Properties

Label 4-50e2-1.1-c2e2-0-2
Degree $4$
Conductor $2500$
Sign $1$
Analytic cond. $1.85613$
Root an. cond. $1.16721$
Motivic weight $2$
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  + 2·2-s + 4·3-s + 2·4-s + 8·6-s − 4·7-s + 8·9-s − 16·11-s + 8·12-s − 6·13-s − 8·14-s − 4·16-s − 14·17-s + 16·18-s − 16·21-s − 32·22-s + 4·23-s − 12·26-s + 36·27-s − 8·28-s + 104·31-s − 8·32-s − 64·33-s − 28·34-s + 16·36-s + 6·37-s − 24·39-s − 16·41-s + ⋯
L(s)  = 1  + 2-s + 4/3·3-s + 1/2·4-s + 4/3·6-s − 4/7·7-s + 8/9·9-s − 1.45·11-s + 2/3·12-s − 0.461·13-s − 4/7·14-s − 1/4·16-s − 0.823·17-s + 8/9·18-s − 0.761·21-s − 1.45·22-s + 4/23·23-s − 0.461·26-s + 4/3·27-s − 2/7·28-s + 3.35·31-s − 1/4·32-s − 1.93·33-s − 0.823·34-s + 4/9·36-s + 6/37·37-s − 0.615·39-s − 0.390·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1.85613\)
Root analytic conductor: \(1.16721\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2500,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.520909804\)
\(L(\frac12)\) \(\approx\) \(2.520909804\)
\(L(2)\) 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$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \)
19$C_2^2$ \( 1 - 322 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \)
31$C_2$ \( ( 1 - 52 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 84 T + 3528 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \)
59$C_2^2$ \( 1 - 6562 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 48 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 124 T + 7688 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 + 28 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 94 T + 4418 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2^2$ \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 9442 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 126 T + 7938 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} \)
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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

−15.28251355163527999351529236127, −15.21317250705393067168465808194, −14.19363308808109727198566100549, −13.96559784724240370666428747266, −13.39879035640600768202794593998, −13.02918218158221418071367170676, −12.34589087151033955625952130667, −11.92153701163236305548920019596, −10.74851921991389014260018770078, −10.41594705560454574578346185499, −9.552185531707097370396717968568, −8.993375708837880510963674875133, −8.151566743344155438981684917321, −7.73418527168254582498586886194, −6.74497640504222998871576364165, −6.05578039413026976077565028111, −4.87141704188521884423827638192, −4.32055384926547141523222318228, −2.82808612201739024620436862174, −2.76410783190052260108059071542, 2.76410783190052260108059071542, 2.82808612201739024620436862174, 4.32055384926547141523222318228, 4.87141704188521884423827638192, 6.05578039413026976077565028111, 6.74497640504222998871576364165, 7.73418527168254582498586886194, 8.151566743344155438981684917321, 8.993375708837880510963674875133, 9.552185531707097370396717968568, 10.41594705560454574578346185499, 10.74851921991389014260018770078, 11.92153701163236305548920019596, 12.34589087151033955625952130667, 13.02918218158221418071367170676, 13.39879035640600768202794593998, 13.96559784724240370666428747266, 14.19363308808109727198566100549, 15.21317250705393067168465808194, 15.28251355163527999351529236127

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