Properties

Label 4-304e2-1.1-c1e2-0-11
Degree $4$
Conductor $92416$
Sign $1$
Analytic cond. $5.89252$
Root an. cond. $1.55802$
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  + 4·3-s + 6·9-s + 10·11-s − 6·17-s − 2·19-s − 9·25-s − 4·27-s + 40·33-s + 20·41-s + 2·43-s − 5·49-s − 24·51-s − 8·57-s + 12·59-s − 24·67-s + 18·73-s − 36·75-s − 37·81-s − 24·83-s + 24·89-s − 16·97-s + 60·99-s + 4·107-s − 20·113-s + 53·121-s + 80·123-s + 127-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s + 3.01·11-s − 1.45·17-s − 0.458·19-s − 9/5·25-s − 0.769·27-s + 6.96·33-s + 3.12·41-s + 0.304·43-s − 5/7·49-s − 3.36·51-s − 1.05·57-s + 1.56·59-s − 2.93·67-s + 2.10·73-s − 4.15·75-s − 4.11·81-s − 2.63·83-s + 2.54·89-s − 1.62·97-s + 6.03·99-s + 0.386·107-s − 1.88·113-s + 4.81·121-s + 7.21·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(92416\)    =    \(2^{8} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5.89252\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{92416} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 92416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.415879964\)
\(L(\frac12)\) \(\approx\) \(3.415879964\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−9.470845631762160929353564163989, −9.084561409834522310688102275122, −8.762972910391102183787591721206, −8.381087888928782814621554216261, −7.62429229152375892340957919668, −7.46802549489449929304458830547, −6.44323514092267885981017350010, −6.39486340421971046323119110374, −5.61135260065455155907566257543, −4.17947794613570531538770424508, −4.15537276955970084853193543262, −3.73571330395388729025783799541, −2.82511261070703297705797963037, −2.22354879090908877160738042469, −1.55214010532178794455093749689, 1.55214010532178794455093749689, 2.22354879090908877160738042469, 2.82511261070703297705797963037, 3.73571330395388729025783799541, 4.15537276955970084853193543262, 4.17947794613570531538770424508, 5.61135260065455155907566257543, 6.39486340421971046323119110374, 6.44323514092267885981017350010, 7.46802549489449929304458830547, 7.62429229152375892340957919668, 8.381087888928782814621554216261, 8.762972910391102183787591721206, 9.084561409834522310688102275122, 9.470845631762160929353564163989

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