Properties

Label 4-930e2-1.1-c1e2-0-7
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4-s − 8·7-s + 9-s − 2·12-s − 8·13-s + 16-s − 8·19-s + 16·21-s + 25-s + 4·27-s − 8·28-s + 2·31-s + 36-s + 16·37-s + 16·39-s − 20·43-s − 2·48-s + 34·49-s − 8·52-s + 16·57-s + 28·61-s − 8·63-s + 64-s + 16·67-s − 8·73-s − 2·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 3.02·7-s + 1/3·9-s − 0.577·12-s − 2.21·13-s + 1/4·16-s − 1.83·19-s + 3.49·21-s + 1/5·25-s + 0.769·27-s − 1.51·28-s + 0.359·31-s + 1/6·36-s + 2.63·37-s + 2.56·39-s − 3.04·43-s − 0.288·48-s + 34/7·49-s − 1.10·52-s + 2.11·57-s + 3.58·61-s − 1.00·63-s + 1/8·64-s + 1.95·67-s − 0.936·73-s − 0.230·75-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: \(1\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−8.140939386265747873328067569806, −7.13014434963814596481540225137, −6.77097595153689876646605531178, −6.72583461739189111327324544278, −6.33719049508577830282130975872, −5.93653080258964708892650187466, −5.21277269593248482625544714347, −5.06637053921005043111949188088, −4.11633325616548011845287467908, −3.86181451968313816241216267886, −2.87299403465911065799617206252, −2.76898105470486532245117845774, −2.16790461724177153358802712730, −0.60676761346407823140058390188, 0, 0.60676761346407823140058390188, 2.16790461724177153358802712730, 2.76898105470486532245117845774, 2.87299403465911065799617206252, 3.86181451968313816241216267886, 4.11633325616548011845287467908, 5.06637053921005043111949188088, 5.21277269593248482625544714347, 5.93653080258964708892650187466, 6.33719049508577830282130975872, 6.72583461739189111327324544278, 6.77097595153689876646605531178, 7.13014434963814596481540225137, 8.140939386265747873328067569806

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