Properties

Label 8-1110e4-1.1-c1e4-0-19
Degree $8$
Conductor $1.518\times 10^{12}$
Sign $1$
Analytic cond. $6171.63$
Root an. cond. $2.97714$
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  + 16·7-s + 2·9-s − 4·13-s − 16-s − 8·19-s + 20·31-s + 24·37-s + 36·43-s + 132·49-s + 16·61-s + 32·63-s − 12·79-s − 5·81-s − 64·91-s − 48·97-s − 40·103-s + 48·109-s − 16·112-s − 8·117-s − 8·121-s + 127-s + 131-s − 128·133-s + 137-s + 139-s − 2·144-s + 149-s + ⋯
L(s)  = 1  + 6.04·7-s + 2/3·9-s − 1.10·13-s − 1/4·16-s − 1.83·19-s + 3.59·31-s + 3.94·37-s + 5.48·43-s + 18.8·49-s + 2.04·61-s + 4.03·63-s − 1.35·79-s − 5/9·81-s − 6.70·91-s − 4.87·97-s − 3.94·103-s + 4.59·109-s − 1.51·112-s − 0.739·117-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s − 11.0·133-s + 0.0854·137-s + 0.0848·139-s − 1/6·144-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(6171.63\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1110} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12.79358652\)
\(L(\frac12)\) \(\approx\) \(12.79358652\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 958 T^{4} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 3442 T^{4} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
79$C_2^2$ \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
89$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
97$C_2^2$ \( ( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.19161426308771528526955445868, −7.00689498302586729877481128948, −6.58832818291790810094777898682, −6.24572512666660200925440454367, −6.22155358515797576018727417546, −5.83146757393035621026640370164, −5.53208497836927928756840890794, −5.38991336622655128978607798956, −5.12421791754311686281628464165, −4.95229467080386844360094903867, −4.65415851961795717985334539737, −4.42963503383769462053577511621, −4.27268727721880451152453244412, −4.24498609967899558218797644174, −4.11857235334476121743580824559, −4.01918135065101729790748411243, −2.82989870082660568266227677528, −2.67021157358878932183785255397, −2.36074257945895288819541044737, −2.30196824802098305971659635939, −2.27528615705376185896537937589, −1.50009799207778545486043385307, −1.20503683304256837796428141553, −1.18344050296027942264272713827, −0.861569195168630739258680216513, 0.861569195168630739258680216513, 1.18344050296027942264272713827, 1.20503683304256837796428141553, 1.50009799207778545486043385307, 2.27528615705376185896537937589, 2.30196824802098305971659635939, 2.36074257945895288819541044737, 2.67021157358878932183785255397, 2.82989870082660568266227677528, 4.01918135065101729790748411243, 4.11857235334476121743580824559, 4.24498609967899558218797644174, 4.27268727721880451152453244412, 4.42963503383769462053577511621, 4.65415851961795717985334539737, 4.95229467080386844360094903867, 5.12421791754311686281628464165, 5.38991336622655128978607798956, 5.53208497836927928756840890794, 5.83146757393035621026640370164, 6.22155358515797576018727417546, 6.24572512666660200925440454367, 6.58832818291790810094777898682, 7.00689498302586729877481128948, 7.19161426308771528526955445868

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