Properties

Label 4-600e2-1.1-c2e2-0-0
Degree $4$
Conductor $360000$
Sign $1$
Analytic cond. $267.283$
Root an. cond. $4.04336$
Motivic weight $2$
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 − 5·9-s − 30·13-s − 46·19-s − 4·21-s − 28·27-s + 66·31-s − 132·37-s − 60·39-s + 14·43-s − 95·49-s − 92·57-s + 78·61-s + 10·63-s − 226·67-s − 116·73-s + 140·79-s − 11·81-s + 60·91-s + 132·93-s + 2·97-s + 52·103-s − 290·109-s − 264·111-s + 150·117-s + 170·121-s + ⋯
L(s)  = 1  + 2/3·3-s − 2/7·7-s − 5/9·9-s − 2.30·13-s − 2.42·19-s − 0.190·21-s − 1.03·27-s + 2.12·31-s − 3.56·37-s − 1.53·39-s + 0.325·43-s − 1.93·49-s − 1.61·57-s + 1.27·61-s + 0.158·63-s − 3.37·67-s − 1.58·73-s + 1.77·79-s − 0.135·81-s + 0.659·91-s + 1.41·93-s + 2/97·97-s + 0.504·103-s − 2.66·109-s − 2.37·111-s + 1.28·117-s + 1.40·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(360000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(267.283\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 360000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2776253813\)
\(L(\frac12)\) \(\approx\) \(0.2776253813\)
\(L(2)\) 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 \( 1 \)
3$C_2$ \( 1 - 2 T + p^{2} T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 170 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 15 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 186 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 23 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 1050 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1034 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 33 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 66 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2010 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 7 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2370 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4266 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 + 3406 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 39 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 113 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 9434 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 58 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 70 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 9550 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 7650 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
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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

−10.71568221953746559351859668597, −10.01031848094890097810970246429, −9.980019776184159154851429878254, −9.373107509670799025211322112375, −8.796789658242839902729075310435, −8.465748554585351195936196324837, −8.287561046623956413144019757188, −7.46114545341304269646281666472, −7.28931689357422175416201784728, −6.49286116847614370838023976425, −6.43016432160658908511892750310, −5.60955888860735139590101540892, −5.02470499469488256969154847949, −4.60844158202719177268599873871, −4.14874789037213047239152701425, −3.24564144807565675570665796214, −2.91049946718576981308270147318, −2.21183001885810142758184276851, −1.81609427952111749276538470203, −0.17318688816335130215891516359, 0.17318688816335130215891516359, 1.81609427952111749276538470203, 2.21183001885810142758184276851, 2.91049946718576981308270147318, 3.24564144807565675570665796214, 4.14874789037213047239152701425, 4.60844158202719177268599873871, 5.02470499469488256969154847949, 5.60955888860735139590101540892, 6.43016432160658908511892750310, 6.49286116847614370838023976425, 7.28931689357422175416201784728, 7.46114545341304269646281666472, 8.287561046623956413144019757188, 8.465748554585351195936196324837, 8.796789658242839902729075310435, 9.373107509670799025211322112375, 9.980019776184159154851429878254, 10.01031848094890097810970246429, 10.71568221953746559351859668597

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