Properties

Label 4-1400e2-1.1-c1e2-0-8
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $124.971$
Root an. cond. $3.34350$
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  + 2·2-s + 2·3-s + 2·4-s + 4·6-s − 3·9-s − 6·11-s + 4·12-s − 4·16-s + 14·17-s − 6·18-s − 12·22-s − 14·27-s − 8·32-s − 12·33-s + 28·34-s − 6·36-s + 4·41-s − 8·43-s − 12·44-s − 8·48-s + 49-s + 28·51-s − 28·54-s + 20·59-s − 8·64-s − 24·66-s + 4·67-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s − 9-s − 1.80·11-s + 1.15·12-s − 16-s + 3.39·17-s − 1.41·18-s − 2.55·22-s − 2.69·27-s − 1.41·32-s − 2.08·33-s + 4.80·34-s − 36-s + 0.624·41-s − 1.21·43-s − 1.80·44-s − 1.15·48-s + 1/7·49-s + 3.92·51-s − 3.81·54-s + 2.60·59-s − 64-s − 2.95·66-s + 0.488·67-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(124.971\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.075790943\)
\(L(\frac12)\) \(\approx\) \(5.075790943\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−7.83592762513447774723326132090, −7.38451703018051703490900915946, −7.05100957889080392304434332168, −6.10912051523666843643835490963, −5.93898740434763322465740513982, −5.45922076877847354484924950710, −5.14936161735373762545181523253, −5.01570061778941535708309486660, −3.95749147690713051772963085752, −3.57752582126466520607951640819, −3.30255970627597751194469732923, −2.76472422088648867371864611007, −2.57959600757021446327298173729, −1.87226019490551330647832359751, −0.64173349430317986722987428715, 0.64173349430317986722987428715, 1.87226019490551330647832359751, 2.57959600757021446327298173729, 2.76472422088648867371864611007, 3.30255970627597751194469732923, 3.57752582126466520607951640819, 3.95749147690713051772963085752, 5.01570061778941535708309486660, 5.14936161735373762545181523253, 5.45922076877847354484924950710, 5.93898740434763322465740513982, 6.10912051523666843643835490963, 7.05100957889080392304434332168, 7.38451703018051703490900915946, 7.83592762513447774723326132090

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