Properties

Label 4-627200-1.1-c1e2-0-8
Degree $4$
Conductor $627200$
Sign $-1$
Analytic cond. $39.9908$
Root an. cond. $2.51472$
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 − 3·9-s − 10·11-s + 6·17-s − 12·19-s + 25-s + 14·27-s + 20·33-s + 16·41-s + 12·43-s + 49-s − 12·51-s + 24·57-s + 16·59-s − 8·67-s + 20·73-s − 2·75-s − 4·81-s − 24·83-s − 32·89-s + 14·97-s + 30·99-s + 4·107-s − 12·113-s + 53·121-s − 32·123-s + 127-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s − 3.01·11-s + 1.45·17-s − 2.75·19-s + 1/5·25-s + 2.69·27-s + 3.48·33-s + 2.49·41-s + 1.82·43-s + 1/7·49-s − 1.68·51-s + 3.17·57-s + 2.08·59-s − 0.977·67-s + 2.34·73-s − 0.230·75-s − 4/9·81-s − 2.63·83-s − 3.39·89-s + 1.42·97-s + 3.01·99-s + 0.386·107-s − 1.12·113-s + 4.81·121-s − 2.88·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(627200\)    =    \(2^{9} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(39.9908\)
Root analytic conductor: \(2.51472\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{627200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 627200,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−8.321037332106296729315848288758, −7.61068895852562797070369378134, −7.47320461628172485062800558626, −6.62516017863013169349321112564, −6.17106315121314565062659046983, −5.71012569325521424791956643699, −5.45734437575009036836059962399, −5.22465963352202663879592762973, −4.44994957570517170152202331319, −4.06182673812785905054132006247, −2.98004661899199231115437847100, −2.60467600555756229149551481858, −2.31436414126049041586401831975, −0.73530382675733263946372082289, 0, 0.73530382675733263946372082289, 2.31436414126049041586401831975, 2.60467600555756229149551481858, 2.98004661899199231115437847100, 4.06182673812785905054132006247, 4.44994957570517170152202331319, 5.22465963352202663879592762973, 5.45734437575009036836059962399, 5.71012569325521424791956643699, 6.17106315121314565062659046983, 6.62516017863013169349321112564, 7.47320461628172485062800558626, 7.61068895852562797070369378134, 8.321037332106296729315848288758

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