Properties

Label 4-1344e2-1.1-c3e2-0-15
Degree $4$
Conductor $1806336$
Sign $1$
Analytic cond. $6288.26$
Root an. cond. $8.90497$
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  + 6·3-s + 6·5-s − 14·7-s + 27·9-s − 26·11-s − 96·13-s + 36·15-s + 78·17-s − 40·19-s − 84·21-s + 22·23-s + 114·25-s + 108·27-s − 204·29-s + 96·31-s − 156·33-s − 84·35-s − 504·37-s − 576·39-s − 102·41-s − 296·43-s + 162·45-s − 780·47-s + 147·49-s + 468·51-s − 192·53-s − 156·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.536·5-s − 0.755·7-s + 9-s − 0.712·11-s − 2.04·13-s + 0.619·15-s + 1.11·17-s − 0.482·19-s − 0.872·21-s + 0.199·23-s + 0.911·25-s + 0.769·27-s − 1.30·29-s + 0.556·31-s − 0.822·33-s − 0.405·35-s − 2.23·37-s − 2.36·39-s − 0.388·41-s − 1.04·43-s + 0.536·45-s − 2.42·47-s + 3/7·49-s + 1.28·51-s − 0.497·53-s − 0.382·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6288.26\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1806336,\ (\ :3/2, 3/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 - p T )^{2} \)
7$C_1$ \( ( 1 + p T )^{2} \)
good5$D_{4}$ \( 1 - 6 T - 78 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 26 T + 2494 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 96 T + 5350 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 78 T + 8314 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 20 T + p^{3} T^{2} )^{2} \)
23$D_{4}$ \( 1 - 22 T + 16030 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 102 T + p^{3} T^{2} )^{2} \)
31$D_{4}$ \( 1 - 96 T - 24386 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 504 T + 163462 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 102 T + 99666 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 296 T + 94646 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 780 T + 326046 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 192 T + 197782 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 212 T + 355942 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 100 T + 370190 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 212 T + 611414 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 534 T + 746334 T^{2} + 534 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1128 T + 1062430 T^{2} - 1128 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 468 T - 92834 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 824 T + 536870 T^{2} - 824 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 2118 T + 2285746 T^{2} + 2118 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 400 T + 214046 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.179760163346610705098244943679, −8.611782201899371217610334746497, −8.101400152727565185947998120773, −8.047601140878086311509698412710, −7.31555490090488496098742203100, −7.15129402998006261671605592714, −6.64217499580888302623195553221, −6.37234713996897943516541989660, −5.39131027119174483239366483908, −5.34064764440222788612520558447, −4.86200205551248960588905930277, −4.36607747138974483316800695582, −3.50660441317413794846140365007, −3.30027792610210279411258461193, −2.86857453650217487667455940126, −2.30356980935062035399057459172, −1.86864537760876230238941328528, −1.27404936513904318896815550117, 0, 0, 1.27404936513904318896815550117, 1.86864537760876230238941328528, 2.30356980935062035399057459172, 2.86857453650217487667455940126, 3.30027792610210279411258461193, 3.50660441317413794846140365007, 4.36607747138974483316800695582, 4.86200205551248960588905930277, 5.34064764440222788612520558447, 5.39131027119174483239366483908, 6.37234713996897943516541989660, 6.64217499580888302623195553221, 7.15129402998006261671605592714, 7.31555490090488496098742203100, 8.047601140878086311509698412710, 8.101400152727565185947998120773, 8.611782201899371217610334746497, 9.179760163346610705098244943679

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