Properties

Label 4-84e3-1.1-c1e2-0-17
Degree $4$
Conductor $592704$
Sign $1$
Analytic cond. $37.7913$
Root an. cond. $2.47940$
Motivic weight $1$
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-s − 3-s − 4-s − 6-s − 7-s − 3·8-s + 9-s − 8·11-s + 12-s + 4·13-s − 14-s − 16-s − 12·17-s + 18-s − 8·19-s + 21-s − 8·22-s + 3·24-s − 6·25-s + 4·26-s − 27-s + 28-s + 4·29-s + 5·32-s + 8·33-s − 12·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 2.41·11-s + 0.288·12-s + 1.10·13-s − 0.267·14-s − 1/4·16-s − 2.91·17-s + 0.235·18-s − 1.83·19-s + 0.218·21-s − 1.70·22-s + 0.612·24-s − 6/5·25-s + 0.784·26-s − 0.192·27-s + 0.188·28-s + 0.742·29-s + 0.883·32-s + 1.39·33-s − 2.05·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(592704\)    =    \(2^{6} \cdot 3^{3} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(37.7913\)
Root analytic conductor: \(2.47940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 592704,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 + T \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.123916451859490481483222873179, −7.48171223884197946068743795147, −6.81422856027217448872168795499, −6.47604188152629865155494776849, −6.00140300914464627775014931904, −5.73267711633132091527857634664, −5.08548839023881336508605977410, −4.61914309368534933978095799917, −4.19315915887555872024919481174, −3.94328917656394473634657471263, −2.78781035361267366934893473837, −2.64839746138688614832708523229, −1.80984559120091606425042866771, 0, 0, 1.80984559120091606425042866771, 2.64839746138688614832708523229, 2.78781035361267366934893473837, 3.94328917656394473634657471263, 4.19315915887555872024919481174, 4.61914309368534933978095799917, 5.08548839023881336508605977410, 5.73267711633132091527857634664, 6.00140300914464627775014931904, 6.47604188152629865155494776849, 6.81422856027217448872168795499, 7.48171223884197946068743795147, 8.123916451859490481483222873179

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