Properties

Label 4-21e2-1.1-c4e2-0-1
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $4.71223$
Root an. cond. $1.47335$
Motivic weight $4$
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  − 2·2-s + 9·3-s + 16·4-s + 18·5-s − 18·6-s + 77·7-s − 88·8-s + 54·9-s − 36·10-s − 194·11-s + 144·12-s − 154·14-s + 162·15-s + 176·16-s − 420·17-s − 108·18-s + 453·19-s + 288·20-s + 693·21-s + 388·22-s + 112·23-s − 792·24-s − 409·25-s + 243·27-s + 1.23e3·28-s + 2.08e3·29-s − 324·30-s + ⋯
L(s)  = 1  − 1/2·2-s + 3-s + 4-s + 0.719·5-s − 1/2·6-s + 11/7·7-s − 1.37·8-s + 2/3·9-s − 0.359·10-s − 1.60·11-s + 12-s − 0.785·14-s + 0.719·15-s + 0.687·16-s − 1.45·17-s − 1/3·18-s + 1.25·19-s + 0.719·20-s + 11/7·21-s + 0.801·22-s + 0.211·23-s − 1.37·24-s − 0.654·25-s + 1/3·27-s + 11/7·28-s + 2.47·29-s − 0.359·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4.71223\)
Root analytic conductor: \(1.47335\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.051814526\)
\(L(\frac12)\) \(\approx\) \(2.051814526\)
\(L(3)\) 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$C_2$ \( 1 - p^{2} T + p^{3} T^{2} \)
7$C_2$ \( 1 - 11 p T + p^{4} T^{2} \)
good2$C_2^2$ \( 1 + p T - 3 p^{2} T^{2} + p^{5} T^{3} + p^{8} T^{4} \)
5$C_2^2$ \( 1 - 18 T + 733 T^{2} - 18 p^{4} T^{3} + p^{8} T^{4} \)
11$C_2^2$ \( 1 + 194 T + 22995 T^{2} + 194 p^{4} T^{3} + p^{8} T^{4} \)
13$C_2^2$ \( 1 - 30047 T^{2} + p^{8} T^{4} \)
17$C_2^2$ \( 1 + 420 T + 142321 T^{2} + 420 p^{4} T^{3} + p^{8} T^{4} \)
19$C_2^2$ \( 1 - 453 T + 198724 T^{2} - 453 p^{4} T^{3} + p^{8} T^{4} \)
23$C_2^2$ \( 1 - 112 T - 267297 T^{2} - 112 p^{4} T^{3} + p^{8} T^{4} \)
29$C_2$ \( ( 1 - 1040 T + p^{4} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2019 T + 2282308 T^{2} + 2019 p^{4} T^{3} + p^{8} T^{4} \)
37$C_2^2$ \( 1 - 1075 T - 718536 T^{2} - 1075 p^{4} T^{3} + p^{8} T^{4} \)
41$C_2^2$ \( 1 - 3945974 T^{2} + p^{8} T^{4} \)
43$C_2$ \( ( 1 + 1087 T + p^{4} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 3750 T + 9567181 T^{2} - 3750 p^{4} T^{3} + p^{8} T^{4} \)
53$C_2^2$ \( 1 - 2200 T - 3050481 T^{2} - 2200 p^{4} T^{3} + p^{8} T^{4} \)
59$C_2^2$ \( 1 + 9264 T + 40724593 T^{2} + 9264 p^{4} T^{3} + p^{8} T^{4} \)
61$C_2^2$ \( 1 - 1212 T + 14335489 T^{2} - 1212 p^{4} T^{3} + p^{8} T^{4} \)
67$C_2^2$ \( 1 + 2375 T - 14510496 T^{2} + 2375 p^{4} T^{3} + p^{8} T^{4} \)
71$C_2$ \( ( 1 + 8938 T + p^{4} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 15807 T + 111685324 T^{2} - 15807 p^{4} T^{3} + p^{8} T^{4} \)
79$C_2^2$ \( 1 + 8147 T + 27423528 T^{2} + 8147 p^{4} T^{3} + p^{8} T^{4} \)
83$C_2^2$ \( 1 - 50356694 T^{2} + p^{8} T^{4} \)
89$C_2^2$ \( 1 - 23628 T + 248836369 T^{2} - 23628 p^{4} T^{3} + p^{8} T^{4} \)
97$C_2^2$ \( 1 - 164817362 T^{2} + p^{8} T^{4} \)
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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

−17.78340213118514644992673709190, −17.46974214424617603279787188109, −16.20309404186310495517491317802, −15.77343176117851996270227029567, −15.16541431360247892732185662370, −14.71232519218962986372349419454, −13.75806710007529918113110688400, −13.50148474438913402447037482135, −12.38183352896072947858929958054, −11.62602351913222686438257381884, −10.86756305348377399011198434557, −10.32117975401905520903699440027, −9.240818625419382141274034358616, −8.687032698455008374623286530672, −7.83594857995684727959195736051, −7.23153127298501343977157287942, −5.93617298363017033538052244961, −4.86556202543996125709901766866, −2.81921384214715572436655270343, −1.97587835613929894864156584210, 1.97587835613929894864156584210, 2.81921384214715572436655270343, 4.86556202543996125709901766866, 5.93617298363017033538052244961, 7.23153127298501343977157287942, 7.83594857995684727959195736051, 8.687032698455008374623286530672, 9.240818625419382141274034358616, 10.32117975401905520903699440027, 10.86756305348377399011198434557, 11.62602351913222686438257381884, 12.38183352896072947858929958054, 13.50148474438913402447037482135, 13.75806710007529918113110688400, 14.71232519218962986372349419454, 15.16541431360247892732185662370, 15.77343176117851996270227029567, 16.20309404186310495517491317802, 17.46974214424617603279787188109, 17.78340213118514644992673709190

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