Properties

Label 4-40e4-1.1-c1e2-0-30
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

Downloads

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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

\(L(1)\) \(\approx\) \(4.558060843\)
\(L(\frac12)\) \(\approx\) \(4.558060843\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−9.335471871715548963134810547260, −9.199986036096361040644173445779, −8.710792514662502558128758756072, −8.379833289083604216675323831739, −7.957129164992029050136585001242, −7.84634544188999616789983521419, −7.37277993114567478708472277995, −7.11121889197624676284593511399, −6.18081979363828298268674060771, −6.00074081911287784179597347611, −5.42686353838499235316846106494, −5.14303012029710053545229930093, −4.42221592988129295533729913592, −4.32105652789930491658332492539, −3.37673221799184133562580383159, −3.04649248551751460664454761990, −2.76498067237927063748406916396, −2.22009061319121592116103461872, −1.32389948059462705641449834747, −0.868681328818889429323508678969, 0.868681328818889429323508678969, 1.32389948059462705641449834747, 2.22009061319121592116103461872, 2.76498067237927063748406916396, 3.04649248551751460664454761990, 3.37673221799184133562580383159, 4.32105652789930491658332492539, 4.42221592988129295533729913592, 5.14303012029710053545229930093, 5.42686353838499235316846106494, 6.00074081911287784179597347611, 6.18081979363828298268674060771, 7.11121889197624676284593511399, 7.37277993114567478708472277995, 7.84634544188999616789983521419, 7.957129164992029050136585001242, 8.379833289083604216675323831739, 8.710792514662502558128758756072, 9.199986036096361040644173445779, 9.335471871715548963134810547260

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