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Properties

Label 4-20e2-1.1-c31e2-0-0
Degree $4$
Conductor $400$
Sign $1$
Analytic cond. $14824.1$
Root an. cond. $11.0342$
Motivic weight $31$
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  − 6.55e4·2-s + 2.14e9·4-s + 3.18e10·5-s − 2.09e15·10-s − 8.51e16·13-s − 4.61e18·16-s + 2.88e19·17-s + 6.85e19·20-s − 3.63e21·25-s + 5.58e21·26-s + 3.02e23·32-s − 1.89e24·34-s − 5.33e24·37-s + 3.50e25·41-s + 2.38e26·50-s − 1.82e26·52-s − 1.50e27·53-s − 1.69e28·61-s − 9.90e27·64-s − 2.71e27·65-s + 6.20e28·68-s − 1.70e29·73-s + 3.49e29·74-s − 1.47e29·80-s − 3.81e29·81-s − 2.29e30·82-s + 9.21e29·85-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.467·5-s − 0.661·10-s − 0.461·13-s − 16-s + 2.44·17-s + 0.467·20-s − 0.781·25-s + 0.652·26-s + 1.41·32-s − 3.45·34-s − 2.62·37-s + 3.51·41-s + 1.10·50-s − 0.461·52-s − 2.82·53-s − 3.59·61-s − 64-s − 0.215·65-s + 2.44·68-s − 2.23·73-s + 3.71·74-s − 0.467·80-s − 81-s − 4.97·82-s + 1.14·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(14824.1\)
Root analytic conductor: \(11.0342\)
Motivic weight: \(31\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 400,\ (\ :31/2, 31/2),\ 1)\)

Particular Values

\(L(16)\) \(\approx\) \(0.9711221112\)
\(L(\frac12)\) \(\approx\) \(0.9711221112\)
\(L(\frac{33}{2})\) 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$C_2$ \( 1 + p^{16} T + p^{31} T^{2} \)
5$C_2$ \( 1 - 31898749516 T + p^{31} T^{2} \)
good3$C_2^2$ \( 1 + p^{62} T^{4} \)
7$C_2^2$ \( 1 + p^{62} T^{4} \)
11$C_2$ \( ( 1 - p^{31} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 214906916298965908 T + p^{31} T^{2} )( 1 + 300091304100379422 T + p^{31} T^{2} ) \)
17$C_2$ \( ( 1 - 22812973752742283294 T + p^{31} T^{2} )( 1 - 6060024580156067464 T + p^{31} T^{2} ) \)
19$C_2$ \( ( 1 + p^{31} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{62} T^{4} \)
29$C_2$ \( ( 1 - \)\(66\!\cdots\!10\)\( T + p^{31} T^{2} )( 1 + \)\(66\!\cdots\!10\)\( T + p^{31} T^{2} ) \)
31$C_2$ \( ( 1 - p^{31} T^{2} )^{2} \)
37$C_2$ \( ( 1 + \)\(16\!\cdots\!36\)\( T + p^{31} T^{2} )( 1 + \)\(37\!\cdots\!66\)\( T + p^{31} T^{2} ) \)
41$C_2$ \( ( 1 - \)\(17\!\cdots\!92\)\( T + p^{31} T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{62} T^{4} \)
47$C_2^2$ \( 1 + p^{62} T^{4} \)
53$C_2$ \( ( 1 + \)\(74\!\cdots\!22\)\( T + p^{31} T^{2} )( 1 + \)\(76\!\cdots\!52\)\( T + p^{31} T^{2} ) \)
59$C_2$ \( ( 1 + p^{31} T^{2} )^{2} \)
61$C_2$ \( ( 1 + \)\(84\!\cdots\!88\)\( T + p^{31} T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{62} T^{4} \)
71$C_2$ \( ( 1 - p^{31} T^{2} )^{2} \)
73$C_2$ \( ( 1 + \)\(19\!\cdots\!92\)\( T + p^{31} T^{2} )( 1 + \)\(15\!\cdots\!62\)\( T + p^{31} T^{2} ) \)
79$C_2$ \( ( 1 + p^{31} T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{62} T^{4} \)
89$C_2$ \( ( 1 - \)\(32\!\cdots\!10\)\( T + p^{31} T^{2} )( 1 + \)\(32\!\cdots\!10\)\( T + p^{31} T^{2} ) \)
97$C_2$ \( ( 1 - \)\(11\!\cdots\!94\)\( T + p^{31} T^{2} )( 1 + \)\(48\!\cdots\!76\)\( T + p^{31} T^{2} ) \)
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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

−12.31313828957809545937099333810, −11.56134098519152291969230106180, −10.72725967322219670616600884163, −10.41824279365299446657303439624, −9.622418267737363476842434653341, −9.506418009097397651187457455688, −8.759172409139762395610369067363, −7.987173096351900894259896427318, −7.51327066369299530245182825341, −7.24025227401347069550953581736, −5.97631313797644162146294714103, −5.90253262916179763378331984547, −4.89158390075898819260596551709, −4.30915145092707936442873082751, −3.23196054130810555808991704802, −2.93244797172377612071765681335, −1.78514215404809631864117448255, −1.65970023678133433519129160771, −0.927551323264629383991045927073, −0.30488811815165137548130493347, 0.30488811815165137548130493347, 0.927551323264629383991045927073, 1.65970023678133433519129160771, 1.78514215404809631864117448255, 2.93244797172377612071765681335, 3.23196054130810555808991704802, 4.30915145092707936442873082751, 4.89158390075898819260596551709, 5.90253262916179763378331984547, 5.97631313797644162146294714103, 7.24025227401347069550953581736, 7.51327066369299530245182825341, 7.987173096351900894259896427318, 8.759172409139762395610369067363, 9.506418009097397651187457455688, 9.622418267737363476842434653341, 10.41824279365299446657303439624, 10.72725967322219670616600884163, 11.56134098519152291969230106180, 12.31313828957809545937099333810

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