Properties

Label 8-98e4-1.1-c3e4-0-0
Degree $8$
Conductor $92236816$
Sign $1$
Analytic cond. $1117.81$
Root an. cond. $2.40461$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 4·4-s − 16·8-s + 4·9-s + 28·11-s − 64·16-s + 16·18-s + 112·22-s − 280·23-s − 142·25-s − 1.14e3·29-s − 64·32-s + 16·36-s + 76·37-s − 136·43-s + 112·44-s − 1.12e3·46-s − 568·50-s + 148·53-s − 4.57e3·58-s + 192·64-s − 1.36e3·67-s + 2.35e3·71-s − 64·72-s + 304·74-s − 2.44e3·79-s + 729·81-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 0.707·8-s + 4/27·9-s + 0.767·11-s − 16-s + 0.209·18-s + 1.08·22-s − 2.53·23-s − 1.13·25-s − 7.32·29-s − 0.353·32-s + 2/27·36-s + 0.337·37-s − 0.482·43-s + 0.383·44-s − 3.58·46-s − 1.60·50-s + 0.383·53-s − 10.3·58-s + 3/8·64-s − 2.49·67-s + 3.93·71-s − 0.104·72-s + 0.477·74-s − 3.47·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(92236816\)    =    \(2^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1117.81\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{98} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 92236816,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.08853239934\)
\(L(\frac12)\) \(\approx\) \(0.08853239934\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
7 \( 1 \)
good3$C_2^3$ \( 1 - 4 T^{2} - 713 T^{4} - 4 p^{6} T^{6} + p^{12} T^{8} \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 233 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )( 1 + 18 T + 233 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} ) \)
11$C_2^2$ \( ( 1 - 14 T - 1135 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 1802 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 9824 T^{2} + 72373407 T^{4} - 9824 p^{6} T^{6} + p^{12} T^{8} \)
19$C_2^3$ \( 1 - 13716 T^{2} + 141082775 T^{4} - 13716 p^{6} T^{6} + p^{12} T^{8} \)
23$C_2^2$ \( ( 1 + 140 T + 7433 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 286 T + p^{3} T^{2} )^{4} \)
31$C_2^3$ \( 1 - 50870 T^{2} + 1700253219 T^{4} - 50870 p^{6} T^{6} + p^{12} T^{8} \)
37$C_2^2$ \( ( 1 - 38 T - 49209 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 122000 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 34 T + p^{3} T^{2} )^{4} \)
47$C_2^3$ \( 1 + 66154 T^{2} - 6402863613 T^{4} + 66154 p^{6} T^{6} + p^{12} T^{8} \)
53$C_2^2$ \( ( 1 - 74 T - 143401 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 222260 T^{2} + 7218973959 T^{4} - 222260 p^{6} T^{6} + p^{12} T^{8} \)
61$C_2^3$ \( 1 - 453762 T^{2} + 154379578283 T^{4} - 453762 p^{6} T^{6} + p^{12} T^{8} \)
67$C_2^2$ \( ( 1 + 684 T + 167093 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 588 T + p^{3} T^{2} )^{4} \)
73$C_2^3$ \( 1 - 705072 T^{2} + 345792298895 T^{4} - 705072 p^{6} T^{6} + p^{12} T^{8} \)
79$C_2^2$ \( ( 1 + 1220 T + 995361 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 964772 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 1028000 T^{2} + 559802709039 T^{4} - 1028000 p^{6} T^{6} + p^{12} T^{8} \)
97$C_2^2$ \( ( 1 - 375456 T^{2} + p^{6} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.752198057551338075762201187225, −9.701053079281238344731863713559, −9.339528961436438735939793019691, −8.811104789708765217865738701935, −8.748889376088170182067053094379, −8.362076369799738357289211538753, −7.64110441500088767462483831442, −7.59455135522075174003361028642, −7.56303018381291545856212536387, −7.05480315751233636208392848719, −6.65691682630307021780731236259, −5.97000276392372530434002676072, −5.81452154018555564508351637710, −5.76271804229069354324198601339, −5.67248152748547664248865299410, −4.75231984050374488329652596511, −4.73046188495361210215353289188, −3.98747034651942221345603214594, −3.79113970056088417285437527670, −3.64676334064836411995569995788, −3.42490691313573738738284499624, −2.18806336988336164400032956730, −2.08253272346669541163945260014, −1.62109349757962770749185104657, −0.06325900694067852553135960071, 0.06325900694067852553135960071, 1.62109349757962770749185104657, 2.08253272346669541163945260014, 2.18806336988336164400032956730, 3.42490691313573738738284499624, 3.64676334064836411995569995788, 3.79113970056088417285437527670, 3.98747034651942221345603214594, 4.73046188495361210215353289188, 4.75231984050374488329652596511, 5.67248152748547664248865299410, 5.76271804229069354324198601339, 5.81452154018555564508351637710, 5.97000276392372530434002676072, 6.65691682630307021780731236259, 7.05480315751233636208392848719, 7.56303018381291545856212536387, 7.59455135522075174003361028642, 7.64110441500088767462483831442, 8.362076369799738357289211538753, 8.748889376088170182067053094379, 8.811104789708765217865738701935, 9.339528961436438735939793019691, 9.701053079281238344731863713559, 9.752198057551338075762201187225

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