Properties

Label 4-570e2-1.1-c1e2-0-7
Degree $4$
Conductor $324900$
Sign $1$
Analytic cond. $20.7159$
Root an. cond. $2.13341$
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 − 3-s + 5-s − 6-s − 2·7-s − 8-s + 10-s + 12·11-s − 5·13-s − 2·14-s − 15-s − 16-s + 8·19-s + 2·21-s + 12·22-s − 6·23-s + 24-s − 5·26-s + 27-s − 6·29-s − 30-s + 10·31-s − 12·33-s − 2·35-s + 22·37-s + 8·38-s + 5·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 3.61·11-s − 1.38·13-s − 0.534·14-s − 0.258·15-s − 1/4·16-s + 1.83·19-s + 0.436·21-s + 2.55·22-s − 1.25·23-s + 0.204·24-s − 0.980·26-s + 0.192·27-s − 1.11·29-s − 0.182·30-s + 1.79·31-s − 2.08·33-s − 0.338·35-s + 3.61·37-s + 1.29·38-s + 0.800·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(20.7159\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.379293172\)
\(L(\frac12)\) \(\approx\) \(2.379293172\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−11.36982033637894669791468707796, −10.43208155591510905407402733024, −9.743021234390817129009060569428, −9.659254930327780879183849864916, −9.395858286633943689357190397993, −9.188891425167415939346695417418, −8.044637873633085113552142149137, −8.025325438266765972930577397627, −7.00940691442395959487259347976, −6.59633765986081560389838601878, −6.56773307320166279813348456709, −5.80967610718789759363736432246, −5.67636816256507058361596928467, −4.73158996731551982760756152523, −4.40412073668219423240403247846, −3.87765881284319077037800196526, −3.34580520528684081280523554154, −2.68301985498319999005389990464, −1.66617431971586454607601298688, −0.891909592953534315446614522707, 0.891909592953534315446614522707, 1.66617431971586454607601298688, 2.68301985498319999005389990464, 3.34580520528684081280523554154, 3.87765881284319077037800196526, 4.40412073668219423240403247846, 4.73158996731551982760756152523, 5.67636816256507058361596928467, 5.80967610718789759363736432246, 6.56773307320166279813348456709, 6.59633765986081560389838601878, 7.00940691442395959487259347976, 8.025325438266765972930577397627, 8.044637873633085113552142149137, 9.188891425167415939346695417418, 9.395858286633943689357190397993, 9.659254930327780879183849864916, 9.743021234390817129009060569428, 10.43208155591510905407402733024, 11.36982033637894669791468707796

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