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Properties

Label 4-20e2-1.1-c39e2-0-0
Degree $4$
Conductor $400$
Sign $1$
Analytic cond. $37125.2$
Root an. cond. $13.8808$
Motivic weight $39$
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  − 1.04e6·2-s + 5.49e11·4-s − 6.13e13·5-s + 6.43e19·10-s + 1.47e22·13-s − 3.02e23·16-s − 2.20e24·17-s − 3.37e25·20-s + 1.95e27·25-s − 1.54e28·26-s + 3.16e29·32-s + 2.31e30·34-s − 6.31e30·37-s + 1.04e32·41-s − 2.04e33·50-s + 8.09e33·52-s + 7.12e33·53-s − 9.93e34·61-s − 1.66e35·64-s − 9.04e35·65-s − 1.21e36·68-s + 3.65e36·73-s + 6.62e36·74-s + 1.85e37·80-s − 1.64e37·81-s − 1.09e38·82-s + 1.35e38·85-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.43·5-s + 2.03·10-s + 2.79·13-s − 16-s − 2.24·17-s − 1.43·20-s + 1.07·25-s − 3.95·26-s + 1.41·32-s + 3.16·34-s − 1.66·37-s + 3.70·41-s − 1.51·50-s + 2.79·52-s + 1.69·53-s − 1.52·61-s − 64-s − 4.02·65-s − 2.24·68-s + 1.68·73-s + 2.34·74-s + 1.43·80-s − 81-s − 5.23·82-s + 3.22·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+39/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(37125.2\)
Root analytic conductor: \(13.8808\)
Motivic weight: \(39\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 400,\ (\ :39/2, 39/2),\ 1)\)

Particular Values

\(L(20)\) \(\approx\) \(0.7370347839\)
\(L(\frac12)\) \(\approx\) \(0.7370347839\)
\(L(\frac{41}{2})\) 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$C_2$ \( 1 + p^{20} T + p^{39} T^{2} \)
5$C_2$ \( 1 + 61397374177364 T + p^{39} T^{2} \)
good3$C_2^2$ \( 1 + p^{78} T^{4} \)
7$C_2^2$ \( 1 + p^{78} T^{4} \)
11$C_2$ \( ( 1 - p^{39} T^{2} )^{2} \)
13$C_2$ \( ( 1 - \)\(85\!\cdots\!08\)\( T + p^{39} T^{2} )( 1 - \)\(62\!\cdots\!38\)\( T + p^{39} T^{2} ) \)
17$C_2$ \( ( 1 + \)\(25\!\cdots\!06\)\( T + p^{39} T^{2} )( 1 + \)\(19\!\cdots\!76\)\( T + p^{39} T^{2} ) \)
19$C_2$ \( ( 1 + p^{39} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{78} T^{4} \)
29$C_2$ \( ( 1 - \)\(42\!\cdots\!10\)\( T + p^{39} T^{2} )( 1 + \)\(42\!\cdots\!10\)\( T + p^{39} T^{2} ) \)
31$C_2$ \( ( 1 - p^{39} T^{2} )^{2} \)
37$C_2$ \( ( 1 - \)\(11\!\cdots\!54\)\( T + p^{39} T^{2} )( 1 + \)\(75\!\cdots\!76\)\( T + p^{39} T^{2} ) \)
41$C_2$ \( ( 1 - \)\(52\!\cdots\!12\)\( T + p^{39} T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{78} T^{4} \)
47$C_2^2$ \( 1 + p^{78} T^{4} \)
53$C_2$ \( ( 1 - \)\(83\!\cdots\!68\)\( T + p^{39} T^{2} )( 1 + \)\(11\!\cdots\!62\)\( T + p^{39} T^{2} ) \)
59$C_2$ \( ( 1 + p^{39} T^{2} )^{2} \)
61$C_2$ \( ( 1 + \)\(49\!\cdots\!08\)\( T + p^{39} T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{78} T^{4} \)
71$C_2$ \( ( 1 - p^{39} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(42\!\cdots\!78\)\( T + p^{39} T^{2} )( 1 + \)\(62\!\cdots\!92\)\( T + p^{39} T^{2} ) \)
79$C_2$ \( ( 1 + p^{39} T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{78} T^{4} \)
89$C_2$ \( ( 1 - \)\(42\!\cdots\!10\)\( T + p^{39} T^{2} )( 1 + \)\(42\!\cdots\!10\)\( T + p^{39} T^{2} ) \)
97$C_2$ \( ( 1 - \)\(62\!\cdots\!64\)\( T + p^{39} T^{2} )( 1 + \)\(90\!\cdots\!06\)\( T + p^{39} T^{2} ) \)
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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

−11.38727676345029643800101343613, −10.70964954106043905563756639421, −10.63428638138278146124767899411, −9.411303421262881771616232876214, −8.921764367269766155283983092776, −8.510458603743087613609698469807, −8.344751677329255819921988775828, −7.31709056288763372521083458042, −7.28741658353485725976959684787, −6.18360980211469711737165581072, −6.16686215700771385176555242468, −4.88376892400170652946187040738, −4.17799439747541032613381388132, −3.97768235823089908850642437348, −3.34966190310235358408199433046, −2.45759987813358710573594887598, −1.92860336885851893114202511230, −1.15927407511143595530540134399, −0.811387365309802552130347647548, −0.28716590660609210971502304263, 0.28716590660609210971502304263, 0.811387365309802552130347647548, 1.15927407511143595530540134399, 1.92860336885851893114202511230, 2.45759987813358710573594887598, 3.34966190310235358408199433046, 3.97768235823089908850642437348, 4.17799439747541032613381388132, 4.88376892400170652946187040738, 6.16686215700771385176555242468, 6.18360980211469711737165581072, 7.28741658353485725976959684787, 7.31709056288763372521083458042, 8.344751677329255819921988775828, 8.510458603743087613609698469807, 8.921764367269766155283983092776, 9.411303421262881771616232876214, 10.63428638138278146124767899411, 10.70964954106043905563756639421, 11.38727676345029643800101343613

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