# 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 $1$

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## 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