Properties

Label 4-1155200-1.1-c1e2-0-3
Degree $4$
Conductor $1155200$
Sign $-1$
Analytic cond. $73.6565$
Root an. cond. $2.92956$
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-s − 6·3-s + 4-s − 6·6-s + 8-s + 21·9-s − 8·11-s − 6·12-s + 16-s − 6·17-s + 21·18-s + 2·19-s − 8·22-s − 6·24-s + 25-s − 54·27-s + 32-s + 48·33-s − 6·34-s + 21·36-s + 2·38-s − 12·41-s + 12·43-s − 8·44-s − 6·48-s + 11·49-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s − 3.46·3-s + 1/2·4-s − 2.44·6-s + 0.353·8-s + 7·9-s − 2.41·11-s − 1.73·12-s + 1/4·16-s − 1.45·17-s + 4.94·18-s + 0.458·19-s − 1.70·22-s − 1.22·24-s + 1/5·25-s − 10.3·27-s + 0.176·32-s + 8.35·33-s − 1.02·34-s + 7/2·36-s + 0.324·38-s − 1.87·41-s + 1.82·43-s − 1.20·44-s − 0.866·48-s + 11/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1155200\)    =    \(2^{7} \cdot 5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(73.6565\)
Root analytic conductor: \(2.92956\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1155200,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−7.34314124395432920163711504829, −7.28045028948955453013969697676, −6.59012976667376581440253973920, −6.52004252877761961716316307615, −5.66768121415484817418669355380, −5.63644764861715146016439123215, −5.39131388312139597378979321025, −4.67101925823320154679184379858, −4.64308824493623454579381988034, −4.13748877232546049391346244336, −3.15015586052048758770880964700, −2.43028667706051710543744405368, −1.67148059123636456155322536163, −0.67099720957898753355831718421, 0, 0.67099720957898753355831718421, 1.67148059123636456155322536163, 2.43028667706051710543744405368, 3.15015586052048758770880964700, 4.13748877232546049391346244336, 4.64308824493623454579381988034, 4.67101925823320154679184379858, 5.39131388312139597378979321025, 5.63644764861715146016439123215, 5.66768121415484817418669355380, 6.52004252877761961716316307615, 6.59012976667376581440253973920, 7.28045028948955453013969697676, 7.34314124395432920163711504829

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