40·74-s + 8·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.707·8-s + 4/3·9-s + 1.20·11-s − 16-s − 1.88·18-s − 1.70·22-s + 1.66·23-s + 2/5·25-s + 1.48·29-s + 0.353·32-s + 2/3·36-s − 3.28·37-s + 1.21·43-s + 0.603·44-s − 2.35·46-s − 0.565·50-s + 0.549·53-s − 2.10·58-s + 3/8·64-s − 2.93·67-s − 5.69·71-s + 0.942·72-s + 4.64·74-s + 0.900·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(92236816\)    =    \(2^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(0.374983\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 92236816,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5140674076\)
\(L(\frac12)\) \(\approx\) \(0.5140674076\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7 \( 1 \)
good3$C_2^3$ \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
11$C_2^2$ \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2^3$ \( 1 - 32 T^{2} + 735 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^3$ \( 1 + 12 T^{2} - 217 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
31$C_2^3$ \( 1 + 10 T^{2} - 861 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 116 T^{2} + 9975 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^3$ \( 1 - 114 T^{2} + 9275 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 - 144 T^{2} + 15407 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + 96 T^{2} + p^{2} T^{4} )^{2} \)
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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

−10.36797756766802892753769301082, −9.987839736584902093994878144698, −9.813535206047039858663936301779, −9.287691601674165385886858031133, −9.037326691025925374064102858460, −8.934524743459123306035708801029, −8.806492093525821167723727426256, −8.299776803105340755450334301620, −8.227231595240179265721530576344, −7.41183225848240975897507852246, −7.27840207632221511058928819666, −7.26568928480256498708708581767, −6.89861159000452862380207722778, −6.57471526883428119238872318887, −5.94916008639523159694127953550, −5.89273232430019475483528776608, −5.18977681863156718533409327610, −4.65071630495262576914634603754, −4.61592896308588504442376428583, −4.24208572076520311235406283664, −3.55355603051224765841058400538, −3.23598228469011589466332382812, −2.55959814979660381660161526943, −1.42533684749379634478021716895, −1.40536345694091675296508686664, 1.40536345694091675296508686664, 1.42533684749379634478021716895, 2.55959814979660381660161526943, 3.23598228469011589466332382812, 3.55355603051224765841058400538, 4.24208572076520311235406283664, 4.61592896308588504442376428583, 4.65071630495262576914634603754, 5.18977681863156718533409327610, 5.89273232430019475483528776608, 5.94916008639523159694127953550, 6.57471526883428119238872318887, 6.89861159000452862380207722778, 7.26568928480256498708708581767, 7.27840207632221511058928819666, 7.41183225848240975897507852246, 8.227231595240179265721530576344, 8.299776803105340755450334301620, 8.806492093525821167723727426256, 8.934524743459123306035708801029, 9.037326691025925374064102858460, 9.287691601674165385886858031133, 9.813535206047039858663936301779, 9.987839736584902093994878144698, 10.36797756766802892753769301082

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