Properties

Label 4-96e2-1.1-c1e2-0-0
Degree $4$
Conductor $9216$
Sign $1$
Analytic cond. $0.587620$
Root an. cond. $0.875536$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 12·13-s − 6·25-s − 4·37-s − 14·49-s − 20·61-s − 12·73-s + 9·81-s + 36·97-s + 12·109-s − 36·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 9-s + 3.32·13-s − 6/5·25-s − 0.657·37-s − 2·49-s − 2.56·61-s − 1.40·73-s + 81-s + 3.65·97-s + 1.14·109-s − 3.32·117-s − 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9216\)    =    \(2^{10} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(0.587620\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9216,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9923475956\)
\(L(\frac12)\) \(\approx\) \(0.9923475956\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−11.74104183190432762188729414275, −10.90769214371221130983350005998, −10.79424580459691548950831956254, −9.988093934240586032989036843847, −8.955386231165229198073332132052, −8.944850116469648867972974935175, −8.155593376401658883389001410970, −7.77199473906097062385997282225, −6.62912355829218159331214255175, −5.98788019309426758968327152208, −5.87146418848833687506982135026, −4.69735185642958092576120623346, −3.67478222653086463350186782835, −3.23795791772265289886094041940, −1.61821392697617915675557729976, 1.61821392697617915675557729976, 3.23795791772265289886094041940, 3.67478222653086463350186782835, 4.69735185642958092576120623346, 5.87146418848833687506982135026, 5.98788019309426758968327152208, 6.62912355829218159331214255175, 7.77199473906097062385997282225, 8.155593376401658883389001410970, 8.944850116469648867972974935175, 8.955386231165229198073332132052, 9.988093934240586032989036843847, 10.79424580459691548950831956254, 10.90769214371221130983350005998, 11.74104183190432762188729414275

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