Properties

Label 4-21675-1.1-c1e2-0-1
Degree $4$
Conductor $21675$
Sign $-1$
Analytic cond. $1.38201$
Root an. cond. $1.08424$
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-s − 4-s − 2·7-s + 12-s − 3·16-s − 6·17-s + 4·19-s + 2·21-s − 6·23-s − 5·25-s + 4·27-s + 2·28-s − 8·37-s + 3·48-s − 2·49-s + 6·51-s − 4·57-s − 12·59-s + 7·64-s + 6·68-s + 6·69-s + 16·73-s + 5·75-s − 4·76-s − 7·81-s − 2·84-s + 6·92-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 0.755·7-s + 0.288·12-s − 3/4·16-s − 1.45·17-s + 0.917·19-s + 0.436·21-s − 1.25·23-s − 25-s + 0.769·27-s + 0.377·28-s − 1.31·37-s + 0.433·48-s − 2/7·49-s + 0.840·51-s − 0.529·57-s − 1.56·59-s + 7/8·64-s + 0.727·68-s + 0.722·69-s + 1.87·73-s + 0.577·75-s − 0.458·76-s − 7/9·81-s − 0.218·84-s + 0.625·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21675\)    =    \(3 \cdot 5^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(1.38201\)
Root analytic conductor: \(1.08424\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 21675,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( 1 + p T^{2} \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−10.64633384544385515680215023732, −9.869886958627660833094493302004, −9.518419820493640327168381820105, −9.066172426018826794796087207332, −8.372850595683571950500622849438, −7.88521058464608532391749024144, −6.89517010030384008110494764675, −6.71103886476802877027475809592, −5.94515789337932256796297517782, −5.36848475646101974564338342579, −4.59172082351411232785791696242, −4.03324695467829362811994557437, −3.13165478752451171735177074911, −2.02226205952559193812136055481, 0, 2.02226205952559193812136055481, 3.13165478752451171735177074911, 4.03324695467829362811994557437, 4.59172082351411232785791696242, 5.36848475646101974564338342579, 5.94515789337932256796297517782, 6.71103886476802877027475809592, 6.89517010030384008110494764675, 7.88521058464608532391749024144, 8.372850595683571950500622849438, 9.066172426018826794796087207332, 9.518419820493640327168381820105, 9.869886958627660833094493302004, 10.64633384544385515680215023732

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