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Properties

Label 4-1305e2-1.1-c0e2-0-0
Degree $4$
Conductor $1703025$
Sign $1$
Analytic cond. $0.424165$
Root an. cond. $0.807019$
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  − 2·7-s + 2·13-s − 16-s + 2·23-s − 25-s + 2·49-s + 2·53-s + 2·67-s − 2·83-s − 4·91-s − 2·103-s − 2·107-s + 2·112-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + 163-s + 167-s + 2·169-s + 173-s + 2·175-s + ⋯
L(s)  = 1  − 2·7-s + 2·13-s − 16-s + 2·23-s − 25-s + 2·49-s + 2·53-s + 2·67-s − 2·83-s − 4·91-s − 2·103-s − 2·107-s + 2·112-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + 163-s + 167-s + 2·169-s + 173-s + 2·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1703025\)    =    \(3^{4} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(0.424165\)
Root analytic conductor: \(0.807019\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1703025,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8537831502\)
\(L(\frac12)\) \(\approx\) \(0.8537831502\)
\(L(1)\) 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 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
29$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.838494888648135015458781081418, −9.656032733823675555912549513925, −9.321189407297538903634678514303, −8.713964683257326198593180607220, −8.650729164496488770198983932516, −8.235167077384079541482623192683, −7.32418807031249877466122374417, −7.20836192124090762125305406272, −6.54658043699249328741430300525, −6.53920060225715955635184696216, −5.99638669612119762934698929537, −5.51342830395093572867571532635, −5.15327193117871385703819638045, −4.31278416644434131834467802188, −3.76001498082527382282625424920, −3.72951469275092571050344583144, −2.83925183425261035585244694899, −2.72896600927028467486062128253, −1.70273985209171435836332150353, −0.823605612749546619287545682450, 0.823605612749546619287545682450, 1.70273985209171435836332150353, 2.72896600927028467486062128253, 2.83925183425261035585244694899, 3.72951469275092571050344583144, 3.76001498082527382282625424920, 4.31278416644434131834467802188, 5.15327193117871385703819638045, 5.51342830395093572867571532635, 5.99638669612119762934698929537, 6.53920060225715955635184696216, 6.54658043699249328741430300525, 7.20836192124090762125305406272, 7.32418807031249877466122374417, 8.235167077384079541482623192683, 8.650729164496488770198983932516, 8.713964683257326198593180607220, 9.321189407297538903634678514303, 9.656032733823675555912549513925, 9.838494888648135015458781081418

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