Properties

Label 4-2352e2-1.1-c1e2-0-7
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $352.718$
Root an. cond. $4.33368$
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  − 2·3-s + 3·9-s + 4·19-s + 7·25-s − 4·27-s − 18·29-s + 2·31-s − 4·37-s + 18·53-s − 8·57-s − 6·59-s − 14·75-s + 5·81-s − 30·83-s + 36·87-s − 4·93-s + 8·103-s − 8·109-s + 8·111-s − 12·113-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 0.917·19-s + 7/5·25-s − 0.769·27-s − 3.34·29-s + 0.359·31-s − 0.657·37-s + 2.47·53-s − 1.05·57-s − 0.781·59-s − 1.61·75-s + 5/9·81-s − 3.29·83-s + 3.85·87-s − 0.414·93-s + 0.788·103-s − 0.766·109-s + 0.759·111-s − 1.12·113-s − 0.454·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(352.718\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9598221381\)
\(L(\frac12)\) \(\approx\) \(0.9598221381\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 155 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \)
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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

−9.399313986225159804161803806916, −8.689566754130849929814897972891, −8.619054464138582699759206127593, −7.87714954488260912208687688887, −7.44570103384186901012538675461, −7.23480578719337984755746912612, −6.94111526070505957458140205928, −6.46142402800890392576405098808, −5.93059242114323604959642214856, −5.54110775059362398383594301348, −5.41254595532140251450104078766, −4.95379099453019211167493133182, −4.41899804655451168354070894288, −3.91557220328210265752888404067, −3.62439317450712667913210896472, −2.96422302701552219325839491158, −2.41829803001243573852728731649, −1.66366257881429197187179758114, −1.24236527673125871696258896508, −0.38878806783810151063413324181, 0.38878806783810151063413324181, 1.24236527673125871696258896508, 1.66366257881429197187179758114, 2.41829803001243573852728731649, 2.96422302701552219325839491158, 3.62439317450712667913210896472, 3.91557220328210265752888404067, 4.41899804655451168354070894288, 4.95379099453019211167493133182, 5.41254595532140251450104078766, 5.54110775059362398383594301348, 5.93059242114323604959642214856, 6.46142402800890392576405098808, 6.94111526070505957458140205928, 7.23480578719337984755746912612, 7.44570103384186901012538675461, 7.87714954488260912208687688887, 8.619054464138582699759206127593, 8.689566754130849929814897972891, 9.399313986225159804161803806916

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