Properties

Label 8-1216e4-1.1-c1e4-0-16
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $8888.79$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4·9-s + 16·13-s + 12·17-s + 14·25-s + 24·29-s − 8·37-s + 26·49-s − 24·53-s − 44·73-s − 6·81-s + 56·109-s + 64·117-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4/3·9-s + 4.43·13-s + 2.91·17-s + 14/5·25-s + 4.45·29-s − 1.31·37-s + 26/7·49-s − 3.29·53-s − 5.14·73-s − 2/3·81-s + 5.36·109-s + 5.91·117-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(8888.79\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.71138462\)
\(L(\frac12)\) \(\approx\) \(10.71138462\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
good3$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 59 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 166 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
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

−7.04662657994024753516024805834, −6.69127609520359241104221571575, −6.32442775985774238655683943201, −6.25373749940938337028082815544, −6.24415098127566892862887630842, −5.89725232891791439454123346175, −5.66762329119144006741507770395, −5.57804828508678475203475401770, −5.14034891632527265725200915838, −4.73617757741179802522364513615, −4.61606933427779664118107664219, −4.58067596317813162544530753290, −4.29152722655187238719740439526, −3.75931286424274109137801982329, −3.73093099926058045251888252420, −3.32351565617078422486380381689, −3.18388995059805578943054428431, −3.15570216796774307805720613851, −2.70329741715616803652970730915, −2.43832786147612692722391104901, −1.64470188708365613517861056945, −1.34973367956283087424626317560, −1.14145275964948875984146320967, −1.12551900768180990217582159057, −0.878931248515100674979406216293, 0.878931248515100674979406216293, 1.12551900768180990217582159057, 1.14145275964948875984146320967, 1.34973367956283087424626317560, 1.64470188708365613517861056945, 2.43832786147612692722391104901, 2.70329741715616803652970730915, 3.15570216796774307805720613851, 3.18388995059805578943054428431, 3.32351565617078422486380381689, 3.73093099926058045251888252420, 3.75931286424274109137801982329, 4.29152722655187238719740439526, 4.58067596317813162544530753290, 4.61606933427779664118107664219, 4.73617757741179802522364513615, 5.14034891632527265725200915838, 5.57804828508678475203475401770, 5.66762329119144006741507770395, 5.89725232891791439454123346175, 6.24415098127566892862887630842, 6.25373749940938337028082815544, 6.32442775985774238655683943201, 6.69127609520359241104221571575, 7.04662657994024753516024805834

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