Properties

Label 4-21e2-1.1-c2e2-0-0
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  − 3·2-s − 3·3-s + 4·4-s + 9·5-s + 9·6-s + 13·7-s − 9·8-s + 6·9-s − 27·10-s − 15·11-s − 12·12-s − 39·14-s − 27·15-s + 27·16-s + 18·17-s − 18·18-s − 18·19-s + 36·20-s − 39·21-s + 45·22-s + 27·24-s + 29·25-s − 9·27-s + 52·28-s − 18·29-s + 81·30-s − 21·31-s + ⋯
L(s)  = 1  − 3/2·2-s − 3-s + 4-s + 9/5·5-s + 3/2·6-s + 13/7·7-s − 9/8·8-s + 2/3·9-s − 2.69·10-s − 1.36·11-s − 12-s − 2.78·14-s − 9/5·15-s + 1.68·16-s + 1.05·17-s − 18-s − 0.947·19-s + 9/5·20-s − 1.85·21-s + 2.04·22-s + 9/8·24-s + 1.15·25-s − 1/3·27-s + 13/7·28-s − 0.620·29-s + 2.69·30-s − 0.677·31-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

\(L(\frac{3}{2})\) \(\approx\) \(0.3922178941\)
\(L(\frac12)\) \(\approx\) \(0.3922178941\)
\(L(2)\) 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$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 - 13 T + p^{2} T^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 15 T + 104 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \)
17$C_2^2$ \( 1 - 18 T + 397 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2^2$ \( 1 + 18 T + 469 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - p^{2} T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 21 T + 1108 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \)
37$C_2^2$ \( 1 + 10 T - 1269 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 3254 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
53$C_2^2$ \( 1 + 33 T - 1720 T^{2} + 33 p^{2} T^{3} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 27 T + 3724 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 156 T + 11833 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 76 T + 1287 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 - 84 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 108 T + 9217 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2^2$ \( 1 - 43 T - 4392 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 + 505 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 + 126 T + 13213 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \)
97$C_2^2$ \( 1 + 15529 T^{2} + p^{4} T^{4} \)
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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

−18.01378780804988666403641028519, −17.80953954122890582278245943390, −17.18327266549577149604830218989, −17.01605716152957519814599177618, −16.20846091400625240615993077031, −15.05999612091091947481807743981, −14.73980320008974528417600094502, −13.78513331752079752861621520575, −13.03946601412904954172944165990, −12.21360208699840445746088239509, −11.31909615551257920893083697409, −10.76218975895576606865220093901, −9.964304215128232789363218962583, −9.737276605148152447619533187655, −8.420391863141385097673542379556, −8.117815582577185859987229594189, −6.78769112702779397449751604521, −5.44308394060920118053063766177, −5.38247025637035665847715214257, −1.87891378443402310107045640709, 1.87891378443402310107045640709, 5.38247025637035665847715214257, 5.44308394060920118053063766177, 6.78769112702779397449751604521, 8.117815582577185859987229594189, 8.420391863141385097673542379556, 9.737276605148152447619533187655, 9.964304215128232789363218962583, 10.76218975895576606865220093901, 11.31909615551257920893083697409, 12.21360208699840445746088239509, 13.03946601412904954172944165990, 13.78513331752079752861621520575, 14.73980320008974528417600094502, 15.05999612091091947481807743981, 16.20846091400625240615993077031, 17.01605716152957519814599177618, 17.18327266549577149604830218989, 17.80953954122890582278245943390, 18.01378780804988666403641028519

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