Properties

Label 4-252e2-1.1-c1e2-0-3
Degree $4$
Conductor $63504$
Sign $1$
Analytic cond. $4.04907$
Root an. cond. $1.41853$
Motivic weight $1$
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  + 3·5-s − 4·7-s − 3·11-s + 4·13-s + 3·17-s + 19-s + 3·23-s + 5·25-s + 12·29-s + 7·31-s − 12·35-s + 37-s − 12·41-s − 8·43-s − 9·47-s + 9·49-s + 3·53-s − 9·55-s + 9·59-s + 61-s + 12·65-s + 7·67-s + 73-s + 12·77-s + 13·79-s − 24·83-s + 9·85-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.51·7-s − 0.904·11-s + 1.10·13-s + 0.727·17-s + 0.229·19-s + 0.625·23-s + 25-s + 2.22·29-s + 1.25·31-s − 2.02·35-s + 0.164·37-s − 1.87·41-s − 1.21·43-s − 1.31·47-s + 9/7·49-s + 0.412·53-s − 1.21·55-s + 1.17·59-s + 0.128·61-s + 1.48·65-s + 0.855·67-s + 0.117·73-s + 1.36·77-s + 1.46·79-s − 2.63·83-s + 0.976·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(63504\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4.04907\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{252} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 63504,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.584800867\)
\(L(\frac12)\) \(\approx\) \(1.584800867\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−12.18603632272705957033828963018, −12.05721099174022547234879819740, −11.24913716824266258475995230084, −10.70185343984639502147945569607, −10.03010945048315609642531373718, −10.00067533311313666691979033844, −9.749015658413658137623813785412, −8.896270539611800093161382839532, −8.337764788676738567030829354261, −8.211079863766860731141599490917, −7.02057016779190290075933797540, −6.69695841326881302583476853477, −6.30770230010870993556249534334, −5.74162173220900054190464728814, −5.19497382469695529042734941497, −4.60851187715082967625889052792, −3.35586292307381058452228809307, −3.16409515102708516884592284828, −2.32080880964095605130769555739, −1.09900860833751981997528731422, 1.09900860833751981997528731422, 2.32080880964095605130769555739, 3.16409515102708516884592284828, 3.35586292307381058452228809307, 4.60851187715082967625889052792, 5.19497382469695529042734941497, 5.74162173220900054190464728814, 6.30770230010870993556249534334, 6.69695841326881302583476853477, 7.02057016779190290075933797540, 8.211079863766860731141599490917, 8.337764788676738567030829354261, 8.896270539611800093161382839532, 9.749015658413658137623813785412, 10.00067533311313666691979033844, 10.03010945048315609642531373718, 10.70185343984639502147945569607, 11.24913716824266258475995230084, 12.05721099174022547234879819740, 12.18603632272705957033828963018

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