Properties

Label 4-40e4-1.1-c1e2-0-2
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
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  + 2·3-s − 2·7-s + 2·9-s − 8·11-s − 6·13-s + 6·17-s − 4·21-s − 6·23-s + 6·27-s − 4·29-s − 16·33-s − 6·37-s − 12·39-s + 12·41-s − 6·43-s − 18·47-s + 2·49-s + 12·51-s + 10·53-s − 4·63-s − 18·67-s − 12·69-s − 10·73-s + 16·77-s + 11·81-s − 6·83-s − 8·87-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 2/3·9-s − 2.41·11-s − 1.66·13-s + 1.45·17-s − 0.872·21-s − 1.25·23-s + 1.15·27-s − 0.742·29-s − 2.78·33-s − 0.986·37-s − 1.92·39-s + 1.87·41-s − 0.914·43-s − 2.62·47-s + 2/7·49-s + 1.68·51-s + 1.37·53-s − 0.503·63-s − 2.19·67-s − 1.44·69-s − 1.17·73-s + 1.82·77-s + 11/9·81-s − 0.658·83-s − 0.857·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9008537337\)
\(L(\frac12)\) \(\approx\) \(0.9008537337\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
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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

−9.766610285977486184781536614746, −9.285152345849520618027460352812, −8.567861705820082026402441543274, −8.557942136292391511888144813265, −7.86472366519306361174980449743, −7.67582681400150810228972534377, −7.44244386212684433193993942438, −7.06202566632976138060279908020, −6.39474630611944680003834418111, −5.80522194710482539098642063165, −5.57794756560665241606885970344, −4.91208335368472627602154805259, −4.79556262901023302769902494080, −4.06173416028383938755914560932, −3.34824530075107269887668813136, −3.11186415975879097431111809649, −2.66418800632813597259367022339, −2.28193055021139060301086659347, −1.62850274420999023814527468902, −0.31641206405672648544789529484, 0.31641206405672648544789529484, 1.62850274420999023814527468902, 2.28193055021139060301086659347, 2.66418800632813597259367022339, 3.11186415975879097431111809649, 3.34824530075107269887668813136, 4.06173416028383938755914560932, 4.79556262901023302769902494080, 4.91208335368472627602154805259, 5.57794756560665241606885970344, 5.80522194710482539098642063165, 6.39474630611944680003834418111, 7.06202566632976138060279908020, 7.44244386212684433193993942438, 7.67582681400150810228972534377, 7.86472366519306361174980449743, 8.557942136292391511888144813265, 8.567861705820082026402441543274, 9.285152345849520618027460352812, 9.766610285977486184781536614746

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