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Properties

Label 4-20e2-1.1-c29e2-0-0
Degree $4$
Conductor $400$
Sign $1$
Analytic cond. $11354.1$
Root an. cond. $10.3226$
Motivic weight $29$
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  + 3.27e4·2-s + 5.36e8·4-s − 1.74e10·5-s − 5.70e14·10-s − 2.12e16·13-s − 2.88e17·16-s − 7.02e16·17-s − 9.34e18·20-s + 1.16e20·25-s − 6.97e20·26-s − 9.44e21·32-s − 2.30e21·34-s + 1.00e23·37-s − 6.39e23·41-s + 3.82e24·50-s − 1.14e25·52-s + 1.53e25·53-s + 8.31e25·61-s − 1.54e26·64-s + 3.70e26·65-s − 3.77e25·68-s + 2.89e27·73-s + 3.30e27·74-s + 5.01e27·80-s − 4.71e27·81-s − 2.09e28·82-s + 1.22e27·85-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.27·5-s − 1.80·10-s − 1.49·13-s − 16-s − 0.101·17-s − 1.27·20-s + 0.626·25-s − 2.12·26-s − 1.41·32-s − 0.143·34-s + 1.84·37-s − 2.63·41-s + 0.885·50-s − 1.49·52-s + 1.52·53-s + 1.07·61-s − 64-s + 1.91·65-s − 0.101·68-s + 2.77·73-s + 2.60·74-s + 1.27·80-s − 81-s − 3.72·82-s + 0.129·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+29/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(11354.1\)
Root analytic conductor: \(10.3226\)
Motivic weight: \(29\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 400,\ (\ :29/2, 29/2),\ 1)\)

Particular Values

\(L(15)\) \(\approx\) \(1.877519062\)
\(L(\frac12)\) \(\approx\) \(1.877519062\)
\(L(\frac{31}{2})\) 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$C_2$ \( 1 - p^{15} T + p^{29} T^{2} \)
5$C_2$ \( 1 + 17403927764 T + p^{29} T^{2} \)
good3$C_2^2$ \( 1 + p^{58} T^{4} \)
7$C_2^2$ \( 1 + p^{58} T^{4} \)
11$C_2$ \( ( 1 - p^{29} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6381221746069194 T + p^{29} T^{2} )( 1 + 27666451069227116 T + p^{29} T^{2} ) \)
17$C_2$ \( ( 1 - 946035151295084008 T + p^{29} T^{2} )( 1 + 1016312836158774082 T + p^{29} T^{2} ) \)
19$C_2$ \( ( 1 + p^{29} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{58} T^{4} \)
29$C_2$ \( ( 1 - \)\(12\!\cdots\!90\)\( T + p^{29} T^{2} )( 1 + \)\(12\!\cdots\!90\)\( T + p^{29} T^{2} ) \)
31$C_2$ \( ( 1 - p^{29} T^{2} )^{2} \)
37$C_2$ \( ( 1 - \)\(10\!\cdots\!18\)\( T + p^{29} T^{2} )( 1 + \)\(84\!\cdots\!72\)\( T + p^{29} T^{2} ) \)
41$C_2$ \( ( 1 + \)\(31\!\cdots\!88\)\( T + p^{29} T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{58} T^{4} \)
47$C_2^2$ \( 1 + p^{58} T^{4} \)
53$C_2$ \( ( 1 - \)\(19\!\cdots\!84\)\( T + p^{29} T^{2} )( 1 + \)\(43\!\cdots\!26\)\( T + p^{29} T^{2} ) \)
59$C_2$ \( ( 1 + p^{29} T^{2} )^{2} \)
61$C_2$ \( ( 1 - \)\(41\!\cdots\!92\)\( T + p^{29} T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{58} T^{4} \)
71$C_2$ \( ( 1 - p^{29} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(17\!\cdots\!94\)\( T + p^{29} T^{2} )( 1 - \)\(11\!\cdots\!04\)\( T + p^{29} T^{2} ) \)
79$C_2$ \( ( 1 + p^{29} T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{58} T^{4} \)
89$C_2$ \( ( 1 - \)\(17\!\cdots\!10\)\( T + p^{29} T^{2} )( 1 + \)\(17\!\cdots\!10\)\( T + p^{29} T^{2} ) \)
97$C_2$ \( ( 1 - \)\(11\!\cdots\!98\)\( T + p^{29} T^{2} )( 1 + \)\(54\!\cdots\!92\)\( T + p^{29} T^{2} ) \)
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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

−12.41525891572583677550380570652, −12.18255064663268020479467532676, −11.48544166942449833725773495055, −11.23250681760558608836230314826, −10.14117764784385567163295132663, −9.641957718353609687692608024219, −8.679403324652103858312117947604, −8.139757025158823959254894011865, −7.29797769531024606821536077720, −7.00361537030602667172947339033, −6.18415343156483442981800078723, −5.43083830690775713984125973818, −4.72999599368185863603777713741, −4.54054587084046956366390475320, −3.58528072311433573301676766354, −3.44119312494775067452718314027, −2.44180354550926064676694787008, −2.14714451714300453306290496207, −0.906227696321985246120463489752, −0.26860631814727616941238385263, 0.26860631814727616941238385263, 0.906227696321985246120463489752, 2.14714451714300453306290496207, 2.44180354550926064676694787008, 3.44119312494775067452718314027, 3.58528072311433573301676766354, 4.54054587084046956366390475320, 4.72999599368185863603777713741, 5.43083830690775713984125973818, 6.18415343156483442981800078723, 7.00361537030602667172947339033, 7.29797769531024606821536077720, 8.139757025158823959254894011865, 8.679403324652103858312117947604, 9.641957718353609687692608024219, 10.14117764784385567163295132663, 11.23250681760558608836230314826, 11.48544166942449833725773495055, 12.18255064663268020479467532676, 12.41525891572583677550380570652

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