Properties

Label 4-930e2-1.1-c1e2-0-18
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
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  + 2·2-s + 2·3-s + 3·4-s − 2·5-s + 4·6-s − 7-s + 4·8-s + 3·9-s − 4·10-s + 5·11-s + 6·12-s + 12·13-s − 2·14-s − 4·15-s + 5·16-s − 8·17-s + 6·18-s + 19-s − 6·20-s − 2·21-s + 10·22-s + 5·23-s + 8·24-s + 3·25-s + 24·26-s + 4·27-s − 3·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.377·7-s + 1.41·8-s + 9-s − 1.26·10-s + 1.50·11-s + 1.73·12-s + 3.32·13-s − 0.534·14-s − 1.03·15-s + 5/4·16-s − 1.94·17-s + 1.41·18-s + 0.229·19-s − 1.34·20-s − 0.436·21-s + 2.13·22-s + 1.04·23-s + 1.63·24-s + 3/5·25-s + 4.70·26-s + 0.769·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.717567198\)
\(L(\frac12)\) \(\approx\) \(8.717567198\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 5 T + 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 9 T + 90 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 92 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 17 T + 198 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 9 T + 150 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 5 T + 168 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
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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

−10.38493600913261989215416619688, −9.961760495444729164998456053953, −9.074285732843307171420637346452, −8.878026530850102149779363867399, −8.691787669625172274776889988903, −8.358868109108267568748612311827, −7.66018531301354046553157548275, −7.14162691728828851538160210390, −6.64973642707212625780897949381, −6.60988606686902975734274433061, −5.97098794366318272387691963461, −5.62931002771655339073818882117, −4.47451138677009031822657834858, −4.41488377043327344288914559931, −3.97913276916945141287673413797, −3.52424350148340957764605013347, −3.17058699423702372500460411658, −2.65989475046172098984043666958, −1.52002144829545026651106709851, −1.31343505123261870533423959538, 1.31343505123261870533423959538, 1.52002144829545026651106709851, 2.65989475046172098984043666958, 3.17058699423702372500460411658, 3.52424350148340957764605013347, 3.97913276916945141287673413797, 4.41488377043327344288914559931, 4.47451138677009031822657834858, 5.62931002771655339073818882117, 5.97098794366318272387691963461, 6.60988606686902975734274433061, 6.64973642707212625780897949381, 7.14162691728828851538160210390, 7.66018531301354046553157548275, 8.358868109108267568748612311827, 8.691787669625172274776889988903, 8.878026530850102149779363867399, 9.074285732843307171420637346452, 9.961760495444729164998456053953, 10.38493600913261989215416619688

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