Properties

Label 4-33075-1.1-c1e2-0-5
Degree $4$
Conductor $33075$
Sign $-1$
Analytic cond. $2.10889$
Root an. cond. $1.20507$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·4-s − 2·5-s + 9-s − 3·12-s − 2·15-s + 5·16-s − 4·17-s + 6·20-s + 3·25-s + 27-s − 3·36-s − 20·37-s − 20·41-s + 8·43-s − 2·45-s − 16·47-s + 5·48-s − 7·49-s − 4·51-s + 8·59-s + 6·60-s − 3·64-s + 24·67-s + 12·68-s + 3·75-s − 10·80-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s − 0.894·5-s + 1/3·9-s − 0.866·12-s − 0.516·15-s + 5/4·16-s − 0.970·17-s + 1.34·20-s + 3/5·25-s + 0.192·27-s − 1/2·36-s − 3.28·37-s − 3.12·41-s + 1.21·43-s − 0.298·45-s − 2.33·47-s + 0.721·48-s − 49-s − 0.560·51-s + 1.04·59-s + 0.774·60-s − 3/8·64-s + 2.93·67-s + 1.45·68-s + 0.346·75-s − 1.11·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33075\)    =    \(3^{3} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(2.10889\)
Root analytic conductor: \(1.20507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 33075,\ (\ :1/2, 1/2),\ -1)\)

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

−9.948652838212898735141314289502, −9.738851148469479131038939478590, −8.838960281218345298135626667274, −8.554588868470699894187777077269, −8.413264856684473334036504393340, −7.68993943182315499613922198544, −6.76955885158912340897756950945, −6.74126848758315603293550810292, −5.28294324247232318062401992920, −5.04860090093668311608273372623, −4.36459226831434420870816413097, −3.63412818236731287070311550143, −3.30593125964790138739196334719, −1.84339419921478037654642459241, 0, 1.84339419921478037654642459241, 3.30593125964790138739196334719, 3.63412818236731287070311550143, 4.36459226831434420870816413097, 5.04860090093668311608273372623, 5.28294324247232318062401992920, 6.74126848758315603293550810292, 6.76955885158912340897756950945, 7.68993943182315499613922198544, 8.413264856684473334036504393340, 8.554588868470699894187777077269, 8.838960281218345298135626667274, 9.738851148469479131038939478590, 9.948652838212898735141314289502

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