Properties

Label 4-882e2-1.1-c3e2-0-7
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $2708.12$
Root an. cond. $7.21385$
Motivic weight $3$
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 + 22·5-s + 8·8-s − 44·10-s + 26·11-s − 108·13-s − 16·16-s + 74·17-s − 116·19-s − 52·22-s − 58·23-s + 125·25-s + 216·26-s − 416·29-s + 252·31-s − 148·34-s − 50·37-s + 232·38-s + 176·40-s − 252·41-s + 328·43-s + 116·46-s − 444·47-s − 250·50-s + 12·53-s + 572·55-s + 832·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.96·5-s + 0.353·8-s − 1.39·10-s + 0.712·11-s − 2.30·13-s − 1/4·16-s + 1.05·17-s − 1.40·19-s − 0.503·22-s − 0.525·23-s + 25-s + 1.62·26-s − 2.66·29-s + 1.46·31-s − 0.746·34-s − 0.222·37-s + 0.990·38-s + 0.695·40-s − 0.959·41-s + 1.16·43-s + 0.371·46-s − 1.37·47-s − 0.707·50-s + 0.0311·53-s + 1.40·55-s + 1.88·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2708.12\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.073294981\)
\(L(\frac12)\) \(\approx\) \(1.073294981\)
\(L(\frac{5}{2})\) 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)$
bad2$C_2$ \( 1 + p T + p^{2} T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 22 T + 359 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 26 T - 655 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 54 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 74 T + 563 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 116 T + 6597 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 58 T - 8803 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 208 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 252 T + 33713 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 50 T - 48153 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 126 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 444 T + 93313 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 12 T - 148733 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 124 T - 190003 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 162 T - 200737 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 860 T + 438837 T^{2} - 860 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 238 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 p T - 69 p^{2} T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 984 T + 475217 T^{2} - 984 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 656 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 954 T + 205147 T^{2} + 954 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 526 T + p^{3} T^{2} )^{2} \)
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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

−9.843893248102421437300247120544, −9.773323800750509909203951255043, −9.166955206717308768430877443598, −8.971298422553563290256116191898, −8.367528446415071956087657778688, −7.67911354794549157461584547997, −7.65213484358953468906103494392, −6.97080313401990448693295940294, −6.43824423174904754658920166588, −6.16871000485566047389421838542, −5.63303940277525784098928761012, −5.08595911866038596763805950688, −4.94728763631310851907447983212, −4.04695018025439706339452306554, −3.65087541392379821884583597952, −2.66996326035878123240456937164, −2.12032342791897659795775838008, −1.98744051570252632130547861794, −1.26563670048215610476525433481, −0.29578349224155031315273062569, 0.29578349224155031315273062569, 1.26563670048215610476525433481, 1.98744051570252632130547861794, 2.12032342791897659795775838008, 2.66996326035878123240456937164, 3.65087541392379821884583597952, 4.04695018025439706339452306554, 4.94728763631310851907447983212, 5.08595911866038596763805950688, 5.63303940277525784098928761012, 6.16871000485566047389421838542, 6.43824423174904754658920166588, 6.97080313401990448693295940294, 7.65213484358953468906103494392, 7.67911354794549157461584547997, 8.367528446415071956087657778688, 8.971298422553563290256116191898, 9.166955206717308768430877443598, 9.773323800750509909203951255043, 9.843893248102421437300247120544

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