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Properties

Label 4-3700e2-1.1-c0e2-0-4
Degree $4$
Conductor $13690000$
Sign $1$
Analytic cond. $3.40971$
Root an. cond. $1.35887$
Motivic weight $0$
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  − 2·3-s + 9-s − 2·17-s − 2·19-s + 2·23-s + 2·27-s + 2·29-s + 2·37-s + 49-s + 4·51-s + 4·57-s − 4·69-s + 2·71-s − 2·73-s + 2·79-s − 4·81-s − 4·87-s + 2·89-s + 2·109-s − 4·111-s + 2·113-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·147-s + ⋯
L(s)  = 1  − 2·3-s + 9-s − 2·17-s − 2·19-s + 2·23-s + 2·27-s + 2·29-s + 2·37-s + 49-s + 4·51-s + 4·57-s − 4·69-s + 2·71-s − 2·73-s + 2·79-s − 4·81-s − 4·87-s + 2·89-s + 2·109-s − 4·111-s + 2·113-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·147-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13690000\)    =    \(2^{4} \cdot 5^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(3.40971\)
Root analytic conductor: \(1.35887\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13690000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5571138403\)
\(L(\frac12)\) \(\approx\) \(0.5571138403\)
\(L(1)\) 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 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
31$C_2^2$ \( 1 + T^{4} \)
41$C_2^2$ \( 1 - T^{2} + T^{4} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 - T^{2} + T^{4} \)
53$C_2^2$ \( 1 - T^{2} + T^{4} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
83$C_2^2$ \( 1 - T^{2} + T^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−8.806047669109706001302528736883, −8.644295001614735714528539047498, −8.192193873738952994386791853030, −7.85476421633931697263178446010, −7.03391609464518320040789405262, −6.82856724457902302161903205338, −6.68762847193377242158639834182, −6.23119066964973085101426622890, −5.90768709554716534381180103750, −5.76392770802143345308451537500, −4.89680619208651750435935843951, −4.86370543048069108749780194246, −4.44416874382017672815107362721, −4.33915410891510458722443829643, −3.38276539324956148445915809154, −2.96291699918422353806167933084, −2.28307403539795704646354649833, −2.18042437468174401015478612365, −0.919009937968217120767539851521, −0.64917023946729775354303897339, 0.64917023946729775354303897339, 0.919009937968217120767539851521, 2.18042437468174401015478612365, 2.28307403539795704646354649833, 2.96291699918422353806167933084, 3.38276539324956148445915809154, 4.33915410891510458722443829643, 4.44416874382017672815107362721, 4.86370543048069108749780194246, 4.89680619208651750435935843951, 5.76392770802143345308451537500, 5.90768709554716534381180103750, 6.23119066964973085101426622890, 6.68762847193377242158639834182, 6.82856724457902302161903205338, 7.03391609464518320040789405262, 7.85476421633931697263178446010, 8.192193873738952994386791853030, 8.644295001614735714528539047498, 8.806047669109706001302528736883

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