Properties

Label 4-570e2-1.1-c1e2-0-8
Degree $4$
Conductor $324900$
Sign $-1$
Analytic cond. $20.7159$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 4-s + 6·9-s + 3·12-s − 8·13-s + 16-s + 25-s − 9·27-s + 6·31-s − 6·36-s + 24·39-s + 2·43-s − 3·48-s − 13·49-s + 8·52-s + 14·61-s − 64-s − 2·67-s + 22·73-s − 3·75-s − 4·79-s + 9·81-s − 18·93-s − 4·97-s − 100-s + 6·103-s + 9·108-s + ⋯
L(s)  = 1  − 1.73·3-s − 1/2·4-s + 2·9-s + 0.866·12-s − 2.21·13-s + 1/4·16-s + 1/5·25-s − 1.73·27-s + 1.07·31-s − 36-s + 3.84·39-s + 0.304·43-s − 0.433·48-s − 1.85·49-s + 1.10·52-s + 1.79·61-s − 1/8·64-s − 0.244·67-s + 2.57·73-s − 0.346·75-s − 0.450·79-s + 81-s − 1.86·93-s − 0.406·97-s − 0.0999·100-s + 0.591·103-s + 0.866·108-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(20.7159\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 324900,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
47$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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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

−8.423750859067621020819700519116, −8.053679357355144104301306054895, −7.45719836178724796307308743509, −7.07884849215165704159052290980, −6.58232251735198184992741163367, −6.22686327568455797717656023439, −5.50948840117199443366279171320, −5.12074175618203345426100603325, −4.81117382544609968730737854617, −4.39609791534731370925307388072, −3.68230536375975179078305808596, −2.78176703262497160389905531938, −2.04857327983780980529070294923, −0.927555344765925089362367726058, 0, 0.927555344765925089362367726058, 2.04857327983780980529070294923, 2.78176703262497160389905531938, 3.68230536375975179078305808596, 4.39609791534731370925307388072, 4.81117382544609968730737854617, 5.12074175618203345426100603325, 5.50948840117199443366279171320, 6.22686327568455797717656023439, 6.58232251735198184992741163367, 7.07884849215165704159052290980, 7.45719836178724796307308743509, 8.053679357355144104301306054895, 8.423750859067621020819700519116

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