Properties

Label 4-231e2-1.1-c1e2-0-10
Degree $4$
Conductor $53361$
Sign $1$
Analytic cond. $3.40234$
Root an. cond. $1.35814$
Motivic weight $1$
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  + 2-s + 2·3-s − 2·4-s + 2·5-s + 2·6-s + 2·7-s − 3·8-s + 3·9-s + 2·10-s + 2·11-s − 4·12-s − 2·13-s + 2·14-s + 4·15-s + 16-s + 6·17-s + 3·18-s − 4·20-s + 4·21-s + 2·22-s − 2·23-s − 6·24-s − 7·25-s − 2·26-s + 4·27-s − 4·28-s + 10·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 4-s + 0.894·5-s + 0.816·6-s + 0.755·7-s − 1.06·8-s + 9-s + 0.632·10-s + 0.603·11-s − 1.15·12-s − 0.554·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 0.894·20-s + 0.872·21-s + 0.426·22-s − 0.417·23-s − 1.22·24-s − 7/5·25-s − 0.392·26-s + 0.769·27-s − 0.755·28-s + 1.85·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(53361\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3.40234\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{231} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 53361,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.780822134\)
\(L(\frac12)\) \(\approx\) \(2.780822134\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 24 T + 273 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 18 T + 207 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 238 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 18 T + 242 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−12.41423514810206826750858491646, −12.16162005777975070071729625160, −11.84397984568412955694849699834, −10.92398971985737103802843587805, −10.13712429767744275515780916615, −10.06551283195180995532021266754, −9.455778596234711352130152784998, −9.089557933844557850256785970912, −8.514660235077902213508523967115, −8.176539531658907131663210060913, −7.51605687488813425627016717498, −7.04800587197851484422507878598, −6.04744333367486623437160774081, −5.70314086406454250044583988535, −4.80904691565255811516487049391, −4.72873507257686043114874995132, −3.60969118437385846600239433648, −3.50835522546679541734792285367, −2.29149634956536874671352143062, −1.53195910092535386464608359172, 1.53195910092535386464608359172, 2.29149634956536874671352143062, 3.50835522546679541734792285367, 3.60969118437385846600239433648, 4.72873507257686043114874995132, 4.80904691565255811516487049391, 5.70314086406454250044583988535, 6.04744333367486623437160774081, 7.04800587197851484422507878598, 7.51605687488813425627016717498, 8.176539531658907131663210060913, 8.514660235077902213508523967115, 9.089557933844557850256785970912, 9.455778596234711352130152784998, 10.06551283195180995532021266754, 10.13712429767744275515780916615, 10.92398971985737103802843587805, 11.84397984568412955694849699834, 12.16162005777975070071729625160, 12.41423514810206826750858491646

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