Properties

Label 4-1440e2-1.1-c1e2-0-30
Degree $4$
Conductor $2073600$
Sign $1$
Analytic cond. $132.214$
Root an. cond. $3.39093$
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  + 4·5-s + 12·11-s − 8·19-s + 11·25-s − 16·31-s + 12·41-s + 10·49-s + 48·55-s − 12·59-s − 12·61-s + 8·71-s + 16·79-s + 28·89-s − 32·95-s − 16·101-s − 4·109-s + 86·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·155-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1.78·5-s + 3.61·11-s − 1.83·19-s + 11/5·25-s − 2.87·31-s + 1.87·41-s + 10/7·49-s + 6.47·55-s − 1.56·59-s − 1.53·61-s + 0.949·71-s + 1.80·79-s + 2.96·89-s − 3.28·95-s − 1.59·101-s − 0.383·109-s + 7.81·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2073600\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(132.214\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1440} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2073600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.396345926\)
\(L(\frac12)\) \(\approx\) \(4.396345926\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.356617050526758854595170777932, −9.317303426941891077866547460954, −9.107199685103969193972183334531, −8.922545014685414204835721066245, −8.291088575153106100783681739681, −7.64935600880662294895007811941, −6.94959191496572905790491693111, −6.94645429673339141052476489839, −6.28663596806027908069130909675, −6.10493581300060518518083163955, −5.89977247781588452141840122716, −5.26259218881425705491593562626, −4.52740208341722857855538290370, −4.27630864557566518018787456996, −3.69671675833016336473403765613, −3.39455638567554239278118331441, −2.35336805806330563720610955749, −1.94363061949545825485439880304, −1.57524137386085790187701855969, −0.905665913566149566844231977803, 0.905665913566149566844231977803, 1.57524137386085790187701855969, 1.94363061949545825485439880304, 2.35336805806330563720610955749, 3.39455638567554239278118331441, 3.69671675833016336473403765613, 4.27630864557566518018787456996, 4.52740208341722857855538290370, 5.26259218881425705491593562626, 5.89977247781588452141840122716, 6.10493581300060518518083163955, 6.28663596806027908069130909675, 6.94645429673339141052476489839, 6.94959191496572905790491693111, 7.64935600880662294895007811941, 8.291088575153106100783681739681, 8.922545014685414204835721066245, 9.107199685103969193972183334531, 9.317303426941891077866547460954, 9.356617050526758854595170777932

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