Properties

Label 4-1152e2-1.1-c1e2-0-68
Degree $4$
Conductor $1327104$
Sign $1$
Analytic cond. $84.6173$
Root an. cond. $3.03294$
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  + 3·3-s + 2·5-s + 2·7-s + 6·9-s + 5·11-s − 4·13-s + 6·15-s + 2·17-s + 10·19-s + 6·21-s + 4·23-s + 5·25-s + 9·27-s + 6·29-s + 15·33-s + 4·35-s − 20·37-s − 12·39-s + 3·41-s − 9·43-s + 12·45-s − 8·47-s + 7·49-s + 6·51-s − 24·53-s + 10·55-s + 30·57-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.894·5-s + 0.755·7-s + 2·9-s + 1.50·11-s − 1.10·13-s + 1.54·15-s + 0.485·17-s + 2.29·19-s + 1.30·21-s + 0.834·23-s + 25-s + 1.73·27-s + 1.11·29-s + 2.61·33-s + 0.676·35-s − 3.28·37-s − 1.92·39-s + 0.468·41-s − 1.37·43-s + 1.78·45-s − 1.16·47-s + 49-s + 0.840·51-s − 3.29·53-s + 1.34·55-s + 3.97·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1327104\)    =    \(2^{14} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(84.6173\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1152} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1327104,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.960644633\)
\(L(\frac12)\) \(\approx\) \(6.960644633\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - p T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.842255698011731244122718462807, −9.422407378600581589533706213669, −9.155121609230913338127624555655, −9.017510078621136817434569056754, −8.312340256855943237234880421254, −8.094096382755166649587472808428, −7.60229709452287359313201812630, −7.12222571070718409979944954488, −6.78122581230283263530123901787, −6.57004312260806554987150265198, −5.56131059307928788633747670112, −5.29909377841398454580362999555, −4.66981252003690900419935481202, −4.55091354842265803655492984900, −3.43447847891362849400522701933, −3.31469324502343981536153621659, −2.94574633124404475276512474120, −2.09441785705918153697506992538, −1.46895385339426385223304336900, −1.31441656201680360567380905189, 1.31441656201680360567380905189, 1.46895385339426385223304336900, 2.09441785705918153697506992538, 2.94574633124404475276512474120, 3.31469324502343981536153621659, 3.43447847891362849400522701933, 4.55091354842265803655492984900, 4.66981252003690900419935481202, 5.29909377841398454580362999555, 5.56131059307928788633747670112, 6.57004312260806554987150265198, 6.78122581230283263530123901787, 7.12222571070718409979944954488, 7.60229709452287359313201812630, 8.094096382755166649587472808428, 8.312340256855943237234880421254, 9.017510078621136817434569056754, 9.155121609230913338127624555655, 9.422407378600581589533706213669, 9.842255698011731244122718462807

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