Properties

Label 4-585e2-1.1-c1e2-0-9
Degree $4$
Conductor $342225$
Sign $1$
Analytic cond. $21.8205$
Root an. cond. $2.16130$
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  + 2-s + 2·4-s + 5-s − 2·7-s + 5·8-s − 3·9-s + 10-s + 11-s + 13-s − 2·14-s + 5·16-s + 4·17-s − 3·18-s − 6·19-s + 2·20-s + 22-s + 26-s − 4·28-s − 5·29-s + 31-s + 10·32-s + 4·34-s − 2·35-s − 6·36-s − 10·37-s − 6·38-s + 5·40-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s + 0.447·5-s − 0.755·7-s + 1.76·8-s − 9-s + 0.316·10-s + 0.301·11-s + 0.277·13-s − 0.534·14-s + 5/4·16-s + 0.970·17-s − 0.707·18-s − 1.37·19-s + 0.447·20-s + 0.213·22-s + 0.196·26-s − 0.755·28-s − 0.928·29-s + 0.179·31-s + 1.76·32-s + 0.685·34-s − 0.338·35-s − 36-s − 1.64·37-s − 0.973·38-s + 0.790·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(342225\)    =    \(3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(21.8205\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{585} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 342225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.463508629\)
\(L(\frac12)\) \(\approx\) \(3.463508629\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + p T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
13$C_2$ \( 1 - T + T^{2} \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 10 T + 17 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−10.73630668251011404097820302396, −10.51811195697517297734044584970, −10.39112812815007298568815751325, −9.677481595227111620669874779139, −9.204035616814526049861858735042, −8.556570091677377216414156840985, −8.460876341539892122819138262147, −7.56466385163742694478419648572, −7.30949724160604065071813470644, −6.84334040578923709037252472116, −6.17574306498307210053639030406, −6.06711973469745507013148682859, −5.26612833496443986194770641292, −5.19545965994405052571672526068, −3.98285346276636788456939161534, −3.97057079452193291999780406948, −3.22852679528122311446459928508, −2.26531065152294420070403497531, −2.21732948289237067147445627025, −0.969386349372659091894649488479, 0.969386349372659091894649488479, 2.21732948289237067147445627025, 2.26531065152294420070403497531, 3.22852679528122311446459928508, 3.97057079452193291999780406948, 3.98285346276636788456939161534, 5.19545965994405052571672526068, 5.26612833496443986194770641292, 6.06711973469745507013148682859, 6.17574306498307210053639030406, 6.84334040578923709037252472116, 7.30949724160604065071813470644, 7.56466385163742694478419648572, 8.460876341539892122819138262147, 8.556570091677377216414156840985, 9.204035616814526049861858735042, 9.677481595227111620669874779139, 10.39112812815007298568815751325, 10.51811195697517297734044584970, 10.73630668251011404097820302396

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