Properties

Label 4-197568-1.1-c1e2-0-9
Degree $4$
Conductor $197568$
Sign $1$
Analytic cond. $12.5971$
Root an. cond. $1.88394$
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 + 9-s − 8·23-s − 6·25-s + 12·29-s − 20·37-s + 24·43-s + 49-s + 12·53-s − 63-s + 24·67-s + 8·71-s + 16·79-s + 81-s + 32·107-s + 28·109-s − 28·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 8·161-s + 163-s + ⋯
L(s)  = 1  − 0.377·7-s + 1/3·9-s − 1.66·23-s − 6/5·25-s + 2.22·29-s − 3.28·37-s + 3.65·43-s + 1/7·49-s + 1.64·53-s − 0.125·63-s + 2.93·67-s + 0.949·71-s + 1.80·79-s + 1/9·81-s + 3.09·107-s + 2.68·109-s − 2.63·113-s − 2·121-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 + 0.630·161-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(197568\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(12.5971\)
Root analytic conductor: \(1.88394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 197568,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.490961623\)
\(L(\frac12)\) \(\approx\) \(1.490961623\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−8.863659982024156066652066757005, −8.820314983809012661425051705328, −8.122692960827942227844556545585, −7.65994621019403063529610936786, −7.25581286887766067123055482424, −6.46532614984807246340960494164, −6.42136658293515809263418102895, −5.57316575173703563161974438677, −5.25019213638990204722565722077, −4.45555499882348184354185481660, −3.85201838137013588199771793801, −3.56921408081238054859871707771, −2.46249542885781697802628622502, −2.06515692945354055956624655608, −0.78183674791366035764093383220, 0.78183674791366035764093383220, 2.06515692945354055956624655608, 2.46249542885781697802628622502, 3.56921408081238054859871707771, 3.85201838137013588199771793801, 4.45555499882348184354185481660, 5.25019213638990204722565722077, 5.57316575173703563161974438677, 6.42136658293515809263418102895, 6.46532614984807246340960494164, 7.25581286887766067123055482424, 7.65994621019403063529610936786, 8.122692960827942227844556545585, 8.820314983809012661425051705328, 8.863659982024156066652066757005

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