Properties

Label 4-882e2-1.1-c3e2-0-36
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·5-s − 8·8-s + 4·10-s − 8·11-s − 84·13-s − 16·16-s − 2·17-s + 124·19-s − 16·22-s + 76·23-s + 125·25-s − 168·26-s − 508·29-s + 72·31-s − 4·34-s − 398·37-s + 248·38-s − 16·40-s − 924·41-s + 424·43-s + 152·46-s − 264·47-s + 250·50-s − 162·53-s − 16·55-s − 1.01e3·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.178·5-s − 0.353·8-s + 0.126·10-s − 0.219·11-s − 1.79·13-s − 1/4·16-s − 0.0285·17-s + 1.49·19-s − 0.155·22-s + 0.689·23-s + 25-s − 1.26·26-s − 3.25·29-s + 0.417·31-s − 0.0201·34-s − 1.76·37-s + 1.05·38-s − 0.0632·40-s − 3.51·41-s + 1.50·43-s + 0.487·46-s − 0.819·47-s + 0.707·50-s − 0.419·53-s − 0.0392·55-s − 2.30·58-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: \(2\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 2 T - 121 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 8 T - 1267 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 42 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 4909 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 124 T + 8517 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 76 T - 6391 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 254 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 72 T - 24607 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 398 T + 107751 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 462 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 212 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 264 T - 34127 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 162 T - 122633 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 772 T + 390605 T^{2} + 772 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 30 T - 226081 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 764 T + 282933 T^{2} - 764 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 236 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 418 T - 214293 T^{2} + 418 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 552 T - 188335 T^{2} + 552 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 1036 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 30 T - 704069 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 1190 T + p^{3} T^{2} )^{2} \)
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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

−9.492125427186425551819928777797, −9.348059075884290558662044359225, −8.532536226072913250494392903732, −8.516685608785929237765059703184, −7.54268344911022424217676066981, −7.36567973164797967155498064495, −7.06702745227632497344772886662, −6.58568620512480671500002888223, −5.86329474625193205020151169480, −5.25050797376140655598605551675, −5.23424739091545880754349025379, −4.86326167096767896252400760059, −4.13769172186625743973252755457, −3.48804790107523701954512735509, −3.15316868092504720130332980679, −2.62852374635398019471148225153, −1.83686934965998388344058772584, −1.36187223808993675589383215702, 0, 0, 1.36187223808993675589383215702, 1.83686934965998388344058772584, 2.62852374635398019471148225153, 3.15316868092504720130332980679, 3.48804790107523701954512735509, 4.13769172186625743973252755457, 4.86326167096767896252400760059, 5.23424739091545880754349025379, 5.25050797376140655598605551675, 5.86329474625193205020151169480, 6.58568620512480671500002888223, 7.06702745227632497344772886662, 7.36567973164797967155498064495, 7.54268344911022424217676066981, 8.516685608785929237765059703184, 8.532536226072913250494392903732, 9.348059075884290558662044359225, 9.492125427186425551819928777797

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