Properties

Label 4-2300e2-1.1-c1e2-0-7
Degree $4$
Conductor $5290000$
Sign $1$
Analytic cond. $337.294$
Root an. cond. $4.28550$
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  + 6·9-s + 12·11-s − 4·19-s + 10·29-s + 2·31-s − 14·41-s + 13·49-s − 26·59-s − 16·61-s + 14·71-s + 24·79-s + 27·81-s + 24·89-s + 72·99-s + 14·101-s + 20·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯
L(s)  = 1  + 2·9-s + 3.61·11-s − 0.917·19-s + 1.85·29-s + 0.359·31-s − 2.18·41-s + 13/7·49-s − 3.38·59-s − 2.04·61-s + 1.66·71-s + 2.70·79-s + 3·81-s + 2.54·89-s + 7.23·99-s + 1.39·101-s + 1.91·109-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 + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5290000\)    =    \(2^{4} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(337.294\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5290000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.807626980\)
\(L(\frac12)\) \(\approx\) \(4.807626980\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 141 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.277260456880027484865366352228, −8.938667363762585019139301567959, −8.479385405703029372376348648787, −8.156328092982725537936892402085, −7.44603973681503725946351077598, −7.27660868738329037422494157338, −6.71998897891180712927148343543, −6.51283795569457881604670024255, −6.25553465764323785701986718471, −6.05088304034239224982757521415, −4.84717135309007106600490248413, −4.74061882018660392885311021756, −4.43193840923078154212539765016, −3.95798706527925428401905688710, −3.43824857914363925674989405022, −3.37611267518398884253952123913, −2.09251379558286140916139089384, −1.86147466988428088294065180594, −1.16729723209106116756486000253, −0.945146199103864715787955007581, 0.945146199103864715787955007581, 1.16729723209106116756486000253, 1.86147466988428088294065180594, 2.09251379558286140916139089384, 3.37611267518398884253952123913, 3.43824857914363925674989405022, 3.95798706527925428401905688710, 4.43193840923078154212539765016, 4.74061882018660392885311021756, 4.84717135309007106600490248413, 6.05088304034239224982757521415, 6.25553465764323785701986718471, 6.51283795569457881604670024255, 6.71998897891180712927148343543, 7.27660868738329037422494157338, 7.44603973681503725946351077598, 8.156328092982725537936892402085, 8.479385405703029372376348648787, 8.938667363762585019139301567959, 9.277260456880027484865366352228

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