Properties

Label 4-1400e2-1.1-c0e2-0-6
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  + 2·2-s + 3·4-s − 2·7-s + 4·8-s − 4·14-s + 5·16-s − 6·28-s + 6·32-s + 3·49-s − 8·56-s + 7·64-s − 81-s + 6·98-s − 10·112-s − 4·113-s + 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 2·162-s + 163-s + 167-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s − 2·7-s + 4·8-s − 4·14-s + 5·16-s − 6·28-s + 6·32-s + 3·49-s − 8·56-s + 7·64-s − 81-s + 6·98-s − 10·112-s − 4·113-s + 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 2·162-s + 163-s + 167-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

\(L(\frac{1}{2})\) \(\approx\) \(3.466936218\)
\(L(\frac12)\) \(\approx\) \(3.466936218\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2^2$ \( 1 + T^{4} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
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

−10.10337105892282530438826706684, −9.832090625551487979729742969852, −9.128431405420742852408896083804, −8.913478514364769430029035608709, −8.076202761389319074767545581436, −7.80062367076982391498526349108, −7.21407151200468417538716643114, −6.87119116955904076246851068767, −6.61267764317405742780906068343, −6.20722443103545889364585251590, −5.73041708354120256504712037816, −5.53546840221130247644329135067, −4.89543848695057668836629153368, −4.40950238144763745485493038407, −3.82834728250147071235564423871, −3.65375259727954791325913932929, −3.00160949841360440982666450832, −2.72365004841739407044195635643, −2.15740706204371031633050772181, −1.24202895758901682203803852052, 1.24202895758901682203803852052, 2.15740706204371031633050772181, 2.72365004841739407044195635643, 3.00160949841360440982666450832, 3.65375259727954791325913932929, 3.82834728250147071235564423871, 4.40950238144763745485493038407, 4.89543848695057668836629153368, 5.53546840221130247644329135067, 5.73041708354120256504712037816, 6.20722443103545889364585251590, 6.61267764317405742780906068343, 6.87119116955904076246851068767, 7.21407151200468417538716643114, 7.80062367076982391498526349108, 8.076202761389319074767545581436, 8.913478514364769430029035608709, 9.128431405420742852408896083804, 9.832090625551487979729742969852, 10.10337105892282530438826706684

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