Properties

Label 4-1584e2-1.1-c1e2-0-0
Degree $4$
Conductor $2509056$
Sign $1$
Analytic cond. $159.979$
Root an. cond. $3.55644$
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  − 6·5-s + 17·25-s + 14·37-s − 14·49-s − 12·53-s + 18·89-s − 34·97-s + 42·113-s − 11·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s − 84·185-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 2.68·5-s + 17/5·25-s + 2.30·37-s − 2·49-s − 1.64·53-s + 1.90·89-s − 3.45·97-s + 3.95·113-s − 121-s − 1.60·125-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 6.17·185-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2509056\)    =    \(2^{8} \cdot 3^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(159.979\)
Root analytic conductor: \(3.55644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2509056,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5794859737\)
\(L(\frac12)\) \(\approx\) \(0.5794859737\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
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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

−9.527298031399968695009072617038, −9.268837417625597732973522039058, −8.636712062768920253185706800042, −8.158487958521229707799719767839, −8.065230724546502233192443461282, −7.76427541762664832793694490961, −7.35385081331080536077665418146, −6.95320088838568664856591127885, −6.47058745551317505987186758689, −6.06369749379060009767879803291, −5.48711054168476884362949155134, −4.77413391941721413371555392182, −4.50280814318955923080287379462, −4.24757181404877892307860964303, −3.63200033997258241259008051521, −3.28772959447658657385014592230, −2.89569895439985644400896794507, −2.05847254630549191612834740248, −1.12974241868404701799265172265, −0.34302960194193152541892960370, 0.34302960194193152541892960370, 1.12974241868404701799265172265, 2.05847254630549191612834740248, 2.89569895439985644400896794507, 3.28772959447658657385014592230, 3.63200033997258241259008051521, 4.24757181404877892307860964303, 4.50280814318955923080287379462, 4.77413391941721413371555392182, 5.48711054168476884362949155134, 6.06369749379060009767879803291, 6.47058745551317505987186758689, 6.95320088838568664856591127885, 7.35385081331080536077665418146, 7.76427541762664832793694490961, 8.065230724546502233192443461282, 8.158487958521229707799719767839, 8.636712062768920253185706800042, 9.268837417625597732973522039058, 9.527298031399968695009072617038

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