Properties

Label 4-930e2-1.1-c1e2-0-14
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 3·4-s + 5-s + 2·6-s + 3·7-s + 4·8-s + 2·10-s − 5·11-s + 3·12-s − 6·13-s + 6·14-s + 15-s + 5·16-s − 8·17-s + 4·19-s + 3·20-s + 3·21-s − 10·22-s + 16·23-s + 4·24-s − 12·26-s − 27-s + 9·28-s − 2·29-s + 2·30-s + 7·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.447·5-s + 0.816·6-s + 1.13·7-s + 1.41·8-s + 0.632·10-s − 1.50·11-s + 0.866·12-s − 1.66·13-s + 1.60·14-s + 0.258·15-s + 5/4·16-s − 1.94·17-s + 0.917·19-s + 0.670·20-s + 0.654·21-s − 2.13·22-s + 3.33·23-s + 0.816·24-s − 2.35·26-s − 0.192·27-s + 1.70·28-s − 0.371·29-s + 0.365·30-s + 1.25·31-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: induced by $\chi_{930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.197709179\)
\(L(\frac12)\) \(\approx\) \(7.197709179\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
31$C_2$ \( 1 - 7 T + p T^{2} \)
good7$C_2^2$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
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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

−10.30478829125647750313917427193, −10.12007160013701459727914195549, −9.347403971485160717193803877670, −8.844716599488246756391889802239, −8.824277912017998811290746832557, −7.979285671058630070901909676871, −7.50807883207395849046759655625, −7.36984083209812424480092532595, −6.93026350648617093171363735046, −6.44200231245006661185708823773, −5.52674926280873650344152585339, −5.46959263109360948546350358763, −4.89841119727973533320497935899, −4.61181462844985176584149588873, −4.32394364756848824847662722274, −3.30873572242856530579676389920, −2.63069543971932632781557599185, −2.62841328024383300917228078445, −2.08174974302417422411168557278, −0.996437145237133674360229016805, 0.996437145237133674360229016805, 2.08174974302417422411168557278, 2.62841328024383300917228078445, 2.63069543971932632781557599185, 3.30873572242856530579676389920, 4.32394364756848824847662722274, 4.61181462844985176584149588873, 4.89841119727973533320497935899, 5.46959263109360948546350358763, 5.52674926280873650344152585339, 6.44200231245006661185708823773, 6.93026350648617093171363735046, 7.36984083209812424480092532595, 7.50807883207395849046759655625, 7.979285671058630070901909676871, 8.824277912017998811290746832557, 8.844716599488246756391889802239, 9.347403971485160717193803877670, 10.12007160013701459727914195549, 10.30478829125647750313917427193

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