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Properties

Label 4-20e2-1.1-c42e2-0-0
Degree $4$
Conductor $400$
Sign $1$
Analytic cond. $49932.3$
Root an. cond. $14.9484$
Motivic weight $42$
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  − 4.39e12·4-s + 7.74e14·5-s − 2.18e20·9-s + 1.93e25·16-s − 3.40e27·20-s + 3.71e29·25-s − 1.97e31·29-s + 9.62e32·36-s + 2.94e34·41-s − 1.69e35·45-s − 6.23e35·49-s − 7.67e37·61-s − 8.50e37·64-s + 1.49e40·80-s + 3.59e40·81-s + 3.48e40·89-s − 1.63e42·100-s + 2.47e42·101-s + 2.31e43·109-s + 8.69e43·116-s + 1.09e44·121-s + 1.11e44·125-s + 127-s + 131-s + 137-s + 139-s − 4.23e45·144-s + ⋯
L(s)  = 1  − 4-s + 1.62·5-s − 2·9-s + 16-s − 1.62·20-s + 1.63·25-s − 3.85·29-s + 2·36-s + 3.99·41-s − 3.24·45-s − 2·49-s − 2.47·61-s − 64-s + 1.62·80-s + 3·81-s + 0.402·89-s − 1.63·100-s + 2.00·101-s + 3.79·109-s + 3.85·116-s + 2·121-s + 1.03·125-s − 2·144-s − 6.25·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+21)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(49932.3\)
Root analytic conductor: \(14.9484\)
Motivic weight: \(42\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 400,\ (\ :21, 21),\ 1)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(2.743237294\)
\(L(\frac12)\) \(\approx\) \(2.743237294\)
\(L(22)\) 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$C_2$ \( 1 + p^{42} T^{2} \)
5$C_2$ \( 1 - 774150816151206 T + p^{42} T^{2} \)
good3$C_2$ \( ( 1 + p^{42} T^{2} )^{2} \)
7$C_2$ \( ( 1 + p^{42} T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - p^{21} T )^{2}( 1 + p^{21} T )^{2} \)
13$C_2$ \( ( 1 - \)\(44\!\cdots\!10\)\( T + p^{42} T^{2} )( 1 + \)\(44\!\cdots\!10\)\( T + p^{42} T^{2} ) \)
17$C_2$ \( ( 1 - \)\(89\!\cdots\!70\)\( T + p^{42} T^{2} )( 1 + \)\(89\!\cdots\!70\)\( T + p^{42} T^{2} ) \)
19$C_1$$\times$$C_1$ \( ( 1 - p^{21} T )^{2}( 1 + p^{21} T )^{2} \)
23$C_2$ \( ( 1 + p^{42} T^{2} )^{2} \)
29$C_2$ \( ( 1 + \)\(98\!\cdots\!58\)\( T + p^{42} T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p^{21} T )^{2}( 1 + p^{21} T )^{2} \)
37$C_2$ \( ( 1 - \)\(13\!\cdots\!70\)\( T + p^{42} T^{2} )( 1 + \)\(13\!\cdots\!70\)\( T + p^{42} T^{2} ) \)
41$C_2$ \( ( 1 - \)\(14\!\cdots\!82\)\( T + p^{42} T^{2} )^{2} \)
43$C_2$ \( ( 1 + p^{42} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{42} T^{2} )^{2} \)
53$C_2$ \( ( 1 - \)\(20\!\cdots\!90\)\( T + p^{42} T^{2} )( 1 + \)\(20\!\cdots\!90\)\( T + p^{42} T^{2} ) \)
59$C_1$$\times$$C_1$ \( ( 1 - p^{21} T )^{2}( 1 + p^{21} T )^{2} \)
61$C_2$ \( ( 1 + \)\(38\!\cdots\!78\)\( T + p^{42} T^{2} )^{2} \)
67$C_2$ \( ( 1 + p^{42} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{21} T )^{2}( 1 + p^{21} T )^{2} \)
73$C_2$ \( ( 1 - \)\(21\!\cdots\!90\)\( T + p^{42} T^{2} )( 1 + \)\(21\!\cdots\!90\)\( T + p^{42} T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - p^{21} T )^{2}( 1 + p^{21} T )^{2} \)
83$C_2$ \( ( 1 + p^{42} T^{2} )^{2} \)
89$C_2$ \( ( 1 - \)\(17\!\cdots\!22\)\( T + p^{42} T^{2} )^{2} \)
97$C_2$ \( ( 1 - \)\(29\!\cdots\!30\)\( T + p^{42} T^{2} )( 1 + \)\(29\!\cdots\!30\)\( T + p^{42} T^{2} ) \)
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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

−10.99451611007048869242594584207, −10.74509775711669743200521849254, −9.582195864228988723705090489109, −9.518273876132698674397271656791, −9.056546662909184262905900620083, −8.605549160007564045510308043916, −7.72239117048235136251257750354, −7.45035368358680190460130120401, −6.05961806935506623840869957471, −5.97837028435137863214144445655, −5.72710877677054861451914210423, −5.03784503370266587344437793255, −4.50776621652036173239668837373, −3.63221288442588815554418610348, −3.18222398667037885078233811033, −2.60711454496670229990090332602, −1.92847699189412615898422457219, −1.66487455720974401995409740724, −0.57184463237298998380856193443, −0.47723609252750201092002264796, 0.47723609252750201092002264796, 0.57184463237298998380856193443, 1.66487455720974401995409740724, 1.92847699189412615898422457219, 2.60711454496670229990090332602, 3.18222398667037885078233811033, 3.63221288442588815554418610348, 4.50776621652036173239668837373, 5.03784503370266587344437793255, 5.72710877677054861451914210423, 5.97837028435137863214144445655, 6.05961806935506623840869957471, 7.45035368358680190460130120401, 7.72239117048235136251257750354, 8.605549160007564045510308043916, 9.056546662909184262905900620083, 9.518273876132698674397271656791, 9.582195864228988723705090489109, 10.74509775711669743200521849254, 10.99451611007048869242594584207

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