Properties

Label 4-21e2-1.1-c2e2-0-2
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $0.327422$
Root an. cond. $0.756444$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 4·4-s − 13·7-s − 12·12-s + 46·13-s − 11·19-s − 39·21-s − 25·25-s − 27·27-s + 52·28-s + 13·31-s + 73·37-s + 138·39-s − 122·43-s + 120·49-s − 184·52-s − 33·57-s − 74·61-s + 64·64-s + 13·67-s + 97·73-s − 75·75-s + 44·76-s − 11·79-s − 81·81-s + 156·84-s − 598·91-s + ⋯
L(s)  = 1  + 3-s − 4-s − 1.85·7-s − 12-s + 3.53·13-s − 0.578·19-s − 1.85·21-s − 25-s − 27-s + 13/7·28-s + 0.419·31-s + 1.97·37-s + 3.53·39-s − 2.83·43-s + 2.44·49-s − 3.53·52-s − 0.578·57-s − 1.21·61-s + 64-s + 0.194·67-s + 1.32·73-s − 75-s + 0.578·76-s − 0.139·79-s − 81-s + 13/7·84-s − 6.57·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.327422\)
Root analytic conductor: \(0.756444\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7789239932\)
\(L(\frac12)\) \(\approx\) \(0.7789239932\)
\(L(2)\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
7$C_2$ \( 1 + 13 T + p^{2} T^{2} \)
good2$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
5$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
11$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
13$C_2$ \( ( 1 - 23 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
19$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \)
23$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
31$C_2$ \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \)
37$C_2$ \( ( 1 - 47 T + p^{2} T^{2} )( 1 - 26 T + p^{2} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
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$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
59$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
61$C_2$ \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \)
67$C_2$ \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 109 T + p^{2} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 - 143 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \)
79$C_2$ \( ( 1 - 131 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
89$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2} \)
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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

−18.22591168197302180246445512932, −18.18469278944139048887368297192, −16.84208441965893813213973103800, −16.41480989137551148618680286450, −15.53367582927521313680246387256, −15.42721410848876837977762970691, −14.19262124482623320311386052803, −13.54581719826265743074080423457, −13.25824231990411862652613133590, −13.05608419011204828296859916744, −11.67230751947332469346862091920, −10.85867357147380919384377244288, −9.853380481351000633855114089440, −9.252588891088115612641962602212, −8.616907772579954865039179977931, −8.157996158947256526575565289898, −6.45358831043530219069890543980, −6.00577066649538924854715699344, −3.95940027798559613597211892625, −3.36578704399175758378478904088, 3.36578704399175758378478904088, 3.95940027798559613597211892625, 6.00577066649538924854715699344, 6.45358831043530219069890543980, 8.157996158947256526575565289898, 8.616907772579954865039179977931, 9.252588891088115612641962602212, 9.853380481351000633855114089440, 10.85867357147380919384377244288, 11.67230751947332469346862091920, 13.05608419011204828296859916744, 13.25824231990411862652613133590, 13.54581719826265743074080423457, 14.19262124482623320311386052803, 15.42721410848876837977762970691, 15.53367582927521313680246387256, 16.41480989137551148618680286450, 16.84208441965893813213973103800, 18.18469278944139048887368297192, 18.22591168197302180246445512932

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