Properties

Label 4-1620e2-1.1-c1e2-0-9
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $167.334$
Root an. cond. $3.59663$
Motivic weight $1$
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-s − 2·7-s − 2·13-s + 12·17-s − 8·19-s + 6·23-s + 6·29-s + 4·31-s + 2·35-s + 4·37-s + 6·41-s + 10·43-s − 6·47-s + 7·49-s + 12·53-s + 12·59-s − 2·61-s + 2·65-s − 2·67-s + 24·71-s + 4·73-s − 8·79-s + 6·83-s − 12·85-s + 12·89-s + 4·91-s + 8·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 0.554·13-s + 2.91·17-s − 1.83·19-s + 1.25·23-s + 1.11·29-s + 0.718·31-s + 0.338·35-s + 0.657·37-s + 0.937·41-s + 1.52·43-s − 0.875·47-s + 49-s + 1.64·53-s + 1.56·59-s − 0.256·61-s + 0.248·65-s − 0.244·67-s + 2.84·71-s + 0.468·73-s − 0.900·79-s + 0.658·83-s − 1.30·85-s + 1.27·89-s + 0.419·91-s + 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(167.334\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.298589934\)
\(L(\frac12)\) \(\approx\) \(2.298589934\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−9.458662684714851864404080516706, −9.444033007247618621045438384566, −8.621809818729676507247985036694, −8.490249047453280027012231826294, −7.920019884955751442423591008300, −7.71566523213339628157756761594, −7.13867389319388351197198050246, −6.89646202038817991171433881562, −6.32755156734376256877235140711, −6.00586804970885833955879536878, −5.46693619519736486074087230697, −5.18225187955212691209606686313, −4.45586560799290528528078787307, −4.22556459238636769494077186938, −3.39884688204924552232079268052, −3.39485230222342112397070753471, −2.47721962337422856461379687633, −2.35000714234537269357006211758, −0.985410814038974510103190904706, −0.77448604197313909151175145249, 0.77448604197313909151175145249, 0.985410814038974510103190904706, 2.35000714234537269357006211758, 2.47721962337422856461379687633, 3.39485230222342112397070753471, 3.39884688204924552232079268052, 4.22556459238636769494077186938, 4.45586560799290528528078787307, 5.18225187955212691209606686313, 5.46693619519736486074087230697, 6.00586804970885833955879536878, 6.32755156734376256877235140711, 6.89646202038817991171433881562, 7.13867389319388351197198050246, 7.71566523213339628157756761594, 7.920019884955751442423591008300, 8.490249047453280027012231826294, 8.621809818729676507247985036694, 9.444033007247618621045438384566, 9.458662684714851864404080516706

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