Properties

Label 4-444528-1.1-c1e2-0-13
Degree $4$
Conductor $444528$
Sign $1$
Analytic cond. $28.3434$
Root an. cond. $2.30734$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 12·11-s + 12·23-s − 10·25-s − 12·29-s + 4·37-s − 8·43-s + 49-s + 12·53-s + 16·67-s − 12·71-s + 12·77-s − 8·79-s + 12·107-s + 28·109-s + 12·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 12·161-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.377·7-s + 3.61·11-s + 2.50·23-s − 2·25-s − 2.22·29-s + 0.657·37-s − 1.21·43-s + 1/7·49-s + 1.64·53-s + 1.95·67-s − 1.42·71-s + 1.36·77-s − 0.900·79-s + 1.16·107-s + 2.68·109-s + 1.12·113-s + 7.81·121-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 + 0.0798·157-s + 0.945·161-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(444528\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(28.3434\)
Root analytic conductor: \(2.30734\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 444528,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.637985935\)
\(L(\frac12)\) \(\approx\) \(2.637985935\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.729935221736740713914818138949, −8.300043091642042619777544931378, −7.44499063873378776873294009662, −7.12758834797060137294534233139, −6.93446262615281462668117301613, −6.11056271857714724112691707067, −6.03492627650465326217694729948, −5.32133833421984252130695309604, −4.63917458628061830491997962672, −4.15927538579926872458234038883, −3.53462971911322619523210724383, −3.52771007221053761224054171247, −2.20338451459941845252791108478, −1.58867103529813382887688476545, −1.00219484121657416616236254344, 1.00219484121657416616236254344, 1.58867103529813382887688476545, 2.20338451459941845252791108478, 3.52771007221053761224054171247, 3.53462971911322619523210724383, 4.15927538579926872458234038883, 4.63917458628061830491997962672, 5.32133833421984252130695309604, 6.03492627650465326217694729948, 6.11056271857714724112691707067, 6.93446262615281462668117301613, 7.12758834797060137294534233139, 7.44499063873378776873294009662, 8.300043091642042619777544931378, 8.729935221736740713914818138949

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