Properties

Label 4-1900e2-1.1-c2e2-0-1
Degree $4$
Conductor $3610000$
Sign $1$
Analytic cond. $2680.26$
Root an. cond. $7.19522$
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  − 5·7-s + 18·9-s − 3·11-s + 15·17-s − 38·19-s + 60·23-s − 85·43-s + 75·47-s + 49·49-s − 103·61-s − 90·63-s − 25·73-s + 15·77-s + 243·81-s − 180·83-s − 54·99-s − 204·101-s − 75·119-s + 121·121-s + 127-s + 131-s + 190·133-s + 137-s + 139-s + 149-s + 151-s + 270·153-s + ⋯
L(s)  = 1  − 5/7·7-s + 2·9-s − 0.272·11-s + 0.882·17-s − 2·19-s + 2.60·23-s − 1.97·43-s + 1.59·47-s + 49-s − 1.68·61-s − 1.42·63-s − 0.342·73-s + 0.194·77-s + 3·81-s − 2.16·83-s − 0.545·99-s − 2.01·101-s − 0.630·119-s + 121-s + 0.00787·127-s + 0.00763·131-s + 10/7·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 1.76·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3610000\)    =    \(2^{4} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2680.26\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3610000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.813152038\)
\(L(\frac12)\) \(\approx\) \(2.813152038\)
\(L(2)\) 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 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 + p T )^{2} \)
good3$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
7$C_2^2$ \( 1 + 5 T - 24 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 112 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
17$C_2^2$ \( 1 - 15 T - 64 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \)
23$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_2^2$ \( 1 + 85 T + 5376 T^{2} + 85 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 75 T + 3416 T^{2} - 75 p^{2} T^{3} + p^{4} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2^2$ \( 1 + 103 T + 6888 T^{2} + 103 p^{2} T^{3} + p^{4} T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 + 25 T - 4704 T^{2} + 25 p^{2} T^{3} + p^{4} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2$ \( ( 1 + 90 T + p^{2} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
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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

−9.144666601389194797669518280595, −9.010336556512219187556098014729, −8.313038279620157081545421914458, −8.237067162843546821653440422386, −7.45422462746594013909818768646, −7.18101921575324547552914729216, −6.82191021843979775329222725351, −6.78345162429324655950345296517, −5.93421300898144336301304374404, −5.81465927517559651812578099604, −4.95948762092247983190963069769, −4.82409167807978054393238039799, −4.19422490477833243840068315924, −4.02024930766860252786133704575, −3.22455640702439263994379769461, −3.01080058983811147757540988983, −2.27491851266803815172964022623, −1.63934699019100711331845484937, −1.17391223682618638231420277467, −0.46845207038472105041304254503, 0.46845207038472105041304254503, 1.17391223682618638231420277467, 1.63934699019100711331845484937, 2.27491851266803815172964022623, 3.01080058983811147757540988983, 3.22455640702439263994379769461, 4.02024930766860252786133704575, 4.19422490477833243840068315924, 4.82409167807978054393238039799, 4.95948762092247983190963069769, 5.81465927517559651812578099604, 5.93421300898144336301304374404, 6.78345162429324655950345296517, 6.82191021843979775329222725351, 7.18101921575324547552914729216, 7.45422462746594013909818768646, 8.237067162843546821653440422386, 8.313038279620157081545421914458, 9.010336556512219187556098014729, 9.144666601389194797669518280595

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