Properties

Label 4-201e2-1.1-c2e2-0-0
Degree $4$
Conductor $40401$
Sign $1$
Analytic cond. $29.9959$
Root an. cond. $2.34026$
Motivic weight $2$
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  − 6·3-s − 4·4-s − 2·7-s + 27·9-s + 24·12-s − 23·13-s − 26·19-s + 12·21-s + 50·25-s − 108·27-s + 8·28-s + 13·31-s − 108·36-s − 26·37-s + 138·39-s − 122·43-s + 49·49-s + 92·52-s + 156·57-s − 47·61-s − 54·63-s + 64·64-s − 109·67-s − 143·73-s − 300·75-s + 104·76-s − 131·79-s + ⋯
L(s)  = 1  − 2·3-s − 4-s − 2/7·7-s + 3·9-s + 2·12-s − 1.76·13-s − 1.36·19-s + 4/7·21-s + 2·25-s − 4·27-s + 2/7·28-s + 0.419·31-s − 3·36-s − 0.702·37-s + 3.53·39-s − 2.83·43-s + 49-s + 1.76·52-s + 2.73·57-s − 0.770·61-s − 6/7·63-s + 64-s − 1.62·67-s − 1.95·73-s − 4·75-s + 1.36·76-s − 1.65·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40401 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40401\)    =    \(3^{2} \cdot 67^{2}\)
Sign: $1$
Analytic conductor: \(29.9959\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40401,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1309819644\)
\(L(\frac12)\) \(\approx\) \(0.1309819644\)
\(L(2)\) 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)$
bad3$C_1$ \( ( 1 + p T )^{2} \)
67$C_2$ \( 1 + 109 T + p^{2} T^{2} \)
good2$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
5$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
7$C_2$ \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \)
11$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
13$C_2$ \( ( 1 + T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \)
17$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
19$C_2$ \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \)
23$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
29$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
31$C_2$ \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \)
37$C_2$ \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 73 T + p^{2} T^{2} ) \)
41$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
43$C_2$ \( ( 1 + 61 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2$ \( ( 1 - 74 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \)
71$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
73$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \)
79$C_2$ \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \)
83$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 169 T + p^{2} T^{2} ) \)
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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

−12.37989785180066436827984156223, −12.12590869541105093339618042409, −11.53511166745489115658758968433, −11.02598778836723771974596784903, −10.42346652910354195625259955906, −10.07237000929381844761074315513, −9.875216597608155323357458796976, −8.942703347217719822379619944812, −8.732509847723348679328551233261, −7.77138383248080587331601140001, −7.05300930626831523457829107869, −6.80832184686532910289846919800, −6.26222463153244787473063895177, −5.40100923865472497282781521346, −5.06127290966401478386616742807, −4.50301043024326037944885782471, −4.20568913151062147393486670261, −2.90356805260228794966888239561, −1.59432626674497962336218884986, −0.23989960814668240457223262795, 0.23989960814668240457223262795, 1.59432626674497962336218884986, 2.90356805260228794966888239561, 4.20568913151062147393486670261, 4.50301043024326037944885782471, 5.06127290966401478386616742807, 5.40100923865472497282781521346, 6.26222463153244787473063895177, 6.80832184686532910289846919800, 7.05300930626831523457829107869, 7.77138383248080587331601140001, 8.732509847723348679328551233261, 8.942703347217719822379619944812, 9.875216597608155323357458796976, 10.07237000929381844761074315513, 10.42346652910354195625259955906, 11.02598778836723771974596784903, 11.53511166745489115658758968433, 12.12590869541105093339618042409, 12.37989785180066436827984156223

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