Properties

Label 4-630e2-1.1-c1e2-0-36
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
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  + 4-s − 2·5-s + 2·7-s + 16-s + 12·17-s − 2·20-s + 3·25-s + 2·28-s − 4·35-s + 16·37-s + 16·43-s − 3·49-s − 12·59-s + 64-s − 8·67-s + 12·68-s − 8·79-s − 2·80-s − 24·83-s − 24·85-s − 24·89-s + 3·100-s + 12·101-s + 4·109-s + 2·112-s + 24·119-s + 14·121-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s + 0.755·7-s + 1/4·16-s + 2.91·17-s − 0.447·20-s + 3/5·25-s + 0.377·28-s − 0.676·35-s + 2.63·37-s + 2.43·43-s − 3/7·49-s − 1.56·59-s + 1/8·64-s − 0.977·67-s + 1.45·68-s − 0.900·79-s − 0.223·80-s − 2.63·83-s − 2.60·85-s − 2.54·89-s + 3/10·100-s + 1.19·101-s + 0.383·109-s + 0.188·112-s + 2.20·119-s + 1.27·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.294137716\)
\(L(\frac12)\) \(\approx\) \(2.294137716\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
good11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.361683788765927099886703120446, −8.137806933221736892131373840887, −7.68072566687187284194521624333, −7.42257774366004622063287945506, −7.10097996037311265247610233993, −6.15095560601501333547012713952, −5.76306799226879953191493212190, −5.57449936126795434832243724178, −4.56211949870166416184694202500, −4.42900889702114434346086151421, −3.67917655074737520443823096900, −3.02077554855234761959138967953, −2.68907161640043859094582658785, −1.50258067725827938602043215208, −0.945958207187822378507835712582, 0.945958207187822378507835712582, 1.50258067725827938602043215208, 2.68907161640043859094582658785, 3.02077554855234761959138967953, 3.67917655074737520443823096900, 4.42900889702114434346086151421, 4.56211949870166416184694202500, 5.57449936126795434832243724178, 5.76306799226879953191493212190, 6.15095560601501333547012713952, 7.10097996037311265247610233993, 7.42257774366004622063287945506, 7.68072566687187284194521624333, 8.137806933221736892131373840887, 8.361683788765927099886703120446

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