Properties

Label 4-1400e2-1.1-c0e2-0-1
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $0.488169$
Root an. cond. $0.835877$
Motivic weight $0$
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 + 2·7-s + 8-s − 2·9-s − 2·14-s − 16-s + 2·18-s + 2·23-s − 2·46-s + 3·49-s + 2·56-s − 4·63-s + 64-s + 2·71-s − 2·72-s + 2·79-s + 3·81-s − 3·98-s − 2·112-s − 2·113-s − 121-s + 4·126-s + 127-s − 128-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2-s + 2·7-s + 8-s − 2·9-s − 2·14-s − 16-s + 2·18-s + 2·23-s − 2·46-s + 3·49-s + 2·56-s − 4·63-s + 64-s + 2·71-s − 2·72-s + 2·79-s + 3·81-s − 3·98-s − 2·112-s − 2·113-s − 121-s + 4·126-s + 127-s − 128-s + 131-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.488169\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6676389305\)
\(L(\frac12)\) \(\approx\) \(0.6676389305\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−9.779024569338101610645830766425, −9.420855792220089780829821102286, −8.974190698158375424781899021951, −8.752864014611837125206330724510, −8.374357162294601641240779888785, −8.175960017039092695013840151809, −7.59126366490503593951134228904, −7.51506998379007194065014130210, −6.80216194492556374663850481879, −6.36151742882763868532942070243, −5.74418343413207275652897321280, −5.14470689008525013489662826487, −5.06511350778806645121820480257, −4.77125087247963427399857683850, −3.90274245315059242989351925157, −3.52479085271814207905637985186, −2.53036363345507421042197738697, −2.43350354613194614869655619725, −1.49688086274837542164910117526, −0.890299391740152946017137073877, 0.890299391740152946017137073877, 1.49688086274837542164910117526, 2.43350354613194614869655619725, 2.53036363345507421042197738697, 3.52479085271814207905637985186, 3.90274245315059242989351925157, 4.77125087247963427399857683850, 5.06511350778806645121820480257, 5.14470689008525013489662826487, 5.74418343413207275652897321280, 6.36151742882763868532942070243, 6.80216194492556374663850481879, 7.51506998379007194065014130210, 7.59126366490503593951134228904, 8.175960017039092695013840151809, 8.374357162294601641240779888785, 8.752864014611837125206330724510, 8.974190698158375424781899021951, 9.420855792220089780829821102286, 9.779024569338101610645830766425

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