Properties

Label 8-21e8-1.1-c1e4-0-4
Degree $8$
Conductor $37822859361$
Sign $1$
Analytic cond. $153.766$
Root an. cond. $1.87654$
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  − 3·4-s + 4·16-s + 10·25-s − 12·37-s + 48·43-s − 9·64-s − 8·67-s − 16·79-s − 30·100-s + 36·109-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 36·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s − 144·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 3/2·4-s + 16-s + 2·25-s − 1.97·37-s + 7.31·43-s − 9/8·64-s − 0.977·67-s − 1.80·79-s − 3·100-s + 3.44·109-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.95·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s − 10.9·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(153.766\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.552261951\)
\(L(\frac12)\) \(\approx\) \(1.552261951\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^3$ \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 18 T^{2} - 205 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^3$ \( 1 + 6 T^{2} - 2773 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−8.255994561345551601365887338859, −7.56938523000811734448321691772, −7.53675227555732028248516407599, −7.37985377558115264145451324182, −7.26599583516057262854233745818, −6.97721602225914234708857564190, −6.49738706073693180934442683909, −6.15070334950467214322174731646, −5.93893370170133916050379149532, −5.92317704149626451149910777062, −5.44292922413603749407026200958, −5.25115218957661253206508273131, −4.95051116933132258606418965211, −4.63089559534812741817019161324, −4.27677854760552821654329783543, −4.23004013721510196810552384000, −4.10954644861899663574982754642, −3.38024253899097073966283710723, −3.34638285657231304250068139640, −2.80373696425068003917133744313, −2.60792087845042405538000081433, −2.14009531752356935003789463378, −1.57524628942162217482476555325, −0.900627049416438146662480534076, −0.62738357114098857897071554810, 0.62738357114098857897071554810, 0.900627049416438146662480534076, 1.57524628942162217482476555325, 2.14009531752356935003789463378, 2.60792087845042405538000081433, 2.80373696425068003917133744313, 3.34638285657231304250068139640, 3.38024253899097073966283710723, 4.10954644861899663574982754642, 4.23004013721510196810552384000, 4.27677854760552821654329783543, 4.63089559534812741817019161324, 4.95051116933132258606418965211, 5.25115218957661253206508273131, 5.44292922413603749407026200958, 5.92317704149626451149910777062, 5.93893370170133916050379149532, 6.15070334950467214322174731646, 6.49738706073693180934442683909, 6.97721602225914234708857564190, 7.26599583516057262854233745818, 7.37985377558115264145451324182, 7.53675227555732028248516407599, 7.56938523000811734448321691772, 8.255994561345551601365887338859

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