Properties

Label 4-105e2-1.1-c1e2-0-5
Degree $4$
Conductor $11025$
Sign $-1$
Analytic cond. $0.702963$
Root an. cond. $0.915657$
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  − 2·2-s − 4-s + 8·8-s + 9-s − 8·11-s − 7·16-s − 2·18-s + 16·22-s + 25-s − 4·29-s − 14·32-s − 36-s − 20·37-s + 8·43-s + 8·44-s − 7·49-s − 2·50-s − 20·53-s + 8·58-s + 35·64-s + 24·67-s − 16·71-s + 8·72-s + 40·74-s + 81-s − 16·86-s − 64·88-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 2.82·8-s + 1/3·9-s − 2.41·11-s − 7/4·16-s − 0.471·18-s + 3.41·22-s + 1/5·25-s − 0.742·29-s − 2.47·32-s − 1/6·36-s − 3.28·37-s + 1.21·43-s + 1.20·44-s − 49-s − 0.282·50-s − 2.74·53-s + 1.05·58-s + 35/8·64-s + 2.93·67-s − 1.89·71-s + 0.942·72-s + 4.64·74-s + 1/9·81-s − 1.72·86-s − 6.82·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(0.702963\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{11025} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 11025,\ (\ :1/2, 1/2),\ -1)\)

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_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−10.67892245123374144028858824682, −10.54178668489831977494597901669, −9.948652838212898735141314289502, −9.523451675812284265792380668213, −8.838960281218345298135626667274, −8.413264856684473334036504393340, −7.68993943182315499613922198544, −7.66488013441745380243230842523, −6.74126848758315603293550810292, −5.28294324247232318062401992920, −5.23920392624592057055772361749, −4.36459226831434420870816413097, −3.30593125964790138739196334719, −1.84339419921478037654642459241, 0, 1.84339419921478037654642459241, 3.30593125964790138739196334719, 4.36459226831434420870816413097, 5.23920392624592057055772361749, 5.28294324247232318062401992920, 6.74126848758315603293550810292, 7.66488013441745380243230842523, 7.68993943182315499613922198544, 8.413264856684473334036504393340, 8.838960281218345298135626667274, 9.523451675812284265792380668213, 9.948652838212898735141314289502, 10.54178668489831977494597901669, 10.67892245123374144028858824682

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