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Properties

Label 4-297e2-1.1-c0e2-0-0
Degree $4$
Conductor $88209$
Sign $1$
Analytic cond. $0.0219698$
Root an. cond. $0.384996$
Motivic weight $0$
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  − 4-s − 5-s + 11-s + 20-s + 2·23-s + 25-s + 31-s − 2·37-s − 44-s − 47-s − 49-s + 2·53-s − 55-s − 59-s + 64-s + 67-s + 2·71-s − 4·89-s − 2·92-s + 97-s − 100-s + 103-s − 113-s − 2·115-s − 124-s − 2·125-s + 127-s + ⋯
L(s)  = 1  − 4-s − 5-s + 11-s + 20-s + 2·23-s + 25-s + 31-s − 2·37-s − 44-s − 47-s − 49-s + 2·53-s − 55-s − 59-s + 64-s + 67-s + 2·71-s − 4·89-s − 2·92-s + 97-s − 100-s + 103-s − 113-s − 2·115-s − 124-s − 2·125-s + 127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(88209\)    =    \(3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.0219698\)
Root analytic conductor: \(0.384996\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 88209,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4333644429\)
\(L(\frac12)\) \(\approx\) \(0.4333644429\)
\(L(1)\) 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 \( 1 \)
11$C_2$ \( 1 - T + T^{2} \)
good2$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 + T + T^{2} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_1$ \( ( 1 + T )^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
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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

−12.20025696734899366629885966017, −11.71818531332266187221612733754, −11.34076781514149823181670269102, −10.94443428055690846751617338561, −10.35009411927364674340876402387, −9.796016467305744821634256055648, −9.255379467038931507014275600433, −8.946546214429717118093823632696, −8.320782544654494967207437478526, −8.279897436479740655662250885709, −7.29776356599962325506388115497, −6.84159616035865893324531911570, −6.62769648388861134426301840169, −5.61229297238830216896924244978, −4.83107582838988357916337143368, −4.80518956208838619905369878599, −3.84413042026171845673964585907, −3.56493651180833204370871519282, −2.67917152496581090293312586167, −1.23510019716345420483302233806, 1.23510019716345420483302233806, 2.67917152496581090293312586167, 3.56493651180833204370871519282, 3.84413042026171845673964585907, 4.80518956208838619905369878599, 4.83107582838988357916337143368, 5.61229297238830216896924244978, 6.62769648388861134426301840169, 6.84159616035865893324531911570, 7.29776356599962325506388115497, 8.279897436479740655662250885709, 8.320782544654494967207437478526, 8.946546214429717118093823632696, 9.255379467038931507014275600433, 9.796016467305744821634256055648, 10.35009411927364674340876402387, 10.94443428055690846751617338561, 11.34076781514149823181670269102, 11.71818531332266187221612733754, 12.20025696734899366629885966017

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