Properties

Label 4-21e4-1.1-c0e2-0-0
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $0.0484385$
Root an. cond. $0.469135$
Motivic weight $0$
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  − 16-s + 2·25-s − 4·67-s + 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 16-s + 2·25-s − 4·67-s + 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.0484385\)
Root analytic conductor: \(0.469135\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7081386246\)
\(L(\frac12)\) \(\approx\) \(0.7081386246\)
\(L(1)\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 + T^{4} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$ \( ( 1 + T )^{4} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_1$ \( ( 1 - T )^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−11.75768382820734119807401977288, −10.96857346074114353539039441239, −10.58591257591691448054745128864, −10.51651999613889885513362405791, −9.637006426297288684903192565796, −9.329485489699679464914913646392, −8.737212900068718365040170104805, −8.669173095535633182753122905473, −7.82066734516852205356545624700, −7.44273120123202670099339716443, −6.96600112572225974598710138211, −6.28449274811526203539564201486, −6.21468501338760857365001128365, −5.07896495965005685939838322715, −5.01233410038919233840908818842, −4.31225387507632891502908836307, −3.64577065757865264279999126722, −2.91671596863141148818752592879, −2.36438985690773399053883083814, −1.32837335488096312434276989129, 1.32837335488096312434276989129, 2.36438985690773399053883083814, 2.91671596863141148818752592879, 3.64577065757865264279999126722, 4.31225387507632891502908836307, 5.01233410038919233840908818842, 5.07896495965005685939838322715, 6.21468501338760857365001128365, 6.28449274811526203539564201486, 6.96600112572225974598710138211, 7.44273120123202670099339716443, 7.82066734516852205356545624700, 8.669173095535633182753122905473, 8.737212900068718365040170104805, 9.329485489699679464914913646392, 9.637006426297288684903192565796, 10.51651999613889885513362405791, 10.58591257591691448054745128864, 10.96857346074114353539039441239, 11.75768382820734119807401977288

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